NetworkX Reference Release 2.0.dev20141224000007 Aric

NetworkX Reference Release 2.0.dev20141224000007 Aric
NetworkX Reference
Release 2.0.dev20141229000009
Aric Hagberg, Dan Schult, Pieter Swart
December 29, 2014
CONTENTS
1
Overview
1.1 Who uses NetworkX? . . . . . . .
1.2 Goals . . . . . . . . . . . . . . . .
1.3 The Python programming language
1.4 Free software . . . . . . . . . . . .
1.5 History . . . . . . . . . . . . . . .
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1
1
1
1
2
2
2
Introduction
2.1 NetworkX Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Nodes and Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
3
Graph types
3.1 Which graph class should I use? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Basic graph types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
9
9
4
Algorithms
4.1 Approximation . . . . . .
4.2 Assortativity . . . . . . .
4.3 Bipartite . . . . . . . . .
4.4 Blockmodeling . . . . . .
4.5 Boundary . . . . . . . . .
4.6 Centrality . . . . . . . . .
4.7 Chordal . . . . . . . . . .
4.8 Clique . . . . . . . . . .
4.9 Clustering . . . . . . . .
4.10 Coloring . . . . . . . . .
4.11 Communities . . . . . . .
4.12 Components . . . . . . .
4.13 Connectivity . . . . . . .
4.14 Cores . . . . . . . . . . .
4.15 Cycles . . . . . . . . . .
4.16 Directed Acyclic Graphs .
4.17 Distance Measures . . . .
4.18 Distance-Regular Graphs .
4.19 Dominance . . . . . . . .
4.20 Dominating Sets . . . . .
4.21 Eulerian . . . . . . . . . .
4.22 Flows . . . . . . . . . . .
4.23 Graphical degree sequence
4.24 Hierarchy . . . . . . . . .
4.25 Isolates . . . . . . . . . .
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127
127
132
141
161
162
163
185
188
191
194
195
196
210
227
231
234
237
239
240
242
243
244
267
270
271
i
4.26
4.27
4.28
4.29
4.30
4.31
4.32
4.33
4.34
4.35
4.36
4.37
4.38
4.39
5
6
7
8
9
ii
Isomorphism . . . . . . .
Link Analysis . . . . . . .
Link Prediction . . . . . .
Matching . . . . . . . . .
Maximal independent set .
Minimum Spanning Tree .
Operators . . . . . . . . .
Rich Club . . . . . . . . .
Shortest Paths . . . . . .
Simple Paths . . . . . . .
Swap . . . . . . . . . . .
Traversal . . . . . . . . .
Tree . . . . . . . . . . . .
Vitality . . . . . . . . . .
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272
285
293
299
300
301
303
310
311
329
330
332
339
341
Functions
5.1 Graph . . . . . . . . . .
5.2 Nodes . . . . . . . . . .
5.3 Edges . . . . . . . . . .
5.4 Attributes . . . . . . . .
5.5 Freezing graph structure
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343
343
345
347
348
350
Graph generators
6.1 Atlas . . . . . . .
6.2 Classic . . . . . .
6.3 Expanders . . . .
6.4 Small . . . . . . .
6.5 Random Graphs .
6.6 Degree Sequence .
6.7 Random Clustered
6.8 Directed . . . . .
6.9 Geometric . . . .
6.10 Hybrid . . . . . .
6.11 Bipartite . . . . .
6.12 Line Graph . . . .
6.13 Ego Graph . . . .
6.14 Stochastic . . . . .
6.15 Intersection . . . .
6.16 Social Networks .
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353
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371
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385
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389
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392
392
394
Linear algebra
7.1 Graph Matrix . . . . .
7.2 Laplacian Matrix . . .
7.3 Spectrum . . . . . . .
7.4 Algebraic Connectivity
7.5 Attribute Matrices . .
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395
395
397
399
400
403
Converting to and from other data formats
8.1 To NetworkX Graph . . . . . . . . . .
8.2 Dictionaries . . . . . . . . . . . . . . .
8.3 Lists . . . . . . . . . . . . . . . . . .
8.4 Numpy . . . . . . . . . . . . . . . . .
8.5 Scipy . . . . . . . . . . . . . . . . . .
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409
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Reading and writing graphs
419
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
9.11
9.12
9.13
Adjacency List . . . . .
Multiline Adjacency List
Edge List . . . . . . . .
GEXF . . . . . . . . . .
GML . . . . . . . . . .
Pickle . . . . . . . . . .
GraphML . . . . . . . .
JSON . . . . . . . . . .
LEDA . . . . . . . . . .
YAML . . . . . . . . .
SparseGraph6 . . . . .
Pajek . . . . . . . . . .
GIS Shapefile . . . . . .
10 Drawing
10.1 Matplotlib . . . . . .
10.2 Graphviz AGraph (dot)
10.3 Graphviz with pydot .
10.4 Graph Layout . . . . .
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419
423
426
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448
450
455
457
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459
459
468
471
473
11 Exceptions
12 Utilities
12.1 Helper Functions . . . . . . . .
12.2 Data Structures and Algorithms
12.3 Random Sequence Generators .
12.4 Decorators . . . . . . . . . . .
12.5 Cuthill-Mckee Ordering . . . .
12.6 Context Managers . . . . . . .
477
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479
479
480
480
483
484
486
13 License
489
14 Citing
491
15 Credits
493
16 Glossary
495
Bibliography
497
Python Module Index
507
Index
509
iii
iv
CHAPTER
ONE
OVERVIEW
NetworkX is a Python language software package for the creation, manipulation, and study of the structure, dynamics,
and function of complex networks.
With NetworkX you can load and store networks in standard and nonstandard data formats, generate many types of
random and classic networks, analyze network structure, build network models, design new network algorithms, draw
networks, and much more.
1.1 Who uses NetworkX?
The potential audience for NetworkX includes mathematicians, physicists, biologists, computer scientists, and social
scientists. Good reviews of the state-of-the-art in the science of complex networks are presented in Albert and BarabГЎsi
[BA02], Newman [Newman03], and Dorogovtsev and Mendes [DM03]. See also the classic texts [Bollobas01],
[Diestel97] and [West01] for graph theoretic results and terminology. For basic graph algorithms, we recommend the
texts of Sedgewick, e.g. [Sedgewick01] and [Sedgewick02] and the survey of Brandes and Erlebach [BE05].
1.2 Goals
NetworkX is intended to provide
• tools for the study of the structure and dynamics of social, biological, and infrastructure networks,
• a standard programming interface and graph implementation that is suitable for many applications,
• a rapid development environment for collaborative, multidisciplinary projects,
• an interface to existing numerical algorithms and code written in C, C++, and FORTRAN,
• the ability to painlessly slurp in large nonstandard data sets.
1.3 The Python programming language
Python is a powerful programming language that allows simple and flexible representations of networks, and clear and
concise expressions of network algorithms (and other algorithms too). Python has a vibrant and growing ecosystem
of packages that NetworkX uses to provide more features such as numerical linear algebra and drawing. In addition
Python is also an excellent “glue” language for putting together pieces of software from other languages which allows
reuse of legacy code and engineering of high-performance algorithms [Langtangen04].
Equally important, Python is free, well-supported, and a joy to use.
1
NetworkX Reference, Release 2.0.dev20141229000009
In order to make the most out of NetworkX you will want to know how to write basic programs in Python. Among
the many guides to Python, we recommend the documentation at http://www.python.org and the text by Alex Martelli
[Martelli03].
1.4 Free software
NetworkX is free software; you can redistribute it and/or modify it under the terms of the BSD License. We welcome
contributions from the community. Information on NetworkX development is found at the NetworkX Developer Zone
at Github https://github.com/networkx/networkx
1.5 History
NetworkX was born in May 2002. The original version was designed and written by Aric Hagberg, Dan Schult, and
Pieter Swart in 2002 and 2003. The first public release was in April 2005.
Many people have contributed to the success of NetworkX. Some of the contributors are listed in the credits.
1.5.1 What Next
• A Brief Tour
• Installing
• Reference
• Examples
2
Chapter 1. Overview
CHAPTER
TWO
INTRODUCTION
The structure of NetworkX can be seen by the organization of its source code. The package provides classes for graph
objects, generators to create standard graphs, IO routines for reading in existing datasets, algorithms to analyse the
resulting networks and some basic drawing tools.
Most of the NetworkX API is provided by functions which take a graph object as an argument. Methods of the graph
object are limited to basic manipulation and reporting. This provides modularity of code and documentation. It also
makes it easier for newcomers to learn about the package in stages. The source code for each module is meant to be
easy to read and reading this Python code is actually a good way to learn more about network algorithms, but we have
put a lot of effort into making the documentation sufficient and friendly. If you have suggestions or questions please
contact us by joining the NetworkX Google group.
Classes are named using CamelCase (capital letters at the start of each word). functions, methods and variable names
are lower_case_underscore (lowercase with an underscore representing a space between words).
2.1 NetworkX Basics
After starting Python, import the networkx module with (the recommended way)
>>> import networkx as nx
To save repetition, in the documentation we assume that NetworkX has been imported this way.
If importing networkx fails, it means that Python cannot find the installed module. Check your installation and your
PYTHONPATH.
The following basic graph types are provided as Python classes:
Graph This class implements an undirected graph. It ignores multiple edges between two nodes. It does allow
self-loop edges between a node and itself.
DiGraph Directed graphs, that is, graphs with directed edges. Operations common to directed graphs, (a subclass of
Graph).
MultiGraph A flexible graph class that allows multiple undirected edges between pairs of nodes. The additional
flexibility leads to some degradation in performance, though usually not significant.
MultiDiGraph A directed version of a MultiGraph.
Empty graph-like objects are created with
>>>
>>>
>>>
>>>
G=nx.Graph()
G=nx.DiGraph()
G=nx.MultiGraph()
G=nx.MultiDiGraph()
3
NetworkX Reference, Release 2.0.dev20141229000009
All graph classes allow any hashable object as a node. Hashable objects include strings, tuples, integers, and more.
Arbitrary edge attributes such as weights and labels can be associated with an edge.
The graph internal data structures are based on an adjacency list representation and implemented using Python dictionary datastructures. The graph adjaceny structure is implemented as a Python dictionary of dictionaries; the outer
dictionary is keyed by nodes to values that are themselves dictionaries keyed by neighboring node to the edge attributes associated with that edge. This “dict-of-dicts” structure allows fast addition, deletion, and lookup of nodes
and neighbors in large graphs. The underlying datastructure is accessed directly by methods (the programming interface “API”) in the class definitions. All functions, on the other hand, manipulate graph-like objects solely via
those API methods and not by acting directly on the datastructure. This design allows for possible replacement of the
�dicts-of-dicts’-based datastructure with an alternative datastructure that implements the same methods.
2.1.1 Graphs
The first choice to be made when using NetworkX is what type of graph object to use. A graph (network) is a collection
of nodes together with a collection of edges that are pairs of nodes. Attributes are often associated with nodes and/or
edges. NetworkX graph objects come in different flavors depending on two main properties of the network:
• Directed: Are the edges directed? Does the order of the edge pairs (u,v) matter? A directed graph is specified
by the “Di” prefix in the class name, e.g. DiGraph(). We make this distinction because many classical graph
properties are defined differently for directed graphs.
• Multi-edges: Are multiple edges allowed between each pair of nodes? As you might imagine, multiple edges
requires a different data structure, though tricky users could design edge data objects to support this functionality. We provide a standard data structure and interface for this type of graph using the prefix “Multi”, e.g.
MultiGraph().
The basic graph classes are named: Graph, DiGraph, MultiGraph, and MultiDiGraph
2.2 Nodes and Edges
The next choice you have to make when specifying a graph is what kinds of nodes and edges to use.
If the topology of the network is all you care about then using integers or strings as the nodes makes sense and you
need not worry about edge data. If you have a data structure already in place to describe nodes you can simply use
that structure as your nodes provided it is hashable. If it is not hashable you can use a unique identifier to represent
the node and assign the data as a node attribute.
Edges often have data associated with them. Arbitrary data can associated with edges as an edge attribute. If the data
is numeric and the intent is to represent a weighted graph then use the �weight’ keyword for the attribute. Some of the
graph algorithms, such as Dijkstra’s shortest path algorithm, use this attribute name to get the weight for each edge.
Other attributes can be assigned to an edge by using keyword/value pairs when adding edges. You can use any keyword
except �weight’ to name your attribute and can then easily query the edge data by that attribute keyword.
Once you’ve decided how to encode the nodes and edges, and whether you have an undirected/directed graph with or
without multiedges you are ready to build your network.
2.2.1 Graph Creation
NetworkX graph objects can be created in one of three ways:
• Graph generators – standard algorithms to create network topologies.
• Importing data from pre-existing (usually file) sources.
4
Chapter 2. Introduction
NetworkX Reference, Release 2.0.dev20141229000009
• Adding edges and nodes explicitly.
Explicit addition and removal of nodes/edges is the easiest to describe. Each graph object supplies methods to manipulate the graph. For example,
>>>
>>>
>>>
>>>
import networkx as nx
G=nx.Graph()
G.add_edge(1,2) # default edge data=1
G.add_edge(2,3,weight=0.9) # specify edge data
Edge attributes can be anything:
>>> import math
>>> G.add_edge('y','x',function=math.cos)
>>> G.add_node(math.cos) # any hashable can be a node
You can add many edges at one time:
>>> elist=[('a','b',5.0),('b','c',3.0),('a','c',1.0),('c','d',7.3)]
>>> G.add_weighted_edges_from(elist)
See the /tutorial/index for more examples.
Some basic graph operations such as union and intersection are described in the Operators module documentation.
Graph generators such as binomial_graph and powerlaw_graph are provided in the Graph generators subpackage.
For importing network data from formats such as GML, GraphML, edge list text files see the Reading and writing
graphs subpackage.
2.2.2 Graph Reporting
Class methods are used for the basic reporting functions neighbors, edges and degree. Reporting of lists is often needed
only to iterate through that list so we supply iterator versions of many property reporting methods. For example edges()
and nodes() have corresponding methods edges_iter() and nodes_iter(). Using these methods when you can will save
memory and often time as well.
The basic graph relationship of an edge can be obtained in two basic ways. One can look for neighbors of a node or
one can look for edges incident to a node. We jokingly refer to people who focus on nodes/neighbors as node-centric
and people who focus on edges as edge-centric. The designers of NetworkX tend to be node-centric and view edges
as a relationship between nodes. You can see this by our avoidance of notation like G[u,v] in favor of G[u][v]. Most
data structures for sparse graphs are essentially adjacency lists and so fit this perspective. In the end, of course, it
doesn’t really matter which way you examine the graph. G.edges() removes duplicate representations of each edge
while G.neighbors(n) or G[n] is slightly faster but doesn’t remove duplicates.
Any properties that are more complicated than edges, neighbors and degree are provided by functions. For example
nx.triangles(G,n) gives the number of triangles which include node n as a vertex. These functions are grouped in the
code and documentation under the term algorithms.
2.2.3 Algorithms
A number of graph algorithms are provided with NetworkX. These include shortest path, and breadth first search (see
traversal), clustering and isomorphism algorithms and others. There are many that we have not developed yet too. If
you implement a graph algorithm that might be useful for others please let us know through the NetworkX Google
group or the Github Developer Zone.
As an example here is code to use Dijkstra’s algorithm to find the shortest weighted path:
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>>> G=nx.Graph()
>>> e=[('a','b',0.3),('b','c',0.9),('a','c',0.5),('c','d',1.2)]
>>> G.add_weighted_edges_from(e)
>>> print(nx.dijkstra_path(G,'a','d'))
['a', 'c', 'd']
2.2.4 Drawing
While NetworkX is not designed as a network layout tool, we provide a simple interface to drawing packages and some
simple layout algorithms. We interface to the excellent Graphviz layout tools like dot and neato with the (suggested)
pygraphviz package or the pydot interface. Drawing can be done using external programs or the Matplotlib Python
package. Interactive GUI interfaces are possible though not provided. The drawing tools are provided in the module
drawing.
The basic drawing functions essentially place the nodes on a scatterplot using the positions in a dictionary or computed
with a layout function. The edges are then lines between those dots.
>>> G=nx.cubical_graph()
>>> nx.draw(G)
# default spring_layout
>>> nx.draw(G,pos=nx.spectral_layout(G), nodecolor='r',edge_color='b')
See the examples for more ideas.
2.2.5 Data Structure
NetworkX uses a “dictionary of dictionaries of dictionaries” as the basic network data structure. This allows fast
lookup with reasonable storage for large sparse networks. The keys are nodes so G[u] returns an adjacency dictionary
keyed by neighbor to the edge attribute dictionary. The expression G[u][v] returns the edge attribute dictionary itself.
A dictionary of lists would have also been possible, but not allowed fast edge detection nor convenient storage of edge
data.
Advantages of dict-of-dicts-of-dicts data structure:
• Find edges and remove edges with two dictionary look-ups.
• Prefer to “lists” because of fast lookup with sparse storage.
• Prefer to “sets” since data can be attached to edge.
• G[u][v] returns the edge attribute dictionary.
• n in G tests if node n is in graph G.
• for n in G: iterates through the graph.
• for nbr in G[n]: iterates through neighbors.
As an example, here is a representation of an undirected graph with the edges (�A’,’B’), (�B’,’C’)
>>> G=nx.Graph()
>>> G.add_edge('A','B')
>>> G.add_edge('B','C')
>>> print(G.adj)
{'A': {'B': {}}, 'C': {'B': {}}, 'B': {'A': {}, 'C': {}}}
The data structure gets morphed slightly for each base graph class. For DiGraph two dict-of-dicts-of-dicts structures
are provided, one for successors and one for predecessors. For MultiGraph/MultiDiGraph we use a dict-of-dicts-of-
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dicts-of-dicts 1 where the third dictionary is keyed by an edge key identifier to the fourth dictionary which contains
the edge attributes for that edge between the two nodes.
Graphs use a dictionary of attributes for each edge. We use a dict-of-dicts-of-dicts data structure with the inner
dictionary storing “name-value” relationships for that edge.
>>> G=nx.Graph()
>>> G.add_edge(1,2,color='red',weight=0.84,size=300)
>>> print(G[1][2]['size'])
300
1
“It’s dictionaries all the way down.”
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8
Chapter 2. Introduction
CHAPTER
THREE
GRAPH TYPES
NetworkX provides data structures and methods for storing graphs.
All NetworkX graph classes allow (hashable) Python objects as nodes. and any Python object can be assigned as an
edge attribute.
The choice of graph class depends on the structure of the graph you want to represent.
3.1 Which graph class should I use?
Graph Type
Undirected Simple
Directed Simple
With Self-loops
With Parallel edges
NetworkX Class
Graph
DiGraph
Graph, DiGraph
MultiGraph, MultiDiGraph
3.2 Basic graph types
3.2.1 Graph – Undirected graphs with self loops
Overview
Graph(data=None, **attr)
Base class for undirected graphs.
A Graph stores nodes and edges with optional data, or attributes.
Graphs hold undirected edges. Self loops are allowed but multiple (parallel) edges are not.
Nodes can be arbitrary (hashable) Python objects with optional key/value attributes.
Edges are represented as links between nodes with optional key/value attributes.
Parameters data : input graph
Data to initialize graph. If data=None (default) an empty graph is created. The data can
be an edge list, or any NetworkX graph object. If the corresponding optional Python
packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse
matrix, or a PyGraphviz graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to graph as key=value pairs.
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See also:
DiGraph, MultiGraph, MultiDiGraph
Examples
Create an empty graph structure (a “null graph”) with no nodes and no edges.
>>> G = nx.Graph()
G can be grown in several ways.
Nodes:
Add one node at a time:
>>> G.add_node(1)
Add the nodes from any container (a list, dict, set or even the lines from a file or the nodes from another graph).
>>>
>>>
>>>
>>>
>>>
G.add_nodes_from([2,3])
G.add_nodes_from(range(100,110))
H=nx.Graph()
H.add_path([0,1,2,3,4,5,6,7,8,9])
G.add_nodes_from(H)
In addition to strings and integers any hashable Python object (except None) can represent a node, e.g. a
customized node object, or even another Graph.
>>> G.add_node(H)
Edges:
G can also be grown by adding edges.
Add one edge,
>>> G.add_edge(1, 2)
a list of edges,
>>> G.add_edges_from([(1,2),(1,3)])
or a collection of edges,
>>> G.add_edges_from(H.edges())
If some edges connect nodes not yet in the graph, the nodes are added automatically. There are no errors when
adding nodes or edges that already exist.
Attributes:
Each graph, node, and edge can hold key/value attribute pairs in an associated attribute dictionary (the keys
must be hashable). By default these are empty, but can be added or changed using add_edge, add_node or direct
manipulation of the attribute dictionaries named graph, node and edge respectively.
>>> G = nx.Graph(day="Friday")
>>> G.graph
{'day': 'Friday'}
Add node attributes using add_node(), add_nodes_from() or G.node
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>>> G.add_node(1, time='5pm')
>>> G.add_nodes_from([3], time='2pm')
>>> G.node[1]
{'time': '5pm'}
>>> G.node[1]['room'] = 714
>>> del G.node[1]['room'] # remove attribute
>>> G.nodes(data=True)
[(1, {'time': '5pm'}), (3, {'time': '2pm'})]
Warning: adding a node to G.node does not add it to the graph.
Add edge attributes using add_edge(), add_edges_from(), subscript notation, or G.edge.
>>>
>>>
>>>
>>>
>>>
G.add_edge(1, 2, weight=4.7 )
G.add_edges_from([(3,4),(4,5)], color='red')
G.add_edges_from([(1,2,{'color':'blue'}), (2,3,{'weight':8})])
G[1][2]['weight'] = 4.7
G.edge[1][2]['weight'] = 4
Shortcuts:
Many common graph features allow python syntax to speed reporting.
>>> 1 in G
# check if node in graph
True
>>> [n for n in G if n<3]
# iterate through nodes
[1, 2]
>>> len(G) # number of nodes in graph
5
The fastest way to traverse all edges of a graph is via adjacency_iter(), but the edges() method is often more
convenient.
>>> for n,nbrsdict in G.adjacency_iter():
...
for nbr,eattr in nbrsdict.items():
...
if 'weight' in eattr:
...
(n,nbr,eattr['weight'])
(1, 2, 4)
(2, 1, 4)
(2, 3, 8)
(3, 2, 8)
>>> [ (u,v,edata['weight']) for u,v,edata in G.edges(data=True) if 'weight' in edata ]
[(1, 2, 4), (2, 3, 8)]
Reporting:
Simple graph information is obtained using methods. Iterator versions of many reporting methods exist for
efficiency. Methods exist for reporting nodes(), edges(), neighbors() and degree() as well as the number of
nodes and edges.
For details on these and other miscellaneous methods, see below.
Adding and removing nodes and edges
Graph.__init__([data])
Graph.add_node(n[, attr_dict])
Graph.add_nodes_from(nodes, **attr)
3.2. Basic graph types
Initialize a graph with edges, name, graph attributes.
Add a single node n and update node attributes.
Add multiple nodes.
Continued on next page
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Table 3.1 – continued from previous page
Graph.remove_node(n)
Remove node n.
Graph.remove_nodes_from(nodes)
Remove multiple nodes.
Graph.add_edge(u, v[, attr_dict])
Add an edge between u and v.
Graph.add_edges_from(ebunch[, attr_dict])
Add all the edges in ebunch.
Graph.add_weighted_edges_from(ebunch[, weight]) Add all the edges in ebunch as weighted edges with specified weights
Graph.remove_edge(u, v)
Remove the edge between u and v.
Graph.remove_edges_from(ebunch)
Remove all edges specified in ebunch.
Graph.add_star(nodes, **attr)
Add a star.
Graph.add_path(nodes, **attr)
Add a path.
Graph.add_cycle(nodes, **attr)
Add a cycle.
Graph.clear()
Remove all nodes and edges from the graph.
__init__
Graph.__init__(data=None, **attr)
Initialize a graph with edges, name, graph attributes.
Parameters data : input graph
Data to initialize graph. If data=None (default) an empty graph is created. The data can
be an edge list, or any NetworkX graph object. If the corresponding optional Python
packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse
matrix, or a PyGraphviz graph.
name : string, optional (default=’�)
An optional name for the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to graph as key=value pairs.
See also:
convert
Examples
>>>
>>>
>>>
>>>
G
G
e
G
=
=
=
=
nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
nx.Graph(name='my graph')
[(1,2),(2,3),(3,4)] # list of edges
nx.Graph(e)
Arbitrary graph attribute pairs (key=value) may be assigned
>>> G=nx.Graph(e, day="Friday")
>>> G.graph
{'day': 'Friday'}
add_node
Graph.add_node(n, attr_dict=None, **attr)
Add a single node n and update node attributes.
Parameters n : node
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A node can be any hashable Python object except None.
attr_dict : dictionary, optional (default= no attributes)
Dictionary of node attributes. Key/value pairs will update existing data associated with
the node.
attr : keyword arguments, optional
Set or change attributes using key=value.
See also:
add_nodes_from
Notes
A hashable object is one that can be used as a key in a Python dictionary. This includes strings, numbers, tuples
of strings and numbers, etc.
On many platforms hashable items also include mutables such as NetworkX Graphs, though one should be
careful that the hash doesn’t change on mutables.
Examples
>>>
>>>
>>>
>>>
>>>
>>>
3
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_node(1)
G.add_node('Hello')
K3 = nx.Graph([(0,1),(1,2),(2,0)])
G.add_node(K3)
G.number_of_nodes()
Use keywords set/change node attributes:
>>> G.add_node(1,size=10)
>>> G.add_node(3,weight=0.4,UTM=('13S',382871,3972649))
add_nodes_from
Graph.add_nodes_from(nodes, **attr)
Add multiple nodes.
Parameters nodes : iterable container
A container of nodes (list, dict, set, etc.). OR A container of (node, attribute dict) tuples.
Node attributes are updated using the attribute dict.
attr : keyword arguments, optional (default= no attributes)
Update attributes for all nodes in nodes. Node attributes specified in nodes as a tuple
take precedence over attributes specified generally.
See also:
add_node
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Examples
>>>
>>>
>>>
>>>
>>>
[0,
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_nodes_from('Hello')
K3 = nx.Graph([(0,1),(1,2),(2,0)])
G.add_nodes_from(K3)
sorted(G.nodes(),key=str)
1, 2, 'H', 'e', 'l', 'o']
Use keywords to update specific node attributes for every node.
>>> G.add_nodes_from([1,2], size=10)
>>> G.add_nodes_from([3,4], weight=0.4)
Use (node, attrdict) tuples to update attributes for specific nodes.
>>>
>>>
11
>>>
>>>
>>>
11
G.add_nodes_from([(1,dict(size=11)), (2,{'color':'blue'})])
G.node[1]['size']
H = nx.Graph()
H.add_nodes_from(G.nodes(data=True))
H.node[1]['size']
remove_node
Graph.remove_node(n)
Remove node n.
Removes the node n and all adjacent edges. Attempting to remove a non-existent node will raise an exception.
Parameters n : node
A node in the graph
Raises NetworkXError
If n is not in the graph.
See also:
remove_nodes_from
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> G.edges()
[(0, 1), (1, 2)]
>>> G.remove_node(1)
>>> G.edges()
[]
remove_nodes_from
Graph.remove_nodes_from(nodes)
Remove multiple nodes.
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Parameters nodes : iterable container
A container of nodes (list, dict, set, etc.). If a node in the container is not in the graph it
is silently ignored.
See also:
remove_node
Examples
>>>
>>>
>>>
>>>
[0,
>>>
>>>
[]
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2])
e = G.nodes()
e
1, 2]
G.remove_nodes_from(e)
G.nodes()
add_edge
Graph.add_edge(u, v, attr_dict=None, **attr)
Add an edge between u and v.
The nodes u and v will be automatically added if they are not already in the graph.
Edge attributes can be specified with keywords or by providing a dictionary with key/value pairs. See examples
below.
Parameters u,v : nodes
Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None)
Python objects.
attr_dict : dictionary, optional (default= no attributes)
Dictionary of edge attributes. Key/value pairs will update existing data associated with
the edge.
attr : keyword arguments, optional
Edge data (or labels or objects) can be assigned using keyword arguments.
See also:
add_edges_from add a collection of edges
Notes
Adding an edge that already exists updates the edge data.
Many NetworkX algorithms designed for weighted graphs use as the edge weight a numerical value assigned to
a keyword which by default is �weight’.
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Examples
The following all add the edge e=(1,2) to graph G:
>>>
>>>
>>>
>>>
>>>
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
e = (1,2)
G.add_edge(1, 2)
# explicit two-node form
G.add_edge(*e)
# single edge as tuple of two nodes
G.add_edges_from( [(1,2)] ) # add edges from iterable container
Associate data to edges using keywords:
>>> G.add_edge(1, 2, weight=3)
>>> G.add_edge(1, 3, weight=7, capacity=15, length=342.7)
add_edges_from
Graph.add_edges_from(ebunch, attr_dict=None, **attr)
Add all the edges in ebunch.
Parameters ebunch : container of edges
Each edge given in the container will be added to the graph. The edges must be given
as as 2-tuples (u,v) or 3-tuples (u,v,d) where d is a dictionary containing edge data.
attr_dict : dictionary, optional (default= no attributes)
Dictionary of edge attributes. Key/value pairs will update existing data associated with
each edge.
attr : keyword arguments, optional
Edge data (or labels or objects) can be assigned using keyword arguments.
See also:
add_edge add a single edge
add_weighted_edges_from convenient way to add weighted edges
Notes
Adding the same edge twice has no effect but any edge data will be updated when each duplicate edge is added.
Edge attributes specified in edges as a tuple take precedence over attributes specified generally.
Examples
>>>
>>>
>>>
>>>
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edges_from([(0,1),(1,2)]) # using a list of edge tuples
e = zip(range(0,3),range(1,4))
G.add_edges_from(e) # Add the path graph 0-1-2-3
Associate data to edges
>>> G.add_edges_from([(1,2),(2,3)], weight=3)
>>> G.add_edges_from([(3,4),(1,4)], label='WN2898')
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add_weighted_edges_from
Graph.add_weighted_edges_from(ebunch, weight=’weight’, **attr)
Add all the edges in ebunch as weighted edges with specified weights.
Parameters ebunch : container of edges
Each edge given in the list or container will be added to the graph. The edges must be
given as 3-tuples (u,v,w) where w is a number.
weight : string, optional (default= �weight’)
The attribute name for the edge weights to be added.
attr : keyword arguments, optional (default= no attributes)
Edge attributes to add/update for all edges.
See also:
add_edge add a single edge
add_edges_from add multiple edges
Notes
Adding the same edge twice for Graph/DiGraph simply updates the edge data. For MultiGraph/MultiDiGraph,
duplicate edges are stored.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_weighted_edges_from([(0,1,3.0),(1,2,7.5)])
remove_edge
Graph.remove_edge(u, v)
Remove the edge between u and v.
Parameters u,v: nodes
Remove the edge between nodes u and v.
Raises NetworkXError
If there is not an edge between u and v.
See also:
remove_edges_from remove a collection of edges
Examples
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>>>
>>>
>>>
>>>
>>>
>>>
>>>
G = nx.Graph()
# or DiGraph, etc
G.add_path([0,1,2,3])
G.remove_edge(0,1)
e = (1,2)
G.remove_edge(*e) # unpacks e from an edge tuple
e = (2,3,{'weight':7}) # an edge with attribute data
G.remove_edge(*e[:2]) # select first part of edge tuple
remove_edges_from
Graph.remove_edges_from(ebunch)
Remove all edges specified in ebunch.
Parameters ebunch: list or container of edge tuples
Each edge given in the list or container will be removed from the graph. The edges can
be:
• 2-tuples (u,v) edge between u and v.
• 3-tuples (u,v,k) where k is ignored.
See also:
remove_edge remove a single edge
Notes
Will fail silently if an edge in ebunch is not in the graph.
Examples
>>>
>>>
>>>
>>>
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
ebunch=[(1,2),(2,3)]
G.remove_edges_from(ebunch)
add_star
Graph.add_star(nodes, **attr)
Add a star.
The first node in nodes is the middle of the star. It is connected to all other nodes.
Parameters nodes : iterable container
A container of nodes.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in star.
See also:
add_path, add_cycle
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Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_star([0,1,2,3])
>>> G.add_star([10,11,12],weight=2)
add_path
Graph.add_path(nodes, **attr)
Add a path.
Parameters nodes : iterable container
A container of nodes. A path will be constructed from the nodes (in order) and added
to the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in path.
See also:
add_star, add_cycle
Examples
>>> G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.add_path([10,11,12],weight=7)
add_cycle
Graph.add_cycle(nodes, **attr)
Add a cycle.
Parameters nodes: iterable container
A container of nodes. A cycle will be constructed from the nodes (in order) and added
to the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in cycle.
See also:
add_path, add_star
Examples
>>> G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_cycle([0,1,2,3])
>>> G.add_cycle([10,11,12],weight=7)
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clear
Graph.clear()
Remove all nodes and edges from the graph.
This also removes the name, and all graph, node, and edge attributes.
Examples
>>>
>>>
>>>
>>>
[]
>>>
[]
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.clear()
G.nodes()
G.edges()
Iterating over nodes and edges
Graph.nodes([data])
Graph.nodes_iter([data])
Graph.__iter__()
Graph.edges([nbunch, data])
Graph.edges_iter([nbunch, data])
Graph.get_edge_data(u, v[, default])
Graph.neighbors(n)
Graph.neighbors_iter(n)
Graph.__getitem__(n)
Graph.adjacency_list()
Graph.adjacency_iter()
Graph.nbunch_iter([nbunch])
Return a list of the nodes in the graph.
Return an iterator over the nodes.
Iterate over the nodes.
Return a list of edges.
Return an iterator over the edges.
Return the attribute dictionary associated with edge (u,v).
Return a list of the nodes connected to the node n.
Return an iterator over all neighbors of node n.
Return a dict of neighbors of node n.
Return an adjacency list representation of the graph.
Return an iterator of (node, adjacency dict) tuples for all nodes.
Return an iterator of nodes contained in nbunch that are also in the graph.
nodes
Graph.nodes(data=False)
Return a list of the nodes in the graph.
Parameters data : boolean, optional (default=False)
If False return a list of nodes. If True return a two-tuple of node and node data dictionary
Returns nlist : list
A list of nodes. If data=True a list of two-tuples containing (node, node data dictionary).
Examples
>>>
>>>
>>>
[0,
>>>
20
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2])
G.nodes()
1, 2]
G.add_node(1, time='5pm')
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>>> G.nodes(data=True)
[(0, {}), (1, {'time': '5pm'}), (2, {})]
nodes_iter
Graph.nodes_iter(data=False)
Return an iterator over the nodes.
Parameters data : boolean, optional (default=False)
If False the iterator returns nodes. If True return a two-tuple of node and node data
dictionary
Returns niter : iterator
An iterator over nodes. If data=True the iterator gives two-tuples containing (node, node
data, dictionary)
Notes
If the node data is not required it is simpler and equivalent to use the expression �for n in G’.
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> [d for n,d in G.nodes_iter(data=True)]
[{}, {}, {}]
__iter__
Graph.__iter__()
Iterate over the nodes. Use the expression �for n in G’.
Returns niter : iterator
An iterator over all nodes in the graph.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
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edges
Graph.edges(nbunch=None, data=False)
Return a list of edges.
Edges are returned as tuples with optional data in the order (node, neighbor, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
Return two tuples (u,v) (False) or three-tuples (u,v,data) (True).
Returns edge_list: list of edge tuples
Edges that are adjacent to any node in nbunch, or a list of all edges if nbunch is not
specified.
See also:
edges_iter return an iterator over the edges
Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.edges()
[(0, 1), (1, 2), (2, 3)]
>>> G.edges(data=True) # default edge data is {} (empty dictionary)
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
>>> G.edges([0,3])
[(0, 1), (3, 2)]
>>> G.edges(0)
[(0, 1)]
edges_iter
Graph.edges_iter(nbunch=None, data=False)
Return an iterator over the edges.
Edges are returned as tuples with optional data in the order (node, neighbor, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
If True, return edge attribute dict in 3-tuple (u,v,data).
Returns edge_iter : iterator
An iterator of (u,v) or (u,v,d) tuples of edges.
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See also:
edges return a list of edges
Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.
Examples
>>> G = nx.Graph()
# or MultiGraph, etc
>>> G.add_path([0,1,2,3])
>>> [e for e in G.edges_iter()]
[(0, 1), (1, 2), (2, 3)]
>>> list(G.edges_iter(data=True)) # default data is {} (empty dict)
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
>>> list(G.edges_iter([0,3]))
[(0, 1), (3, 2)]
>>> list(G.edges_iter(0))
[(0, 1)]
get_edge_data
Graph.get_edge_data(u, v, default=None)
Return the attribute dictionary associated with edge (u,v).
Parameters u,v : nodes
default: any Python object (default=None)
Value to return if the edge (u,v) is not found.
Returns edge_dict : dictionary
The edge attribute dictionary.
Notes
It is faster to use G[u][v].
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G[0][1]
{}
Warning: Assigning G[u][v] corrupts the graph data structure. But it is safe to assign attributes to that dictionary,
>>> G[0][1]['weight'] = 7
>>> G[0][1]['weight']
7
>>> G[1][0]['weight']
7
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Examples
>>>
>>>
>>>
{}
>>>
>>>
{}
>>>
0
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.get_edge_data(0,1) # default edge data is {}
e = (0,1)
G.get_edge_data(*e) # tuple form
G.get_edge_data('a','b',default=0) # edge not in graph, return 0
neighbors
Graph.neighbors(n)
Return a list of the nodes connected to the node n.
Parameters n : node
A node in the graph
Returns nlist : list
A list of nodes that are adjacent to n.
Raises NetworkXError
If the node n is not in the graph.
Notes
It is usually more convenient (and faster) to access the adjacency dictionary as G[n]:
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edge('a','b',weight=7)
>>> G['a']
{'b': {'weight': 7}}
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.neighbors(0)
[1]
neighbors_iter
Graph.neighbors_iter(n)
Return an iterator over all neighbors of node n.
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Notes
It is faster to use the idiom “in G[0]”, e.g.
>>> G = nx.path_graph(4)
>>> [n for n in G[0]]
[1]
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> [n for n in G.neighbors_iter(0)]
[1]
__getitem__
Graph.__getitem__(n)
Return a dict of neighbors of node n. Use the expression �G[n]’.
Parameters n : node
A node in the graph.
Returns adj_dict : dictionary
The adjacency dictionary for nodes connected to n.
Notes
G[n] is similar to G.neighbors(n) but the internal data dictionary is returned instead of a list.
Assigning G[n] will corrupt the internal graph data structure. Use G[n] for reading data only.
Examples
>>>
>>>
>>>
{1:
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G[0]
{}}
adjacency_list
Graph.adjacency_list()
Return an adjacency list representation of the graph.
The output adjacency list is in the order of G.nodes(). For directed graphs, only outgoing adjacencies are
included.
Returns adj_list : lists of lists
The adjacency structure of the graph as a list of lists.
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See also:
adjacency_iter
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.adjacency_list() # in order given by G.nodes()
[[1], [0, 2], [1, 3], [2]]
adjacency_iter
Graph.adjacency_iter()
Return an iterator of (node, adjacency dict) tuples for all nodes.
This is the fastest way to look at every edge. For directed graphs, only outgoing adjacencies are included.
Returns adj_iter : iterator
An iterator of (node, adjacency dictionary) for all nodes in the graph.
See also:
adjacency_list
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> [(n,nbrdict) for n,nbrdict in G.adjacency_iter()]
[(0, {1: {}}), (1, {0: {}, 2: {}}), (2, {1: {}, 3: {}}), (3, {2: {}})]
nbunch_iter
Graph.nbunch_iter(nbunch=None)
Return an iterator of nodes contained in nbunch that are also in the graph.
The nodes in nbunch are checked for membership in the graph and if not are silently ignored.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
Returns niter : iterator
An iterator over nodes in nbunch that are also in the graph. If nbunch is None, iterate
over all nodes in the graph.
Raises NetworkXError
If nbunch is not a node or or sequence of nodes. If a node in nbunch is not hashable.
See also:
Graph.__iter__
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Notes
When nbunch is an iterator, the returned iterator yields values directly from nbunch, becoming exhausted when
nbunch is exhausted.
To test whether nbunch is a single node, one can use “if nbunch in self:”, even after processing with this routine.
If nbunch is not a node or a (possibly empty) sequence/iterator or None, a NetworkXError is raised. Also, if any
object in nbunch is not hashable, a NetworkXError is raised.
Information about graph structure
Graph.has_node(n)
Graph.__contains__(n)
Graph.has_edge(u, v)
Graph.order()
Graph.number_of_nodes()
Graph.__len__()
Graph.degree([nbunch, weight])
Graph.degree_iter([nbunch, weight])
Graph.size([weight])
Graph.number_of_edges([u, v])
Graph.nodes_with_selfloops()
Graph.selfloop_edges([data])
Graph.number_of_selfloops()
Return True if the graph contains the node n.
Return True if n is a node, False otherwise.
Return True if the edge (u,v) is in the graph.
Return the number of nodes in the graph.
Return the number of nodes in the graph.
Return the number of nodes.
Return the degree of a node or nodes.
Return an iterator for (node, degree).
Return the number of edges.
Return the number of edges between two nodes.
Return a list of nodes with self loops.
Return a list of selfloop edges.
Return the number of selfloop edges.
has_node
Graph.has_node(n)
Return True if the graph contains the node n.
Parameters n : node
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> G.has_node(0)
True
It is more readable and simpler to use
>>> 0 in G
True
__contains__
Graph.__contains__(n)
Return True if n is a node, False otherwise. Use the expression �n in G’.
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Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> 1 in G
True
has_edge
Graph.has_edge(u, v)
Return True if the edge (u,v) is in the graph.
Parameters u,v : nodes
Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None)
Python objects.
Returns edge_ind : bool
True if edge is in the graph, False otherwise.
Examples
Can be called either using two nodes u,v or edge tuple (u,v)
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.has_edge(0,1) # using two nodes
True
>>> e = (0,1)
>>> G.has_edge(*e) # e is a 2-tuple (u,v)
True
>>> e = (0,1,{'weight':7})
>>> G.has_edge(*e[:2]) # e is a 3-tuple (u,v,data_dictionary)
True
The following syntax are all equivalent:
>>> G.has_edge(0,1)
True
>>> 1 in G[0] # though this gives KeyError if 0 not in G
True
order
Graph.order()
Return the number of nodes in the graph.
Returns nnodes : int
The number of nodes in the graph.
See also:
number_of_nodes, __len__
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number_of_nodes
Graph.number_of_nodes()
Return the number of nodes in the graph.
Returns nnodes : int
The number of nodes in the graph.
See also:
order, __len__
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> len(G)
3
__len__
Graph.__len__()
Return the number of nodes. Use the expression �len(G)’.
Returns nnodes : int
The number of nodes in the graph.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> len(G)
4
degree
Graph.degree(nbunch=None, weight=None)
Return the degree of a node or nodes.
The node degree is the number of edges adjacent to that node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd : dictionary, or number
A dictionary with nodes as keys and degree as values or a number if a single node is
specified.
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Examples
>>>
>>>
>>>
1
>>>
{0:
>>>
[1,
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.degree(0)
G.degree([0,1])
1, 1: 2}
list(G.degree([0,1]).values())
2]
degree_iter
Graph.degree_iter(nbunch=None, weight=None)
Return an iterator for (node, degree).
The node degree is the number of edges adjacent to the node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd_iter : an iterator
The iterator returns two-tuples of (node, degree).
See also:
degree
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> list(G.degree_iter(0)) # node 0 with degree 1
[(0, 1)]
>>> list(G.degree_iter([0,1]))
[(0, 1), (1, 2)]
size
Graph.size(weight=None)
Return the number of edges.
Parameters weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1.
Returns nedges : int
The number of edges or sum of edge weights in the graph.
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See also:
number_of_edges
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.size()
3
>>>
>>>
>>>
>>>
2
>>>
6.0
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge('a','b',weight=2)
G.add_edge('b','c',weight=4)
G.size()
G.size(weight='weight')
number_of_edges
Graph.number_of_edges(u=None, v=None)
Return the number of edges between two nodes.
Parameters u,v : nodes, optional (default=all edges)
If u and v are specified, return the number of edges between u and v. Otherwise return
the total number of all edges.
Returns nedges : int
The number of edges in the graph. If nodes u and v are specified return the number of
edges between those nodes.
See also:
size
Examples
>>>
>>>
>>>
3
>>>
1
>>>
>>>
1
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.number_of_edges()
G.number_of_edges(0,1)
e = (0,1)
G.number_of_edges(*e)
nodes_with_selfloops
Graph.nodes_with_selfloops()
Return a list of nodes with self loops.
A node with a self loop has an edge with both ends adjacent to that node.
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Returns nodelist : list
A list of nodes with self loops.
See also:
selfloop_edges, number_of_selfloops
Examples
>>>
>>>
>>>
>>>
[1]
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge(1,1)
G.add_edge(1,2)
G.nodes_with_selfloops()
selfloop_edges
Graph.selfloop_edges(data=False)
Return a list of selfloop edges.
A selfloop edge has the same node at both ends.
Parameters data : bool, optional (default=False)
Return selfloop edges as two tuples (u,v) (data=False) or three-tuples (u,v,data)
(data=True)
Returns edgelist : list of edge tuples
A list of all selfloop edges.
See also:
nodes_with_selfloops, number_of_selfloops
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edge(1,1)
>>> G.add_edge(1,2)
>>> G.selfloop_edges()
[(1, 1)]
>>> G.selfloop_edges(data=True)
[(1, 1, {})]
number_of_selfloops
Graph.number_of_selfloops()
Return the number of selfloop edges.
A selfloop edge has the same node at both ends.
Returns nloops : int
The number of selfloops.
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See also:
nodes_with_selfloops, selfloop_edges
Examples
>>>
>>>
>>>
>>>
1
G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge(1,1)
G.add_edge(1,2)
G.number_of_selfloops()
Making copies and subgraphs
Graph.copy()
Graph.to_undirected()
Graph.to_directed()
Graph.subgraph(nbunch)
Return a copy of the graph.
Return an undirected copy of the graph.
Return a directed representation of the graph.
Return the subgraph induced on nodes in nbunch.
copy
Graph.copy()
Return a copy of the graph.
Returns G : Graph
A copy of the graph.
See also:
to_directed return a directed copy of the graph.
Notes
This makes a complete copy of the graph including all of the node or edge attributes.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> H = G.copy()
to_undirected
Graph.to_undirected()
Return an undirected copy of the graph.
Returns G : Graph/MultiGraph
A deepcopy of the graph.
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See also:
copy, add_edge, add_edges_from
Notes
This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the
data and references.
This is in contrast to the similar G=DiGraph(D) which returns a shallow copy of the data.
See the Python copy module for
http://docs.python.org/library/copy.html.
more
information
on
shallow
and
deep
copies,
Examples
>>> G = nx.Graph()
# or MultiGraph, etc
>>> G.add_path([0,1])
>>> H = G.to_directed()
>>> H.edges()
[(0, 1), (1, 0)]
>>> G2 = H.to_undirected()
>>> G2.edges()
[(0, 1)]
to_directed
Graph.to_directed()
Return a directed representation of the graph.
Returns G : DiGraph
A directed graph with the same name, same nodes, and with each edge (u,v,data) replaced by two directed edges (u,v,data) and (v,u,data).
Notes
This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the
data and references.
This is in contrast to the similar D=DiGraph(G) which returns a shallow copy of the data.
See the Python copy module for
http://docs.python.org/library/copy.html.
more
information
on
shallow
and
deep
copies,
Examples
>>> G = nx.Graph()
# or MultiGraph, etc
>>> G.add_path([0,1])
>>> H = G.to_directed()
>>> H.edges()
[(0, 1), (1, 0)]
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If already directed, return a (deep) copy
>>> G = nx.DiGraph()
# or MultiDiGraph, etc
>>> G.add_path([0,1])
>>> H = G.to_directed()
>>> H.edges()
[(0, 1)]
subgraph
Graph.subgraph(nbunch)
Return the subgraph induced on nodes in nbunch.
The induced subgraph of the graph contains the nodes in nbunch and the edges between those nodes.
Parameters nbunch : list, iterable
A container of nodes which will be iterated through once.
Returns G : Graph
A subgraph of the graph with the same edge attributes.
Notes
The graph, edge or node attributes just point to the original graph. So changes to the node or edge structure will
not be reflected in the original graph while changes to the attributes will.
To create a subgraph with its own copy of the edge/node attributes use: nx.Graph(G.subgraph(nbunch))
If edge attributes are containers, a deep copy can be obtained using: G.subgraph(nbunch).copy()
For an inplace reduction of a graph to a subgraph you can remove nodes: G.remove_nodes_from([ n in G if n
not in set(nbunch)])
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> H = G.subgraph([0,1,2])
>>> H.edges()
[(0, 1), (1, 2)]
3.2.2 DiGraph - Directed graphs with self loops
Overview
DiGraph(data=None, **attr)
Base class for directed graphs.
A DiGraph stores nodes and edges with optional data, or attributes.
DiGraphs hold directed edges. Self loops are allowed but multiple (parallel) edges are not.
Nodes can be arbitrary (hashable) Python objects with optional key/value attributes.
Edges are represented as links between nodes with optional key/value attributes.
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Parameters data : input graph
Data to initialize graph. If data=None (default) an empty graph is created. The data can
be an edge list, or any NetworkX graph object. If the corresponding optional Python
packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse
matrix, or a PyGraphviz graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to graph as key=value pairs.
See also:
Graph, MultiGraph, MultiDiGraph
Examples
Create an empty graph structure (a “null graph”) with no nodes and no edges.
>>> G = nx.DiGraph()
G can be grown in several ways.
Nodes:
Add one node at a time:
>>> G.add_node(1)
Add the nodes from any container (a list, dict, set or even the lines from a file or the nodes from another graph).
>>>
>>>
>>>
>>>
>>>
G.add_nodes_from([2,3])
G.add_nodes_from(range(100,110))
H=nx.Graph()
H.add_path([0,1,2,3,4,5,6,7,8,9])
G.add_nodes_from(H)
In addition to strings and integers any hashable Python object (except None) can represent a node, e.g. a
customized node object, or even another Graph.
>>> G.add_node(H)
Edges:
G can also be grown by adding edges.
Add one edge,
>>> G.add_edge(1, 2)
a list of edges,
>>> G.add_edges_from([(1,2),(1,3)])
or a collection of edges,
>>> G.add_edges_from(H.edges())
If some edges connect nodes not yet in the graph, the nodes are added automatically. There are no errors when
adding nodes or edges that already exist.
Attributes:
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Each graph, node, and edge can hold key/value attribute pairs in an associated attribute dictionary (the keys
must be hashable). By default these are empty, but can be added or changed using add_edge, add_node or direct
manipulation of the attribute dictionaries named graph, node and edge respectively.
>>> G = nx.DiGraph(day="Friday")
>>> G.graph
{'day': 'Friday'}
Add node attributes using add_node(), add_nodes_from() or G.node
>>> G.add_node(1, time='5pm')
>>> G.add_nodes_from([3], time='2pm')
>>> G.node[1]
{'time': '5pm'}
>>> G.node[1]['room'] = 714
>>> del G.node[1]['room'] # remove attribute
>>> G.nodes(data=True)
[(1, {'time': '5pm'}), (3, {'time': '2pm'})]
Warning: adding a node to G.node does not add it to the graph.
Add edge attributes using add_edge(), add_edges_from(), subscript notation, or G.edge.
>>>
>>>
>>>
>>>
>>>
G.add_edge(1, 2, weight=4.7 )
G.add_edges_from([(3,4),(4,5)], color='red')
G.add_edges_from([(1,2,{'color':'blue'}), (2,3,{'weight':8})])
G[1][2]['weight'] = 4.7
G.edge[1][2]['weight'] = 4
Shortcuts:
Many common graph features allow python syntax to speed reporting.
>>> 1 in G
# check if node in graph
True
>>> [n for n in G if n<3]
# iterate through nodes
[1, 2]
>>> len(G) # number of nodes in graph
5
The fastest way to traverse all edges of a graph is via adjacency_iter(), but the edges() method is often more
convenient.
>>> for n,nbrsdict in G.adjacency_iter():
...
for nbr,eattr in nbrsdict.items():
...
if 'weight' in eattr:
...
(n,nbr,eattr['weight'])
(1, 2, 4)
(2, 3, 8)
>>> [ (u,v,edata['weight']) for u,v,edata in G.edges(data=True) if 'weight' in edata ]
[(1, 2, 4), (2, 3, 8)]
Reporting:
Simple graph information is obtained using methods. Iterator versions of many reporting methods exist for
efficiency. Methods exist for reporting nodes(), edges(), neighbors() and degree() as well as the number of
nodes and edges.
For details on these and other miscellaneous methods, see below.
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Adding and removing nodes and edges
DiGraph.__init__([data])
DiGraph.add_node(n[, attr_dict])
DiGraph.add_nodes_from(nodes, **attr)
DiGraph.remove_node(n)
DiGraph.remove_nodes_from(nbunch)
DiGraph.add_edge(u, v[, attr_dict])
DiGraph.add_edges_from(ebunch[, attr_dict])
DiGraph.add_weighted_edges_from(ebunch[, weight])
DiGraph.remove_edge(u, v)
DiGraph.remove_edges_from(ebunch)
DiGraph.add_star(nodes, **attr)
DiGraph.add_path(nodes, **attr)
DiGraph.add_cycle(nodes, **attr)
DiGraph.clear()
Initialize a graph with edges, name, graph attributes.
Add a single node n and update node attributes.
Add multiple nodes.
Remove node n.
Remove multiple nodes.
Add an edge between u and v.
Add all the edges in ebunch.
Add all the edges in ebunch as weighted edges with specified weig
Remove the edge between u and v.
Remove all edges specified in ebunch.
Add a star.
Add a path.
Add a cycle.
Remove all nodes and edges from the graph.
__init__
DiGraph.__init__(data=None, **attr)
Initialize a graph with edges, name, graph attributes.
Parameters data : input graph
Data to initialize graph. If data=None (default) an empty graph is created. The data can
be an edge list, or any NetworkX graph object. If the corresponding optional Python
packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse
matrix, or a PyGraphviz graph.
name : string, optional (default=’�)
An optional name for the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to graph as key=value pairs.
See also:
convert
Examples
>>>
>>>
>>>
>>>
G
G
e
G
=
=
=
=
nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
nx.Graph(name='my graph')
[(1,2),(2,3),(3,4)] # list of edges
nx.Graph(e)
Arbitrary graph attribute pairs (key=value) may be assigned
>>> G=nx.Graph(e, day="Friday")
>>> G.graph
{'day': 'Friday'}
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add_node
DiGraph.add_node(n, attr_dict=None, **attr)
Add a single node n and update node attributes.
Parameters n : node
A node can be any hashable Python object except None.
attr_dict : dictionary, optional (default= no attributes)
Dictionary of node attributes. Key/value pairs will update existing data associated with
the node.
attr : keyword arguments, optional
Set or change attributes using key=value.
See also:
add_nodes_from
Notes
A hashable object is one that can be used as a key in a Python dictionary. This includes strings, numbers, tuples
of strings and numbers, etc.
On many platforms hashable items also include mutables such as NetworkX Graphs, though one should be
careful that the hash doesn’t change on mutables.
Examples
>>>
>>>
>>>
>>>
>>>
>>>
3
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_node(1)
G.add_node('Hello')
K3 = nx.Graph([(0,1),(1,2),(2,0)])
G.add_node(K3)
G.number_of_nodes()
Use keywords set/change node attributes:
>>> G.add_node(1,size=10)
>>> G.add_node(3,weight=0.4,UTM=('13S',382871,3972649))
add_nodes_from
DiGraph.add_nodes_from(nodes, **attr)
Add multiple nodes.
Parameters nodes : iterable container
A container of nodes (list, dict, set, etc.). OR A container of (node, attribute dict) tuples.
Node attributes are updated using the attribute dict.
attr : keyword arguments, optional (default= no attributes)
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Update attributes for all nodes in nodes. Node attributes specified in nodes as a tuple
take precedence over attributes specified generally.
See also:
add_node
Examples
>>>
>>>
>>>
>>>
>>>
[0,
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_nodes_from('Hello')
K3 = nx.Graph([(0,1),(1,2),(2,0)])
G.add_nodes_from(K3)
sorted(G.nodes(),key=str)
1, 2, 'H', 'e', 'l', 'o']
Use keywords to update specific node attributes for every node.
>>> G.add_nodes_from([1,2], size=10)
>>> G.add_nodes_from([3,4], weight=0.4)
Use (node, attrdict) tuples to update attributes for specific nodes.
>>>
>>>
11
>>>
>>>
>>>
11
G.add_nodes_from([(1,dict(size=11)), (2,{'color':'blue'})])
G.node[1]['size']
H = nx.Graph()
H.add_nodes_from(G.nodes(data=True))
H.node[1]['size']
remove_node
DiGraph.remove_node(n)
Remove node n.
Removes the node n and all adjacent edges. Attempting to remove a non-existent node will raise an exception.
Parameters n : node
A node in the graph
Raises NetworkXError
If n is not in the graph.
See also:
remove_nodes_from
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> G.edges()
[(0, 1), (1, 2)]
>>> G.remove_node(1)
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>>> G.edges()
[]
remove_nodes_from
DiGraph.remove_nodes_from(nbunch)
Remove multiple nodes.
Parameters nodes : iterable container
A container of nodes (list, dict, set, etc.). If a node in the container is not in the graph it
is silently ignored.
See also:
remove_node
Examples
>>>
>>>
>>>
>>>
[0,
>>>
>>>
[]
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2])
e = G.nodes()
e
1, 2]
G.remove_nodes_from(e)
G.nodes()
add_edge
DiGraph.add_edge(u, v, attr_dict=None, **attr)
Add an edge between u and v.
The nodes u and v will be automatically added if they are not already in the graph.
Edge attributes can be specified with keywords or by providing a dictionary with key/value pairs. See examples
below.
Parameters u,v : nodes
Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None)
Python objects.
attr_dict : dictionary, optional (default= no attributes)
Dictionary of edge attributes. Key/value pairs will update existing data associated with
the edge.
attr : keyword arguments, optional
Edge data (or labels or objects) can be assigned using keyword arguments.
See also:
add_edges_from add a collection of edges
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Notes
Adding an edge that already exists updates the edge data.
Many NetworkX algorithms designed for weighted graphs use as the edge weight a numerical value assigned to
a keyword which by default is �weight’.
Examples
The following all add the edge e=(1,2) to graph G:
>>>
>>>
>>>
>>>
>>>
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
e = (1,2)
G.add_edge(1, 2)
# explicit two-node form
G.add_edge(*e)
# single edge as tuple of two nodes
G.add_edges_from( [(1,2)] ) # add edges from iterable container
Associate data to edges using keywords:
>>> G.add_edge(1, 2, weight=3)
>>> G.add_edge(1, 3, weight=7, capacity=15, length=342.7)
add_edges_from
DiGraph.add_edges_from(ebunch, attr_dict=None, **attr)
Add all the edges in ebunch.
Parameters ebunch : container of edges
Each edge given in the container will be added to the graph. The edges must be given
as as 2-tuples (u,v) or 3-tuples (u,v,d) where d is a dictionary containing edge data.
attr_dict : dictionary, optional (default= no attributes)
Dictionary of edge attributes. Key/value pairs will update existing data associated with
each edge.
attr : keyword arguments, optional
Edge data (or labels or objects) can be assigned using keyword arguments.
See also:
add_edge add a single edge
add_weighted_edges_from convenient way to add weighted edges
Notes
Adding the same edge twice has no effect but any edge data will be updated when each duplicate edge is added.
Edge attributes specified in edges as a tuple take precedence over attributes specified generally.
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Examples
>>>
>>>
>>>
>>>
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edges_from([(0,1),(1,2)]) # using a list of edge tuples
e = zip(range(0,3),range(1,4))
G.add_edges_from(e) # Add the path graph 0-1-2-3
Associate data to edges
>>> G.add_edges_from([(1,2),(2,3)], weight=3)
>>> G.add_edges_from([(3,4),(1,4)], label='WN2898')
add_weighted_edges_from
DiGraph.add_weighted_edges_from(ebunch, weight=’weight’, **attr)
Add all the edges in ebunch as weighted edges with specified weights.
Parameters ebunch : container of edges
Each edge given in the list or container will be added to the graph. The edges must be
given as 3-tuples (u,v,w) where w is a number.
weight : string, optional (default= �weight’)
The attribute name for the edge weights to be added.
attr : keyword arguments, optional (default= no attributes)
Edge attributes to add/update for all edges.
See also:
add_edge add a single edge
add_edges_from add multiple edges
Notes
Adding the same edge twice for Graph/DiGraph simply updates the edge data. For MultiGraph/MultiDiGraph,
duplicate edges are stored.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_weighted_edges_from([(0,1,3.0),(1,2,7.5)])
remove_edge
DiGraph.remove_edge(u, v)
Remove the edge between u and v.
Parameters u,v: nodes
Remove the edge between nodes u and v.
Raises NetworkXError
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If there is not an edge between u and v.
See also:
remove_edges_from remove a collection of edges
Examples
>>>
>>>
>>>
>>>
>>>
>>>
>>>
G = nx.Graph()
# or DiGraph, etc
G.add_path([0,1,2,3])
G.remove_edge(0,1)
e = (1,2)
G.remove_edge(*e) # unpacks e from an edge tuple
e = (2,3,{'weight':7}) # an edge with attribute data
G.remove_edge(*e[:2]) # select first part of edge tuple
remove_edges_from
DiGraph.remove_edges_from(ebunch)
Remove all edges specified in ebunch.
Parameters ebunch: list or container of edge tuples
Each edge given in the list or container will be removed from the graph. The edges can
be:
• 2-tuples (u,v) edge between u and v.
• 3-tuples (u,v,k) where k is ignored.
See also:
remove_edge remove a single edge
Notes
Will fail silently if an edge in ebunch is not in the graph.
Examples
>>>
>>>
>>>
>>>
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
ebunch=[(1,2),(2,3)]
G.remove_edges_from(ebunch)
add_star
DiGraph.add_star(nodes, **attr)
Add a star.
The first node in nodes is the middle of the star. It is connected to all other nodes.
Parameters nodes : iterable container
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A container of nodes.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in star.
See also:
add_path, add_cycle
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_star([0,1,2,3])
>>> G.add_star([10,11,12],weight=2)
add_path
DiGraph.add_path(nodes, **attr)
Add a path.
Parameters nodes : iterable container
A container of nodes. A path will be constructed from the nodes (in order) and added
to the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in path.
See also:
add_star, add_cycle
Examples
>>> G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.add_path([10,11,12],weight=7)
add_cycle
DiGraph.add_cycle(nodes, **attr)
Add a cycle.
Parameters nodes: iterable container
A container of nodes. A cycle will be constructed from the nodes (in order) and added
to the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in cycle.
See also:
add_path, add_star
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Examples
>>> G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_cycle([0,1,2,3])
>>> G.add_cycle([10,11,12],weight=7)
clear
DiGraph.clear()
Remove all nodes and edges from the graph.
This also removes the name, and all graph, node, and edge attributes.
Examples
>>>
>>>
>>>
>>>
[]
>>>
[]
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.clear()
G.nodes()
G.edges()
Iterating over nodes and edges
DiGraph.nodes([data])
DiGraph.nodes_iter([data])
DiGraph.__iter__()
DiGraph.edges([nbunch, data])
DiGraph.edges_iter([nbunch, data])
DiGraph.out_edges([nbunch, data])
DiGraph.out_edges_iter([nbunch, data])
DiGraph.in_edges([nbunch, data])
DiGraph.in_edges_iter([nbunch, data])
DiGraph.get_edge_data(u, v[, default])
DiGraph.neighbors(n)
DiGraph.neighbors_iter(n)
DiGraph.__getitem__(n)
DiGraph.successors(n)
DiGraph.successors_iter(n)
DiGraph.predecessors(n)
DiGraph.predecessors_iter(n)
DiGraph.adjacency_list()
DiGraph.adjacency_iter()
DiGraph.nbunch_iter([nbunch])
46
Return a list of the nodes in the graph.
Return an iterator over the nodes.
Iterate over the nodes.
Return a list of edges.
Return an iterator over the edges.
Return a list of edges.
Return an iterator over the edges.
Return a list of the incoming edges.
Return an iterator over the incoming edges.
Return the attribute dictionary associated with edge (u,v).
Return a list of successor nodes of n.
Return an iterator over successor nodes of n.
Return a dict of neighbors of node n.
Return a list of successor nodes of n.
Return an iterator over successor nodes of n.
Return a list of predecessor nodes of n.
Return an iterator over predecessor nodes of n.
Return an adjacency list representation of the graph.
Return an iterator of (node, adjacency dict) tuples for all nodes.
Return an iterator of nodes contained in nbunch that are also in the graph.
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nodes
DiGraph.nodes(data=False)
Return a list of the nodes in the graph.
Parameters data : boolean, optional (default=False)
If False return a list of nodes. If True return a two-tuple of node and node data dictionary
Returns nlist : list
A list of nodes. If data=True a list of two-tuples containing (node, node data dictionary).
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> G.nodes()
[0, 1, 2]
>>> G.add_node(1, time='5pm')
>>> G.nodes(data=True)
[(0, {}), (1, {'time': '5pm'}), (2, {})]
nodes_iter
DiGraph.nodes_iter(data=False)
Return an iterator over the nodes.
Parameters data : boolean, optional (default=False)
If False the iterator returns nodes. If True return a two-tuple of node and node data
dictionary
Returns niter : iterator
An iterator over nodes. If data=True the iterator gives two-tuples containing (node, node
data, dictionary)
Notes
If the node data is not required it is simpler and equivalent to use the expression �for n in G’.
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> [d for n,d in G.nodes_iter(data=True)]
[{}, {}, {}]
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__iter__
DiGraph.__iter__()
Iterate over the nodes. Use the expression �for n in G’.
Returns niter : iterator
An iterator over all nodes in the graph.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
edges
DiGraph.edges(nbunch=None, data=False)
Return a list of edges.
Edges are returned as tuples with optional data in the order (node, neighbor, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
Return two tuples (u,v) (False) or three-tuples (u,v,data) (True).
Returns edge_list: list of edge tuples
Edges that are adjacent to any node in nbunch, or a list of all edges if nbunch is not
specified.
See also:
edges_iter return an iterator over the edges
Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.edges()
[(0, 1), (1, 2), (2, 3)]
>>> G.edges(data=True) # default edge data is {} (empty dictionary)
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
>>> G.edges([0,3])
[(0, 1), (3, 2)]
>>> G.edges(0)
[(0, 1)]
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edges_iter
DiGraph.edges_iter(nbunch=None, data=False)
Return an iterator over the edges.
Edges are returned as tuples with optional data in the order (node, neighbor, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
If True, return edge attribute dict in 3-tuple (u,v,data).
Returns edge_iter : iterator
An iterator of (u,v) or (u,v,d) tuples of edges.
See also:
edges return a list of edges
Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.
Examples
>>> G = nx.DiGraph()
# or MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> [e for e in G.edges_iter()]
[(0, 1), (1, 2), (2, 3)]
>>> list(G.edges_iter(data=True)) # default data is {} (empty dict)
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
>>> list(G.edges_iter([0,2]))
[(0, 1), (2, 3)]
>>> list(G.edges_iter(0))
[(0, 1)]
out_edges
DiGraph.out_edges(nbunch=None, data=False)
Return a list of edges.
Edges are returned as tuples with optional data in the order (node, neighbor, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
Return two tuples (u,v) (False) or three-tuples (u,v,data) (True).
Returns edge_list: list of edge tuples
Edges that are adjacent to any node in nbunch, or a list of all edges if nbunch is not
specified.
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See also:
edges_iter return an iterator over the edges
Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.edges()
[(0, 1), (1, 2), (2, 3)]
>>> G.edges(data=True) # default edge data is {} (empty dictionary)
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
>>> G.edges([0,3])
[(0, 1), (3, 2)]
>>> G.edges(0)
[(0, 1)]
out_edges_iter
DiGraph.out_edges_iter(nbunch=None, data=False)
Return an iterator over the edges.
Edges are returned as tuples with optional data in the order (node, neighbor, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
If True, return edge attribute dict in 3-tuple (u,v,data).
Returns edge_iter : iterator
An iterator of (u,v) or (u,v,d) tuples of edges.
See also:
edges return a list of edges
Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.
Examples
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>>> G = nx.DiGraph()
# or MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> [e for e in G.edges_iter()]
[(0, 1), (1, 2), (2, 3)]
>>> list(G.edges_iter(data=True)) # default data is {} (empty dict)
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
>>> list(G.edges_iter([0,2]))
[(0, 1), (2, 3)]
>>> list(G.edges_iter(0))
[(0, 1)]
in_edges
DiGraph.in_edges(nbunch=None, data=False)
Return a list of the incoming edges.
See also:
edges return a list of edges
in_edges_iter
DiGraph.in_edges_iter(nbunch=None, data=False)
Return an iterator over the incoming edges.
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
If True, return edge attribute dict in 3-tuple (u,v,data).
Returns in_edge_iter : iterator
An iterator of (u,v) or (u,v,d) tuples of incoming edges.
See also:
edges_iter return an iterator of edges
get_edge_data
DiGraph.get_edge_data(u, v, default=None)
Return the attribute dictionary associated with edge (u,v).
Parameters u,v : nodes
default: any Python object (default=None)
Value to return if the edge (u,v) is not found.
Returns edge_dict : dictionary
The edge attribute dictionary.
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Notes
It is faster to use G[u][v].
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G[0][1]
{}
Warning: Assigning G[u][v] corrupts the graph data structure. But it is safe to assign attributes to that dictionary,
>>> G[0][1]['weight'] = 7
>>> G[0][1]['weight']
7
>>> G[1][0]['weight']
7
Examples
>>>
>>>
>>>
{}
>>>
>>>
{}
>>>
0
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.get_edge_data(0,1) # default edge data is {}
e = (0,1)
G.get_edge_data(*e) # tuple form
G.get_edge_data('a','b',default=0) # edge not in graph, return 0
neighbors
DiGraph.neighbors(n)
Return a list of successor nodes of n.
neighbors() and successors() are the same function.
neighbors_iter
DiGraph.neighbors_iter(n)
Return an iterator over successor nodes of n.
neighbors_iter() and successors_iter() are the same.
__getitem__
DiGraph.__getitem__(n)
Return a dict of neighbors of node n. Use the expression �G[n]’.
Parameters n : node
A node in the graph.
Returns adj_dict : dictionary
The adjacency dictionary for nodes connected to n.
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Notes
G[n] is similar to G.neighbors(n) but the internal data dictionary is returned instead of a list.
Assigning G[n] will corrupt the internal graph data structure. Use G[n] for reading data only.
Examples
>>>
>>>
>>>
{1:
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G[0]
{}}
successors
DiGraph.successors(n)
Return a list of successor nodes of n.
neighbors() and successors() are the same function.
successors_iter
DiGraph.successors_iter(n)
Return an iterator over successor nodes of n.
neighbors_iter() and successors_iter() are the same.
predecessors
DiGraph.predecessors(n)
Return a list of predecessor nodes of n.
predecessors_iter
DiGraph.predecessors_iter(n)
Return an iterator over predecessor nodes of n.
adjacency_list
DiGraph.adjacency_list()
Return an adjacency list representation of the graph.
The output adjacency list is in the order of G.nodes(). For directed graphs, only outgoing adjacencies are
included.
Returns adj_list : lists of lists
The adjacency structure of the graph as a list of lists.
See also:
adjacency_iter
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Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.adjacency_list() # in order given by G.nodes()
[[1], [0, 2], [1, 3], [2]]
adjacency_iter
DiGraph.adjacency_iter()
Return an iterator of (node, adjacency dict) tuples for all nodes.
This is the fastest way to look at every edge. For directed graphs, only outgoing adjacencies are included.
Returns adj_iter : iterator
An iterator of (node, adjacency dictionary) for all nodes in the graph.
See also:
adjacency_list
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> [(n,nbrdict) for n,nbrdict in G.adjacency_iter()]
[(0, {1: {}}), (1, {0: {}, 2: {}}), (2, {1: {}, 3: {}}), (3, {2: {}})]
nbunch_iter
DiGraph.nbunch_iter(nbunch=None)
Return an iterator of nodes contained in nbunch that are also in the graph.
The nodes in nbunch are checked for membership in the graph and if not are silently ignored.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
Returns niter : iterator
An iterator over nodes in nbunch that are also in the graph. If nbunch is None, iterate
over all nodes in the graph.
Raises NetworkXError
If nbunch is not a node or or sequence of nodes. If a node in nbunch is not hashable.
See also:
Graph.__iter__
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Notes
When nbunch is an iterator, the returned iterator yields values directly from nbunch, becoming exhausted when
nbunch is exhausted.
To test whether nbunch is a single node, one can use “if nbunch in self:”, even after processing with this routine.
If nbunch is not a node or a (possibly empty) sequence/iterator or None, a NetworkXError is raised. Also, if any
object in nbunch is not hashable, a NetworkXError is raised.
Information about graph structure
DiGraph.has_node(n)
DiGraph.__contains__(n)
DiGraph.has_edge(u, v)
DiGraph.order()
DiGraph.number_of_nodes()
DiGraph.__len__()
DiGraph.degree([nbunch, weight])
DiGraph.degree_iter([nbunch, weight])
DiGraph.in_degree([nbunch, weight])
DiGraph.in_degree_iter([nbunch, weight])
DiGraph.out_degree([nbunch, weight])
DiGraph.out_degree_iter([nbunch, weight])
DiGraph.size([weight])
DiGraph.number_of_edges([u, v])
DiGraph.nodes_with_selfloops()
DiGraph.selfloop_edges([data])
DiGraph.number_of_selfloops()
Return True if the graph contains the node n.
Return True if n is a node, False otherwise.
Return True if the edge (u,v) is in the graph.
Return the number of nodes in the graph.
Return the number of nodes in the graph.
Return the number of nodes.
Return the degree of a node or nodes.
Return an iterator for (node, degree).
Return the in-degree of a node or nodes.
Return an iterator for (node, in-degree).
Return the out-degree of a node or nodes.
Return an iterator for (node, out-degree).
Return the number of edges.
Return the number of edges between two nodes.
Return a list of nodes with self loops.
Return a list of selfloop edges.
Return the number of selfloop edges.
has_node
DiGraph.has_node(n)
Return True if the graph contains the node n.
Parameters n : node
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> G.has_node(0)
True
It is more readable and simpler to use
>>> 0 in G
True
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__contains__
DiGraph.__contains__(n)
Return True if n is a node, False otherwise. Use the expression �n in G’.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> 1 in G
True
has_edge
DiGraph.has_edge(u, v)
Return True if the edge (u,v) is in the graph.
Parameters u,v : nodes
Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None)
Python objects.
Returns edge_ind : bool
True if edge is in the graph, False otherwise.
Examples
Can be called either using two nodes u,v or edge tuple (u,v)
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.has_edge(0,1) # using two nodes
True
>>> e = (0,1)
>>> G.has_edge(*e) # e is a 2-tuple (u,v)
True
>>> e = (0,1,{'weight':7})
>>> G.has_edge(*e[:2]) # e is a 3-tuple (u,v,data_dictionary)
True
The following syntax are all equivalent:
>>> G.has_edge(0,1)
True
>>> 1 in G[0] # though this gives KeyError if 0 not in G
True
order
DiGraph.order()
Return the number of nodes in the graph.
Returns nnodes : int
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The number of nodes in the graph.
See also:
number_of_nodes, __len__
number_of_nodes
DiGraph.number_of_nodes()
Return the number of nodes in the graph.
Returns nnodes : int
The number of nodes in the graph.
See also:
order, __len__
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> len(G)
3
__len__
DiGraph.__len__()
Return the number of nodes. Use the expression �len(G)’.
Returns nnodes : int
The number of nodes in the graph.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> len(G)
4
degree
DiGraph.degree(nbunch=None, weight=None)
Return the degree of a node or nodes.
The node degree is the number of edges adjacent to that node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
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The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd : dictionary, or number
A dictionary with nodes as keys and degree as values or a number if a single node is
specified.
Examples
>>>
>>>
>>>
1
>>>
{0:
>>>
[1,
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.degree(0)
G.degree([0,1])
1, 1: 2}
list(G.degree([0,1]).values())
2]
degree_iter
DiGraph.degree_iter(nbunch=None, weight=None)
Return an iterator for (node, degree).
The node degree is the number of edges adjacent to the node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd_iter : an iterator
The iterator returns two-tuples of (node, degree).
See also:
degree, in_degree, out_degree, in_degree_iter, out_degree_iter
Examples
>>> G = nx.DiGraph()
# or MultiDiGraph
>>> G.add_path([0,1,2,3])
>>> list(G.degree_iter(0)) # node 0 with degree 1
[(0, 1)]
>>> list(G.degree_iter([0,1]))
[(0, 1), (1, 2)]
in_degree
DiGraph.in_degree(nbunch=None, weight=None)
Return the in-degree of a node or nodes.
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The node in-degree is the number of edges pointing in to the node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd : dictionary, or number
A dictionary with nodes as keys and in-degree as values or a number if a single node is
specified.
See also:
degree, out_degree, in_degree_iter
Examples
>>>
>>>
>>>
0
>>>
{0:
>>>
[0,
G = nx.DiGraph()
# or MultiDiGraph
G.add_path([0,1,2,3])
G.in_degree(0)
G.in_degree([0,1])
0, 1: 1}
list(G.in_degree([0,1]).values())
1]
in_degree_iter
DiGraph.in_degree_iter(nbunch=None, weight=None)
Return an iterator for (node, in-degree).
The node in-degree is the number of edges pointing in to the node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd_iter : an iterator
The iterator returns two-tuples of (node, in-degree).
See also:
degree, in_degree, out_degree, out_degree_iter
Examples
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>>> G = nx.DiGraph()
>>> G.add_path([0,1,2,3])
>>> list(G.in_degree_iter(0)) # node 0 with degree 0
[(0, 0)]
>>> list(G.in_degree_iter([0,1]))
[(0, 0), (1, 1)]
out_degree
DiGraph.out_degree(nbunch=None, weight=None)
Return the out-degree of a node or nodes.
The node out-degree is the number of edges pointing out of the node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd : dictionary, or number
A dictionary with nodes as keys and out-degree as values or a number if a single node
is specified.
Examples
>>>
>>>
>>>
1
>>>
{0:
>>>
[1,
G = nx.DiGraph()
# or MultiDiGraph
G.add_path([0,1,2,3])
G.out_degree(0)
G.out_degree([0,1])
1, 1: 1}
list(G.out_degree([0,1]).values())
1]
out_degree_iter
DiGraph.out_degree_iter(nbunch=None, weight=None)
Return an iterator for (node, out-degree).
The node out-degree is the number of edges pointing out of the node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd_iter : an iterator
The iterator returns two-tuples of (node, out-degree).
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See also:
degree, in_degree, out_degree, in_degree_iter
Examples
>>> G = nx.DiGraph()
>>> G.add_path([0,1,2,3])
>>> list(G.out_degree_iter(0)) # node 0 with degree 1
[(0, 1)]
>>> list(G.out_degree_iter([0,1]))
[(0, 1), (1, 1)]
size
DiGraph.size(weight=None)
Return the number of edges.
Parameters weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1.
Returns nedges : int
The number of edges or sum of edge weights in the graph.
See also:
number_of_edges
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.size()
3
>>>
>>>
>>>
>>>
2
>>>
6.0
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge('a','b',weight=2)
G.add_edge('b','c',weight=4)
G.size()
G.size(weight='weight')
number_of_edges
DiGraph.number_of_edges(u=None, v=None)
Return the number of edges between two nodes.
Parameters u,v : nodes, optional (default=all edges)
If u and v are specified, return the number of edges between u and v. Otherwise return
the total number of all edges.
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Returns nedges : int
The number of edges in the graph. If nodes u and v are specified return the number of
edges between those nodes.
See also:
size
Examples
>>>
>>>
>>>
3
>>>
1
>>>
>>>
1
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.number_of_edges()
G.number_of_edges(0,1)
e = (0,1)
G.number_of_edges(*e)
nodes_with_selfloops
DiGraph.nodes_with_selfloops()
Return a list of nodes with self loops.
A node with a self loop has an edge with both ends adjacent to that node.
Returns nodelist : list
A list of nodes with self loops.
See also:
selfloop_edges, number_of_selfloops
Examples
>>>
>>>
>>>
>>>
[1]
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge(1,1)
G.add_edge(1,2)
G.nodes_with_selfloops()
selfloop_edges
DiGraph.selfloop_edges(data=False)
Return a list of selfloop edges.
A selfloop edge has the same node at both ends.
Parameters data : bool, optional (default=False)
Return selfloop edges as two tuples (u,v) (data=False) or three-tuples (u,v,data)
(data=True)
Returns edgelist : list of edge tuples
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A list of all selfloop edges.
See also:
nodes_with_selfloops, number_of_selfloops
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edge(1,1)
>>> G.add_edge(1,2)
>>> G.selfloop_edges()
[(1, 1)]
>>> G.selfloop_edges(data=True)
[(1, 1, {})]
number_of_selfloops
DiGraph.number_of_selfloops()
Return the number of selfloop edges.
A selfloop edge has the same node at both ends.
Returns nloops : int
The number of selfloops.
See also:
nodes_with_selfloops, selfloop_edges
Examples
>>>
>>>
>>>
>>>
1
G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge(1,1)
G.add_edge(1,2)
G.number_of_selfloops()
Making copies and subgraphs
DiGraph.copy()
DiGraph.to_undirected([reciprocal])
DiGraph.to_directed()
DiGraph.subgraph(nbunch)
DiGraph.reverse([copy])
Return a copy of the graph.
Return an undirected representation of the digraph.
Return a directed copy of the graph.
Return the subgraph induced on nodes in nbunch.
Return the reverse of the graph.
copy
DiGraph.copy()
Return a copy of the graph.
Returns G : Graph
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A copy of the graph.
See also:
to_directed return a directed copy of the graph.
Notes
This makes a complete copy of the graph including all of the node or edge attributes.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> H = G.copy()
to_undirected
DiGraph.to_undirected(reciprocal=False)
Return an undirected representation of the digraph.
Parameters reciprocal : bool (optional)
If True only keep edges that appear in both directions in the original digraph.
Returns G : Graph
An undirected graph with the same name and nodes and with edge (u,v,data) if either
(u,v,data) or (v,u,data) is in the digraph. If both edges exist in digraph and their edge
data is different, only one edge is created with an arbitrary choice of which edge data to
use. You must check and correct for this manually if desired.
Notes
If edges in both directions (u,v) and (v,u) exist in the graph, attributes for the new undirected edge will be a
combination of the attributes of the directed edges. The edge data is updated in the (arbitrary) order that the
edges are encountered. For more customized control of the edge attributes use add_edge().
This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the
data and references.
This is in contrast to the similar G=DiGraph(D) which returns a shallow copy of the data.
See the Python copy module for
http://docs.python.org/library/copy.html.
more
information
on
shallow
and
deep
copies,
to_directed
DiGraph.to_directed()
Return a directed copy of the graph.
Returns G : DiGraph
A deepcopy of the graph.
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Notes
This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the
data and references.
This is in contrast to the similar D=DiGraph(G) which returns a shallow copy of the data.
See the Python copy module for
http://docs.python.org/library/copy.html.
more
information
on
shallow
and
deep
copies,
Examples
>>> G = nx.Graph()
# or MultiGraph, etc
>>> G.add_path([0,1])
>>> H = G.to_directed()
>>> H.edges()
[(0, 1), (1, 0)]
If already directed, return a (deep) copy
>>> G = nx.DiGraph()
# or MultiDiGraph, etc
>>> G.add_path([0,1])
>>> H = G.to_directed()
>>> H.edges()
[(0, 1)]
subgraph
DiGraph.subgraph(nbunch)
Return the subgraph induced on nodes in nbunch.
The induced subgraph of the graph contains the nodes in nbunch and the edges between those nodes.
Parameters nbunch : list, iterable
A container of nodes which will be iterated through once.
Returns G : Graph
A subgraph of the graph with the same edge attributes.
Notes
The graph, edge or node attributes just point to the original graph. So changes to the node or edge structure will
not be reflected in the original graph while changes to the attributes will.
To create a subgraph with its own copy of the edge/node attributes use: nx.Graph(G.subgraph(nbunch))
If edge attributes are containers, a deep copy can be obtained using: G.subgraph(nbunch).copy()
For an inplace reduction of a graph to a subgraph you can remove nodes: G.remove_nodes_from([ n in G if n
not in set(nbunch)])
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Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> H = G.subgraph([0,1,2])
>>> H.edges()
[(0, 1), (1, 2)]
reverse
DiGraph.reverse(copy=True)
Return the reverse of the graph.
The reverse is a graph with the same nodes and edges but with the directions of the edges reversed.
Parameters copy : bool optional (default=True)
If True, return a new DiGraph holding the reversed edges. If False, reverse the reverse
graph is created using the original graph (this changes the original graph).
3.2.3 MultiGraph - Undirected graphs with self loops and parallel edges
Overview
MultiGraph(data=None, **attr)
An undirected graph class that can store multiedges.
Multiedges are multiple edges between two nodes. Each edge can hold optional data or attributes.
A MultiGraph holds undirected edges. Self loops are allowed.
Nodes can be arbitrary (hashable) Python objects with optional key/value attributes.
Edges are represented as links between nodes with optional key/value attributes.
Parameters data : input graph
Data to initialize graph. If data=None (default) an empty graph is created. The data can
be an edge list, or any NetworkX graph object. If the corresponding optional Python
packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse
matrix, or a PyGraphviz graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to graph as key=value pairs.
See also:
Graph, DiGraph, MultiDiGraph
Examples
Create an empty graph structure (a “null graph”) with no nodes and no edges.
>>> G = nx.MultiGraph()
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G can be grown in several ways.
Nodes:
Add one node at a time:
>>> G.add_node(1)
Add the nodes from any container (a list, dict, set or even the lines from a file or the nodes from another graph).
>>>
>>>
>>>
>>>
>>>
G.add_nodes_from([2,3])
G.add_nodes_from(range(100,110))
H=nx.Graph()
H.add_path([0,1,2,3,4,5,6,7,8,9])
G.add_nodes_from(H)
In addition to strings and integers any hashable Python object (except None) can represent a node, e.g. a
customized node object, or even another Graph.
>>> G.add_node(H)
Edges:
G can also be grown by adding edges.
Add one edge,
>>> G.add_edge(1, 2)
a list of edges,
>>> G.add_edges_from([(1,2),(1,3)])
or a collection of edges,
>>> G.add_edges_from(H.edges())
If some edges connect nodes not yet in the graph, the nodes are added automatically. If an edge already exists,
an additional edge is created and stored using a key to identify the edge. By default the key is the lowest unused
integer.
>>> G.add_edges_from([(4,5,dict(route=282)), (4,5,dict(route=37))])
>>> G[4]
{3: {0: {}}, 5: {0: {}, 1: {'route': 282}, 2: {'route': 37}}}
Attributes:
Each graph, node, and edge can hold key/value attribute pairs in an associated attribute dictionary (the keys
must be hashable). By default these are empty, but can be added or changed using add_edge, add_node or direct
manipulation of the attribute dictionaries named graph, node and edge respectively.
>>> G = nx.MultiGraph(day="Friday")
>>> G.graph
{'day': 'Friday'}
Add node attributes using add_node(), add_nodes_from() or G.node
>>> G.add_node(1, time='5pm')
>>> G.add_nodes_from([3], time='2pm')
>>> G.node[1]
{'time': '5pm'}
>>> G.node[1]['room'] = 714
>>> del G.node[1]['room'] # remove attribute
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>>> G.nodes(data=True)
[(1, {'time': '5pm'}), (3, {'time': '2pm'})]
Warning: adding a node to G.node does not add it to the graph.
Add edge attributes using add_edge(), add_edges_from(), subscript notation, or G.edge.
>>>
>>>
>>>
>>>
>>>
G.add_edge(1, 2, weight=4.7 )
G.add_edges_from([(3,4),(4,5)], color='red')
G.add_edges_from([(1,2,{'color':'blue'}), (2,3,{'weight':8})])
G[1][2][0]['weight'] = 4.7
G.edge[1][2][0]['weight'] = 4
Shortcuts:
Many common graph features allow python syntax to speed reporting.
>>> 1 in G
# check if node in graph
True
>>> [n for n in G if n<3]
# iterate through nodes
[1, 2]
>>> len(G) # number of nodes in graph
5
>>> G[1] # adjacency dict keyed by neighbor to edge attributes
...
# Note: you should not change this dict manually!
{2: {0: {'weight': 4}, 1: {'color': 'blue'}}}
The fastest way to traverse all edges of a graph is via adjacency_iter(), but the edges() method is often more
convenient.
>>> for n,nbrsdict in G.adjacency_iter():
...
for nbr,keydict in nbrsdict.items():
...
for key,eattr in keydict.items():
...
if 'weight' in eattr:
...
(n,nbr,eattr['weight'])
(1, 2, 4)
(2, 1, 4)
(2, 3, 8)
(3, 2, 8)
>>> [ (u,v,edata['weight']) for u,v,edata in G.edges(data=True) if 'weight' in edata ]
[(1, 2, 4), (2, 3, 8)]
Reporting:
Simple graph information is obtained using methods. Iterator versions of many reporting methods exist for
efficiency. Methods exist for reporting nodes(), edges(), neighbors() and degree() as well as the number of
nodes and edges.
For details on these and other miscellaneous methods, see below.
Adding and removing nodes and edges
MultiGraph.__init__([data])
MultiGraph.add_node(n[, attr_dict])
MultiGraph.add_nodes_from(nodes, **attr)
MultiGraph.remove_node(n)
MultiGraph.remove_nodes_from(nodes)
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Add a single node n and update node attributes.
Add multiple nodes.
Remove node n.
Remove multiple nodes.
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Table 3.9 – continued from previous page
MultiGraph.add_edge(u, v[, key, attr_dict])
Add an edge between u and v.
MultiGraph.add_edges_from(ebunch[, attr_dict])
Add all the edges in ebunch.
MultiGraph.add_weighted_edges_from(ebunch[, ...]) Add all the edges in ebunch as weighted edges with specified weig
MultiGraph.remove_edge(u, v[, key])
Remove an edge between u and v.
MultiGraph.remove_edges_from(ebunch)
Remove all edges specified in ebunch.
MultiGraph.add_star(nodes, **attr)
Add a star.
MultiGraph.add_path(nodes, **attr)
Add a path.
MultiGraph.add_cycle(nodes, **attr)
Add a cycle.
MultiGraph.clear()
Remove all nodes and edges from the graph.
__init__
MultiGraph.__init__(data=None, **attr)
Initialize a graph with edges, name, graph attributes.
Parameters data : input graph
Data to initialize graph. If data=None (default) an empty graph is created. The data can
be an edge list, or any NetworkX graph object. If the corresponding optional Python
packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse
matrix, or a PyGraphviz graph.
name : string, optional (default=’�)
An optional name for the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to graph as key=value pairs.
See also:
convert
Examples
>>>
>>>
>>>
>>>
G
G
e
G
=
=
=
=
nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
nx.Graph(name='my graph')
[(1,2),(2,3),(3,4)] # list of edges
nx.Graph(e)
Arbitrary graph attribute pairs (key=value) may be assigned
>>> G=nx.Graph(e, day="Friday")
>>> G.graph
{'day': 'Friday'}
add_node
MultiGraph.add_node(n, attr_dict=None, **attr)
Add a single node n and update node attributes.
Parameters n : node
A node can be any hashable Python object except None.
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attr_dict : dictionary, optional (default= no attributes)
Dictionary of node attributes. Key/value pairs will update existing data associated with
the node.
attr : keyword arguments, optional
Set or change attributes using key=value.
See also:
add_nodes_from
Notes
A hashable object is one that can be used as a key in a Python dictionary. This includes strings, numbers, tuples
of strings and numbers, etc.
On many platforms hashable items also include mutables such as NetworkX Graphs, though one should be
careful that the hash doesn’t change on mutables.
Examples
>>>
>>>
>>>
>>>
>>>
>>>
3
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_node(1)
G.add_node('Hello')
K3 = nx.Graph([(0,1),(1,2),(2,0)])
G.add_node(K3)
G.number_of_nodes()
Use keywords set/change node attributes:
>>> G.add_node(1,size=10)
>>> G.add_node(3,weight=0.4,UTM=('13S',382871,3972649))
add_nodes_from
MultiGraph.add_nodes_from(nodes, **attr)
Add multiple nodes.
Parameters nodes : iterable container
A container of nodes (list, dict, set, etc.). OR A container of (node, attribute dict) tuples.
Node attributes are updated using the attribute dict.
attr : keyword arguments, optional (default= no attributes)
Update attributes for all nodes in nodes. Node attributes specified in nodes as a tuple
take precedence over attributes specified generally.
See also:
add_node
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Examples
>>>
>>>
>>>
>>>
>>>
[0,
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_nodes_from('Hello')
K3 = nx.Graph([(0,1),(1,2),(2,0)])
G.add_nodes_from(K3)
sorted(G.nodes(),key=str)
1, 2, 'H', 'e', 'l', 'o']
Use keywords to update specific node attributes for every node.
>>> G.add_nodes_from([1,2], size=10)
>>> G.add_nodes_from([3,4], weight=0.4)
Use (node, attrdict) tuples to update attributes for specific nodes.
>>>
>>>
11
>>>
>>>
>>>
11
G.add_nodes_from([(1,dict(size=11)), (2,{'color':'blue'})])
G.node[1]['size']
H = nx.Graph()
H.add_nodes_from(G.nodes(data=True))
H.node[1]['size']
remove_node
MultiGraph.remove_node(n)
Remove node n.
Removes the node n and all adjacent edges. Attempting to remove a non-existent node will raise an exception.
Parameters n : node
A node in the graph
Raises NetworkXError
If n is not in the graph.
See also:
remove_nodes_from
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> G.edges()
[(0, 1), (1, 2)]
>>> G.remove_node(1)
>>> G.edges()
[]
remove_nodes_from
MultiGraph.remove_nodes_from(nodes)
Remove multiple nodes.
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Parameters nodes : iterable container
A container of nodes (list, dict, set, etc.). If a node in the container is not in the graph it
is silently ignored.
See also:
remove_node
Examples
>>>
>>>
>>>
>>>
[0,
>>>
>>>
[]
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2])
e = G.nodes()
e
1, 2]
G.remove_nodes_from(e)
G.nodes()
add_edge
MultiGraph.add_edge(u, v, key=None, attr_dict=None, **attr)
Add an edge between u and v.
The nodes u and v will be automatically added if they are not already in the graph.
Edge attributes can be specified with keywords or by providing a dictionary with key/value pairs. See examples
below.
Parameters u,v : nodes
Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None)
Python objects.
key : hashable identifier, optional (default=lowest unused integer)
Used to distinguish multiedges between a pair of nodes.
attr_dict : dictionary, optional (default= no attributes)
Dictionary of edge attributes. Key/value pairs will update existing data associated with
the edge.
attr : keyword arguments, optional
Edge data (or labels or objects) can be assigned using keyword arguments.
See also:
add_edges_from add a collection of edges
Notes
To replace/update edge data, use the optional key argument to identify a unique edge. Otherwise a new edge
will be created.
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NetworkX algorithms designed for weighted graphs cannot use multigraphs directly because it is not clear
how to handle multiedge weights. Convert to Graph using edge attribute �weight’ to enable weighted graph
algorithms.
Examples
The following all add the edge e=(1,2) to graph G:
>>>
>>>
>>>
>>>
>>>
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
e = (1,2)
G.add_edge(1, 2)
# explicit two-node form
G.add_edge(*e)
# single edge as tuple of two nodes
G.add_edges_from( [(1,2)] ) # add edges from iterable container
Associate data to edges using keywords:
>>> G.add_edge(1, 2, weight=3)
>>> G.add_edge(1, 2, key=0, weight=4)
# update data for key=0
>>> G.add_edge(1, 3, weight=7, capacity=15, length=342.7)
add_edges_from
MultiGraph.add_edges_from(ebunch, attr_dict=None, **attr)
Add all the edges in ebunch.
Parameters ebunch : container of edges
Each edge given in the container will be added to the graph. The edges can be:
• 2-tuples (u,v) or
• 3-tuples (u,v,d) for an edge attribute dict d, or
• 4-tuples (u,v,k,d) for an edge identified by key k
attr_dict : dictionary, optional (default= no attributes)
Dictionary of edge attributes. Key/value pairs will update existing data associated with
each edge.
attr : keyword arguments, optional
Edge data (or labels or objects) can be assigned using keyword arguments.
See also:
add_edge add a single edge
add_weighted_edges_from convenient way to add weighted edges
Notes
Adding the same edge twice has no effect but any edge data will be updated when each duplicate edge is added.
Edge attributes specified in edges as a tuple take precedence over attributes specified generally.
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Examples
>>>
>>>
>>>
>>>
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edges_from([(0,1),(1,2)]) # using a list of edge tuples
e = zip(range(0,3),range(1,4))
G.add_edges_from(e) # Add the path graph 0-1-2-3
Associate data to edges
>>> G.add_edges_from([(1,2),(2,3)], weight=3)
>>> G.add_edges_from([(3,4),(1,4)], label='WN2898')
add_weighted_edges_from
MultiGraph.add_weighted_edges_from(ebunch, weight=’weight’, **attr)
Add all the edges in ebunch as weighted edges with specified weights.
Parameters ebunch : container of edges
Each edge given in the list or container will be added to the graph. The edges must be
given as 3-tuples (u,v,w) where w is a number.
weight : string, optional (default= �weight’)
The attribute name for the edge weights to be added.
attr : keyword arguments, optional (default= no attributes)
Edge attributes to add/update for all edges.
See also:
add_edge add a single edge
add_edges_from add multiple edges
Notes
Adding the same edge twice for Graph/DiGraph simply updates the edge data. For MultiGraph/MultiDiGraph,
duplicate edges are stored.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_weighted_edges_from([(0,1,3.0),(1,2,7.5)])
remove_edge
MultiGraph.remove_edge(u, v, key=None)
Remove an edge between u and v.
Parameters u,v: nodes
Remove an edge between nodes u and v.
key : hashable identifier, optional (default=None)
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Used to distinguish multiple edges between a pair of nodes. If None remove a single
(abritrary) edge between u and v.
Raises NetworkXError
If there is not an edge between u and v, or if there is no edge with the specified key.
See also:
remove_edges_from remove a collection of edges
Examples
>>>
>>>
>>>
>>>
>>>
G = nx.MultiGraph()
G.add_path([0,1,2,3])
G.remove_edge(0,1)
e = (1,2)
G.remove_edge(*e) # unpacks e from an edge tuple
For multiple edges
>>> G = nx.MultiGraph()
# or MultiDiGraph, etc
>>> G.add_edges_from([(1,2),(1,2),(1,2)])
>>> G.remove_edge(1,2) # remove a single (arbitrary) edge
For edges with keys
>>>
>>>
>>>
>>>
G = nx.MultiGraph()
# or MultiDiGraph, etc
G.add_edge(1,2,key='first')
G.add_edge(1,2,key='second')
G.remove_edge(1,2,key='second')
remove_edges_from
MultiGraph.remove_edges_from(ebunch)
Remove all edges specified in ebunch.
Parameters ebunch: list or container of edge tuples
Each edge given in the list or container will be removed from the graph. The edges can
be:
• 2-tuples (u,v) All edges between u and v are removed.
• 3-tuples (u,v,key) The edge identified by key is removed.
• 4-tuples (u,v,key,data) where data is ignored.
See also:
remove_edge remove a single edge
Notes
Will fail silently if an edge in ebunch is not in the graph.
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Examples
>>>
>>>
>>>
>>>
G = nx.MultiGraph() # or MultiDiGraph
G.add_path([0,1,2,3])
ebunch=[(1,2),(2,3)]
G.remove_edges_from(ebunch)
Removing multiple copies of edges
>>> G = nx.MultiGraph()
>>> G.add_edges_from([(1,2),(1,2),(1,2)])
>>> G.remove_edges_from([(1,2),(1,2)])
>>> G.edges()
[(1, 2)]
>>> G.remove_edges_from([(1,2),(1,2)]) # silently ignore extra copy
>>> G.edges() # now empty graph
[]
add_star
MultiGraph.add_star(nodes, **attr)
Add a star.
The first node in nodes is the middle of the star. It is connected to all other nodes.
Parameters nodes : iterable container
A container of nodes.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in star.
See also:
add_path, add_cycle
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_star([0,1,2,3])
>>> G.add_star([10,11,12],weight=2)
add_path
MultiGraph.add_path(nodes, **attr)
Add a path.
Parameters nodes : iterable container
A container of nodes. A path will be constructed from the nodes (in order) and added
to the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in path.
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See also:
add_star, add_cycle
Examples
>>> G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.add_path([10,11,12],weight=7)
add_cycle
MultiGraph.add_cycle(nodes, **attr)
Add a cycle.
Parameters nodes: iterable container
A container of nodes. A cycle will be constructed from the nodes (in order) and added
to the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in cycle.
See also:
add_path, add_star
Examples
>>> G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_cycle([0,1,2,3])
>>> G.add_cycle([10,11,12],weight=7)
clear
MultiGraph.clear()
Remove all nodes and edges from the graph.
This also removes the name, and all graph, node, and edge attributes.
Examples
>>>
>>>
>>>
>>>
[]
>>>
[]
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.clear()
G.nodes()
G.edges()
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Iterating over nodes and edges
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MultiGraph.nodes([data])
MultiGraph.nodes_iter([data])
MultiGraph.__iter__()
MultiGraph.edges([nbunch, data, keys])
MultiGraph.edges_iter([nbunch, data, keys])
MultiGraph.get_edge_data(u, v[, key, default])
MultiGraph.neighbors(n)
MultiGraph.neighbors_iter(n)
MultiGraph.__getitem__(n)
MultiGraph.adjacency_list()
MultiGraph.adjacency_iter()
MultiGraph.nbunch_iter([nbunch])
Return a list of the nodes in the graph.
Return an iterator over the nodes.
Iterate over the nodes.
Return a list of edges.
Return an iterator over the edges.
Return the attribute dictionary associated with edge (u,v).
Return a list of the nodes connected to the node n.
Return an iterator over all neighbors of node n.
Return a dict of neighbors of node n.
Return an adjacency list representation of the graph.
Return an iterator of (node, adjacency dict) tuples for all nodes.
Return an iterator of nodes contained in nbunch that are also in the graph.
nodes
MultiGraph.nodes(data=False)
Return a list of the nodes in the graph.
Parameters data : boolean, optional (default=False)
If False return a list of nodes. If True return a two-tuple of node and node data dictionary
Returns nlist : list
A list of nodes. If data=True a list of two-tuples containing (node, node data dictionary).
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> G.nodes()
[0, 1, 2]
>>> G.add_node(1, time='5pm')
>>> G.nodes(data=True)
[(0, {}), (1, {'time': '5pm'}), (2, {})]
nodes_iter
MultiGraph.nodes_iter(data=False)
Return an iterator over the nodes.
Parameters data : boolean, optional (default=False)
If False the iterator returns nodes. If True return a two-tuple of node and node data
dictionary
Returns niter : iterator
An iterator over nodes. If data=True the iterator gives two-tuples containing (node, node
data, dictionary)
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Notes
If the node data is not required it is simpler and equivalent to use the expression �for n in G’.
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> [d for n,d in G.nodes_iter(data=True)]
[{}, {}, {}]
__iter__
MultiGraph.__iter__()
Iterate over the nodes. Use the expression �for n in G’.
Returns niter : iterator
An iterator over all nodes in the graph.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
edges
MultiGraph.edges(nbunch=None, data=False, keys=False)
Return a list of edges.
Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
Return two tuples (u,v) (False) or three-tuples (u,v,data) (True).
keys : bool, optional (default=False)
Return two tuples (u,v) (False) or three-tuples (u,v,key) (True).
Returns edge_list: list of edge tuples
Edges that are adjacent to any node in nbunch, or a list of all edges if nbunch is not
specified.
See also:
edges_iter return an iterator over the edges
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Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.
Examples
>>> G = nx.MultiGraph() # or MultiDiGraph
>>> G.add_path([0,1,2,3])
>>> G.edges()
[(0, 1), (1, 2), (2, 3)]
>>> G.edges(data=True) # default edge data is {} (empty dictionary)
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
>>> G.edges(keys=True) # default keys are integers
[(0, 1, 0), (1, 2, 0), (2, 3, 0)]
>>> G.edges(data=True,keys=True) # default keys are integers
[(0, 1, 0, {}), (1, 2, 0, {}), (2, 3, 0, {})]
>>> G.edges([0,3])
[(0, 1), (3, 2)]
>>> G.edges(0)
[(0, 1)]
edges_iter
MultiGraph.edges_iter(nbunch=None, data=False, keys=False)
Return an iterator over the edges.
Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
If True, return edge attribute dict with each edge.
keys : bool, optional (default=False)
If True, return edge keys with each edge.
Returns edge_iter : iterator
An iterator of (u,v), (u,v,d) or (u,v,key,d) tuples of edges.
See also:
edges return a list of edges
Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.
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Examples
>>> G = nx.MultiGraph()
# or MultiDiGraph
>>> G.add_path([0,1,2,3])
>>> [e for e in G.edges_iter()]
[(0, 1), (1, 2), (2, 3)]
>>> list(G.edges_iter(data=True)) # default data is {} (empty dict)
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
>>> list(G.edges(keys=True)) # default keys are integers
[(0, 1, 0), (1, 2, 0), (2, 3, 0)]
>>> list(G.edges(data=True,keys=True)) # default keys are integers
[(0, 1, 0, {}), (1, 2, 0, {}), (2, 3, 0, {})]
>>> list(G.edges_iter([0,3]))
[(0, 1), (3, 2)]
>>> list(G.edges_iter(0))
[(0, 1)]
get_edge_data
MultiGraph.get_edge_data(u, v, key=None, default=None)
Return the attribute dictionary associated with edge (u,v).
Parameters u,v : nodes
default: any Python object (default=None)
Value to return if the edge (u,v) is not found.
key : hashable identifier, optional (default=None)
Return data only for the edge with specified key.
Returns edge_dict : dictionary
The edge attribute dictionary.
Notes
It is faster to use G[u][v][key].
>>> G = nx.MultiGraph() # or MultiDiGraph
>>> G.add_edge(0,1,key='a',weight=7)
>>> G[0][1]['a'] # key='a'
{'weight': 7}
Warning: Assigning G[u][v][key] corrupts the graph data structure. But it is safe to assign attributes to that
dictionary,
>>> G[0][1]['a']['weight'] = 10
>>> G[0][1]['a']['weight']
10
>>> G[1][0]['a']['weight']
10
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Examples
>>>
>>>
>>>
{0:
>>>
>>>
{0:
>>>
0
G = nx.MultiGraph() # or MultiDiGraph
G.add_path([0,1,2,3])
G.get_edge_data(0,1)
{}}
e = (0,1)
G.get_edge_data(*e) # tuple form
{}}
G.get_edge_data('a','b',default=0) # edge not in graph, return 0
neighbors
MultiGraph.neighbors(n)
Return a list of the nodes connected to the node n.
Parameters n : node
A node in the graph
Returns nlist : list
A list of nodes that are adjacent to n.
Raises NetworkXError
If the node n is not in the graph.
Notes
It is usually more convenient (and faster) to access the adjacency dictionary as G[n]:
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edge('a','b',weight=7)
>>> G['a']
{'b': {'weight': 7}}
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.neighbors(0)
[1]
neighbors_iter
MultiGraph.neighbors_iter(n)
Return an iterator over all neighbors of node n.
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Notes
It is faster to use the idiom “in G[0]”, e.g.
>>> G = nx.path_graph(4)
>>> [n for n in G[0]]
[1]
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> [n for n in G.neighbors_iter(0)]
[1]
__getitem__
MultiGraph.__getitem__(n)
Return a dict of neighbors of node n. Use the expression �G[n]’.
Parameters n : node
A node in the graph.
Returns adj_dict : dictionary
The adjacency dictionary for nodes connected to n.
Notes
G[n] is similar to G.neighbors(n) but the internal data dictionary is returned instead of a list.
Assigning G[n] will corrupt the internal graph data structure. Use G[n] for reading data only.
Examples
>>>
>>>
>>>
{1:
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G[0]
{}}
adjacency_list
MultiGraph.adjacency_list()
Return an adjacency list representation of the graph.
The output adjacency list is in the order of G.nodes(). For directed graphs, only outgoing adjacencies are
included.
Returns adj_list : lists of lists
The adjacency structure of the graph as a list of lists.
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See also:
adjacency_iter
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.adjacency_list() # in order given by G.nodes()
[[1], [0, 2], [1, 3], [2]]
adjacency_iter
MultiGraph.adjacency_iter()
Return an iterator of (node, adjacency dict) tuples for all nodes.
This is the fastest way to look at every edge. For directed graphs, only outgoing adjacencies are included.
Returns adj_iter : iterator
An iterator of (node, adjacency dictionary) for all nodes in the graph.
See also:
adjacency_list
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> [(n,nbrdict) for n,nbrdict in G.adjacency_iter()]
[(0, {1: {}}), (1, {0: {}, 2: {}}), (2, {1: {}, 3: {}}), (3, {2: {}})]
nbunch_iter
MultiGraph.nbunch_iter(nbunch=None)
Return an iterator of nodes contained in nbunch that are also in the graph.
The nodes in nbunch are checked for membership in the graph and if not are silently ignored.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
Returns niter : iterator
An iterator over nodes in nbunch that are also in the graph. If nbunch is None, iterate
over all nodes in the graph.
Raises NetworkXError
If nbunch is not a node or or sequence of nodes. If a node in nbunch is not hashable.
See also:
Graph.__iter__
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Notes
When nbunch is an iterator, the returned iterator yields values directly from nbunch, becoming exhausted when
nbunch is exhausted.
To test whether nbunch is a single node, one can use “if nbunch in self:”, even after processing with this routine.
If nbunch is not a node or a (possibly empty) sequence/iterator or None, a NetworkXError is raised. Also, if any
object in nbunch is not hashable, a NetworkXError is raised.
Information about graph structure
MultiGraph.has_node(n)
MultiGraph.__contains__(n)
MultiGraph.has_edge(u, v[, key])
MultiGraph.order()
MultiGraph.number_of_nodes()
MultiGraph.__len__()
MultiGraph.degree([nbunch, weight])
MultiGraph.degree_iter([nbunch, weight])
MultiGraph.size([weight])
MultiGraph.number_of_edges([u, v])
MultiGraph.nodes_with_selfloops()
MultiGraph.selfloop_edges([data, keys])
MultiGraph.number_of_selfloops()
Return True if the graph contains the node n.
Return True if n is a node, False otherwise.
Return True if the graph has an edge between nodes u and v.
Return the number of nodes in the graph.
Return the number of nodes in the graph.
Return the number of nodes.
Return the degree of a node or nodes.
Return an iterator for (node, degree).
Return the number of edges.
Return the number of edges between two nodes.
Return a list of nodes with self loops.
Return a list of selfloop edges.
Return the number of selfloop edges.
has_node
MultiGraph.has_node(n)
Return True if the graph contains the node n.
Parameters n : node
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> G.has_node(0)
True
It is more readable and simpler to use
>>> 0 in G
True
__contains__
MultiGraph.__contains__(n)
Return True if n is a node, False otherwise. Use the expression �n in G’.
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Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> 1 in G
True
has_edge
MultiGraph.has_edge(u, v, key=None)
Return True if the graph has an edge between nodes u and v.
Parameters u,v : nodes
Nodes can be, for example, strings or numbers.
key : hashable identifier, optional (default=None)
If specified return True only if the edge with key is found.
Returns edge_ind : bool
True if edge is in the graph, False otherwise.
Examples
Can be called either using two nodes u,v, an edge tuple (u,v), or an edge tuple (u,v,key).
>>> G = nx.MultiGraph()
# or MultiDiGraph
>>> G.add_path([0,1,2,3])
>>> G.has_edge(0,1) # using two nodes
True
>>> e = (0,1)
>>> G.has_edge(*e) # e is a 2-tuple (u,v)
True
>>> G.add_edge(0,1,key='a')
>>> G.has_edge(0,1,key='a') # specify key
True
>>> e=(0,1,'a')
>>> G.has_edge(*e) # e is a 3-tuple (u,v,'a')
True
The following syntax are equivalent:
>>> G.has_edge(0,1)
True
>>> 1 in G[0] # though this gives KeyError if 0 not in G
True
order
MultiGraph.order()
Return the number of nodes in the graph.
Returns nnodes : int
The number of nodes in the graph.
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See also:
number_of_nodes, __len__
number_of_nodes
MultiGraph.number_of_nodes()
Return the number of nodes in the graph.
Returns nnodes : int
The number of nodes in the graph.
See also:
order, __len__
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> len(G)
3
__len__
MultiGraph.__len__()
Return the number of nodes. Use the expression �len(G)’.
Returns nnodes : int
The number of nodes in the graph.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> len(G)
4
degree
MultiGraph.degree(nbunch=None, weight=None)
Return the degree of a node or nodes.
The node degree is the number of edges adjacent to that node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
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Returns nd : dictionary, or number
A dictionary with nodes as keys and degree as values or a number if a single node is
specified.
Examples
>>>
>>>
>>>
1
>>>
{0:
>>>
[1,
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.degree(0)
G.degree([0,1])
1, 1: 2}
list(G.degree([0,1]).values())
2]
degree_iter
MultiGraph.degree_iter(nbunch=None, weight=None)
Return an iterator for (node, degree).
The node degree is the number of edges adjacent to the node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd_iter : an iterator
The iterator returns two-tuples of (node, degree).
See also:
degree
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> list(G.degree_iter(0)) # node 0 with degree 1
[(0, 1)]
>>> list(G.degree_iter([0,1]))
[(0, 1), (1, 2)]
size
MultiGraph.size(weight=None)
Return the number of edges.
Parameters weight : string or None, optional (default=None)
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The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1.
Returns nedges : int
The number of edges or sum of edge weights in the graph.
See also:
number_of_edges
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.size()
3
>>>
>>>
>>>
>>>
2
>>>
6.0
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge('a','b',weight=2)
G.add_edge('b','c',weight=4)
G.size()
G.size(weight='weight')
number_of_edges
MultiGraph.number_of_edges(u=None, v=None)
Return the number of edges between two nodes.
Parameters u,v : nodes, optional (default=all edges)
If u and v are specified, return the number of edges between u and v. Otherwise return
the total number of all edges.
Returns nedges : int
The number of edges in the graph. If nodes u and v are specified return the number of
edges between those nodes.
See also:
size
Examples
>>>
>>>
>>>
3
>>>
1
>>>
>>>
1
90
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.number_of_edges()
G.number_of_edges(0,1)
e = (0,1)
G.number_of_edges(*e)
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nodes_with_selfloops
MultiGraph.nodes_with_selfloops()
Return a list of nodes with self loops.
A node with a self loop has an edge with both ends adjacent to that node.
Returns nodelist : list
A list of nodes with self loops.
See also:
selfloop_edges, number_of_selfloops
Examples
>>>
>>>
>>>
>>>
[1]
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge(1,1)
G.add_edge(1,2)
G.nodes_with_selfloops()
selfloop_edges
MultiGraph.selfloop_edges(data=False, keys=False)
Return a list of selfloop edges.
A selfloop edge has the same node at both ends.
Parameters data : bool, optional (default=False)
Return selfloop edges as two tuples (u,v) (data=False) or three-tuples (u,v,data)
(data=True)
keys : bool, optional (default=False)
If True, return edge keys with each edge.
Returns edgelist : list of edge tuples
A list of all selfloop edges.
See also:
nodes_with_selfloops, number_of_selfloops
Examples
>>> G = nx.MultiGraph()
# or MultiDiGraph
>>> G.add_edge(1,1)
>>> G.add_edge(1,2)
>>> G.selfloop_edges()
[(1, 1)]
>>> G.selfloop_edges(data=True)
[(1, 1, {})]
>>> G.selfloop_edges(keys=True)
[(1, 1, 0)]
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>>> G.selfloop_edges(keys=True, data=True)
[(1, 1, 0, {})]
number_of_selfloops
MultiGraph.number_of_selfloops()
Return the number of selfloop edges.
A selfloop edge has the same node at both ends.
Returns nloops : int
The number of selfloops.
See also:
nodes_with_selfloops, selfloop_edges
Examples
>>>
>>>
>>>
>>>
1
G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge(1,1)
G.add_edge(1,2)
G.number_of_selfloops()
Making copies and subgraphs
MultiGraph.copy()
MultiGraph.to_undirected()
MultiGraph.to_directed()
MultiGraph.subgraph(nbunch)
Return a copy of the graph.
Return an undirected copy of the graph.
Return a directed representation of the graph.
Return the subgraph induced on nodes in nbunch.
copy
MultiGraph.copy()
Return a copy of the graph.
Returns G : Graph
A copy of the graph.
See also:
to_directed return a directed copy of the graph.
Notes
This makes a complete copy of the graph including all of the node or edge attributes.
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Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> H = G.copy()
to_undirected
MultiGraph.to_undirected()
Return an undirected copy of the graph.
Returns G : Graph/MultiGraph
A deepcopy of the graph.
See also:
copy, add_edge, add_edges_from
Notes
This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the
data and references.
This is in contrast to the similar G=DiGraph(D) which returns a shallow copy of the data.
See the Python copy module for
http://docs.python.org/library/copy.html.
more
information
on
shallow
and
deep
copies,
Examples
>>> G = nx.Graph()
# or MultiGraph, etc
>>> G.add_path([0,1])
>>> H = G.to_directed()
>>> H.edges()
[(0, 1), (1, 0)]
>>> G2 = H.to_undirected()
>>> G2.edges()
[(0, 1)]
to_directed
MultiGraph.to_directed()
Return a directed representation of the graph.
Returns G : MultiDiGraph
A directed graph with the same name, same nodes, and with each edge (u,v,data) replaced by two directed edges (u,v,data) and (v,u,data).
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Notes
This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the
data and references.
This is in contrast to the similar D=DiGraph(G) which returns a shallow copy of the data.
See the Python copy module for
http://docs.python.org/library/copy.html.
more
information
on
shallow
and
deep
copies,
Examples
>>> G = nx.Graph()
# or MultiGraph, etc
>>> G.add_path([0,1])
>>> H = G.to_directed()
>>> H.edges()
[(0, 1), (1, 0)]
If already directed, return a (deep) copy
>>> G = nx.DiGraph()
# or MultiDiGraph, etc
>>> G.add_path([0,1])
>>> H = G.to_directed()
>>> H.edges()
[(0, 1)]
subgraph
MultiGraph.subgraph(nbunch)
Return the subgraph induced on nodes in nbunch.
The induced subgraph of the graph contains the nodes in nbunch and the edges between those nodes.
Parameters nbunch : list, iterable
A container of nodes which will be iterated through once.
Returns G : Graph
A subgraph of the graph with the same edge attributes.
Notes
The graph, edge or node attributes just point to the original graph. So changes to the node or edge structure will
not be reflected in the original graph while changes to the attributes will.
To create a subgraph with its own copy of the edge/node attributes use: nx.Graph(G.subgraph(nbunch))
If edge attributes are containers, a deep copy can be obtained using: G.subgraph(nbunch).copy()
For an inplace reduction of a graph to a subgraph you can remove nodes: G.remove_nodes_from([ n in G if n
not in set(nbunch)])
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Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> H = G.subgraph([0,1,2])
>>> H.edges()
[(0, 1), (1, 2)]
3.2.4 MultiDiGraph - Directed graphs with self loops and parallel edges
Overview
MultiDiGraph(data=None, **attr)
A directed graph class that can store multiedges.
Multiedges are multiple edges between two nodes. Each edge can hold optional data or attributes.
A MultiDiGraph holds directed edges. Self loops are allowed.
Nodes can be arbitrary (hashable) Python objects with optional key/value attributes.
Edges are represented as links between nodes with optional key/value attributes.
Parameters data : input graph
Data to initialize graph. If data=None (default) an empty graph is created. The data can
be an edge list, or any NetworkX graph object. If the corresponding optional Python
packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse
matrix, or a PyGraphviz graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to graph as key=value pairs.
See also:
Graph, DiGraph, MultiGraph
Examples
Create an empty graph structure (a “null graph”) with no nodes and no edges.
>>> G = nx.MultiDiGraph()
G can be grown in several ways.
Nodes:
Add one node at a time:
>>> G.add_node(1)
Add the nodes from any container (a list, dict, set or even the lines from a file or the nodes from another graph).
>>>
>>>
>>>
>>>
>>>
G.add_nodes_from([2,3])
G.add_nodes_from(range(100,110))
H=nx.Graph()
H.add_path([0,1,2,3,4,5,6,7,8,9])
G.add_nodes_from(H)
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In addition to strings and integers any hashable Python object (except None) can represent a node, e.g. a
customized node object, or even another Graph.
>>> G.add_node(H)
Edges:
G can also be grown by adding edges.
Add one edge,
>>> G.add_edge(1, 2)
a list of edges,
>>> G.add_edges_from([(1,2),(1,3)])
or a collection of edges,
>>> G.add_edges_from(H.edges())
If some edges connect nodes not yet in the graph, the nodes are added automatically. If an edge already exists,
an additional edge is created and stored using a key to identify the edge. By default the key is the lowest unused
integer.
>>> G.add_edges_from([(4,5,dict(route=282)), (4,5,dict(route=37))])
>>> G[4]
{5: {0: {}, 1: {'route': 282}, 2: {'route': 37}}}
Attributes:
Each graph, node, and edge can hold key/value attribute pairs in an associated attribute dictionary (the keys
must be hashable). By default these are empty, but can be added or changed using add_edge, add_node or direct
manipulation of the attribute dictionaries named graph, node and edge respectively.
>>> G = nx.MultiDiGraph(day="Friday")
>>> G.graph
{'day': 'Friday'}
Add node attributes using add_node(), add_nodes_from() or G.node
>>> G.add_node(1, time='5pm')
>>> G.add_nodes_from([3], time='2pm')
>>> G.node[1]
{'time': '5pm'}
>>> G.node[1]['room'] = 714
>>> del G.node[1]['room'] # remove attribute
>>> G.nodes(data=True)
[(1, {'time': '5pm'}), (3, {'time': '2pm'})]
Warning: adding a node to G.node does not add it to the graph.
Add edge attributes using add_edge(), add_edges_from(), subscript notation, or G.edge.
>>>
>>>
>>>
>>>
>>>
G.add_edge(1, 2, weight=4.7 )
G.add_edges_from([(3,4),(4,5)], color='red')
G.add_edges_from([(1,2,{'color':'blue'}), (2,3,{'weight':8})])
G[1][2][0]['weight'] = 4.7
G.edge[1][2][0]['weight'] = 4
Shortcuts:
Many common graph features allow python syntax to speed reporting.
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>>> 1 in G
# check if node in graph
True
>>> [n for n in G if n<3]
# iterate through nodes
[1, 2]
>>> len(G) # number of nodes in graph
5
>>> G[1] # adjacency dict keyed by neighbor to edge attributes
...
# Note: you should not change this dict manually!
{2: {0: {'weight': 4}, 1: {'color': 'blue'}}}
The fastest way to traverse all edges of a graph is via adjacency_iter(), but the edges() method is often more
convenient.
>>> for n,nbrsdict in G.adjacency_iter():
...
for nbr,keydict in nbrsdict.items():
...
for key,eattr in keydict.items():
...
if 'weight' in eattr:
...
(n,nbr,eattr['weight'])
(1, 2, 4)
(2, 3, 8)
>>> [ (u,v,edata['weight']) for u,v,edata in G.edges(data=True) if 'weight' in edata ]
[(1, 2, 4), (2, 3, 8)]
Reporting:
Simple graph information is obtained using methods. Iterator versions of many reporting methods exist for
efficiency. Methods exist for reporting nodes(), edges(), neighbors() and degree() as well as the number of
nodes and edges.
For details on these and other miscellaneous methods, see below.
Adding and Removing Nodes and Edges
MultiDiGraph.__init__([data])
MultiDiGraph.add_node(n[, attr_dict])
MultiDiGraph.add_nodes_from(nodes, **attr)
MultiDiGraph.remove_node(n)
MultiDiGraph.remove_nodes_from(nbunch)
MultiDiGraph.add_edge(u, v[, key, attr_dict])
MultiDiGraph.add_edges_from(ebunch[, attr_dict])
MultiDiGraph.add_weighted_edges_from(ebunch)
MultiDiGraph.remove_edge(u, v[, key])
MultiDiGraph.remove_edges_from(ebunch)
MultiDiGraph.add_star(nodes, **attr)
MultiDiGraph.add_path(nodes, **attr)
MultiDiGraph.add_cycle(nodes, **attr)
MultiDiGraph.clear()
Initialize a graph with edges, name, graph attributes.
Add a single node n and update node attributes.
Add multiple nodes.
Remove node n.
Remove multiple nodes.
Add an edge between u and v.
Add all the edges in ebunch.
Add all the edges in ebunch as weighted edges with specified weight
Remove an edge between u and v.
Remove all edges specified in ebunch.
Add a star.
Add a path.
Add a cycle.
Remove all nodes and edges from the graph.
__init__
MultiDiGraph.__init__(data=None, **attr)
Initialize a graph with edges, name, graph attributes.
Parameters data : input graph
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Data to initialize graph. If data=None (default) an empty graph is created. The data can
be an edge list, or any NetworkX graph object. If the corresponding optional Python
packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse
matrix, or a PyGraphviz graph.
name : string, optional (default=’�)
An optional name for the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to graph as key=value pairs.
See also:
convert
Examples
>>>
>>>
>>>
>>>
G
G
e
G
=
=
=
=
nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
nx.Graph(name='my graph')
[(1,2),(2,3),(3,4)] # list of edges
nx.Graph(e)
Arbitrary graph attribute pairs (key=value) may be assigned
>>> G=nx.Graph(e, day="Friday")
>>> G.graph
{'day': 'Friday'}
add_node
MultiDiGraph.add_node(n, attr_dict=None, **attr)
Add a single node n and update node attributes.
Parameters n : node
A node can be any hashable Python object except None.
attr_dict : dictionary, optional (default= no attributes)
Dictionary of node attributes. Key/value pairs will update existing data associated with
the node.
attr : keyword arguments, optional
Set or change attributes using key=value.
See also:
add_nodes_from
Notes
A hashable object is one that can be used as a key in a Python dictionary. This includes strings, numbers, tuples
of strings and numbers, etc.
On many platforms hashable items also include mutables such as NetworkX Graphs, though one should be
careful that the hash doesn’t change on mutables.
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Examples
>>>
>>>
>>>
>>>
>>>
>>>
3
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_node(1)
G.add_node('Hello')
K3 = nx.Graph([(0,1),(1,2),(2,0)])
G.add_node(K3)
G.number_of_nodes()
Use keywords set/change node attributes:
>>> G.add_node(1,size=10)
>>> G.add_node(3,weight=0.4,UTM=('13S',382871,3972649))
add_nodes_from
MultiDiGraph.add_nodes_from(nodes, **attr)
Add multiple nodes.
Parameters nodes : iterable container
A container of nodes (list, dict, set, etc.). OR A container of (node, attribute dict) tuples.
Node attributes are updated using the attribute dict.
attr : keyword arguments, optional (default= no attributes)
Update attributes for all nodes in nodes. Node attributes specified in nodes as a tuple
take precedence over attributes specified generally.
See also:
add_node
Examples
>>>
>>>
>>>
>>>
>>>
[0,
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_nodes_from('Hello')
K3 = nx.Graph([(0,1),(1,2),(2,0)])
G.add_nodes_from(K3)
sorted(G.nodes(),key=str)
1, 2, 'H', 'e', 'l', 'o']
Use keywords to update specific node attributes for every node.
>>> G.add_nodes_from([1,2], size=10)
>>> G.add_nodes_from([3,4], weight=0.4)
Use (node, attrdict) tuples to update attributes for specific nodes.
>>>
>>>
11
>>>
>>>
>>>
11
G.add_nodes_from([(1,dict(size=11)), (2,{'color':'blue'})])
G.node[1]['size']
H = nx.Graph()
H.add_nodes_from(G.nodes(data=True))
H.node[1]['size']
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remove_node
MultiDiGraph.remove_node(n)
Remove node n.
Removes the node n and all adjacent edges. Attempting to remove a non-existent node will raise an exception.
Parameters n : node
A node in the graph
Raises NetworkXError
If n is not in the graph.
See also:
remove_nodes_from
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> G.edges()
[(0, 1), (1, 2)]
>>> G.remove_node(1)
>>> G.edges()
[]
remove_nodes_from
MultiDiGraph.remove_nodes_from(nbunch)
Remove multiple nodes.
Parameters nodes : iterable container
A container of nodes (list, dict, set, etc.). If a node in the container is not in the graph it
is silently ignored.
See also:
remove_node
Examples
>>>
>>>
>>>
>>>
[0,
>>>
>>>
[]
100
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2])
e = G.nodes()
e
1, 2]
G.remove_nodes_from(e)
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add_edge
MultiDiGraph.add_edge(u, v, key=None, attr_dict=None, **attr)
Add an edge between u and v.
The nodes u and v will be automatically added if they are not already in the graph.
Edge attributes can be specified with keywords or by providing a dictionary with key/value pairs. See examples
below.
Parameters u,v : nodes
Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None)
Python objects.
key : hashable identifier, optional (default=lowest unused integer)
Used to distinguish multiedges between a pair of nodes.
attr_dict : dictionary, optional (default= no attributes)
Dictionary of edge attributes. Key/value pairs will update existing data associated with
the edge.
attr : keyword arguments, optional
Edge data (or labels or objects) can be assigned using keyword arguments.
See also:
add_edges_from add a collection of edges
Notes
To replace/update edge data, use the optional key argument to identify a unique edge. Otherwise a new edge
will be created.
NetworkX algorithms designed for weighted graphs cannot use multigraphs directly because it is not clear
how to handle multiedge weights. Convert to Graph using edge attribute �weight’ to enable weighted graph
algorithms.
Examples
The following all add the edge e=(1,2) to graph G:
>>>
>>>
>>>
>>>
>>>
G = nx.MultiDiGraph()
e = (1,2)
G.add_edge(1, 2)
# explicit two-node form
G.add_edge(*e)
# single edge as tuple of two nodes
G.add_edges_from( [(1,2)] ) # add edges from iterable container
Associate data to edges using keywords:
>>> G.add_edge(1, 2, weight=3)
>>> G.add_edge(1, 2, key=0, weight=4)
# update data for key=0
>>> G.add_edge(1, 3, weight=7, capacity=15, length=342.7)
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add_edges_from
MultiDiGraph.add_edges_from(ebunch, attr_dict=None, **attr)
Add all the edges in ebunch.
Parameters ebunch : container of edges
Each edge given in the container will be added to the graph. The edges can be:
• 2-tuples (u,v) or
• 3-tuples (u,v,d) for an edge attribute dict d, or
• 4-tuples (u,v,k,d) for an edge identified by key k
attr_dict : dictionary, optional (default= no attributes)
Dictionary of edge attributes. Key/value pairs will update existing data associated with
each edge.
attr : keyword arguments, optional
Edge data (or labels or objects) can be assigned using keyword arguments.
See also:
add_edge add a single edge
add_weighted_edges_from convenient way to add weighted edges
Notes
Adding the same edge twice has no effect but any edge data will be updated when each duplicate edge is added.
Edge attributes specified in edges as a tuple take precedence over attributes specified generally.
Examples
>>>
>>>
>>>
>>>
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edges_from([(0,1),(1,2)]) # using a list of edge tuples
e = zip(range(0,3),range(1,4))
G.add_edges_from(e) # Add the path graph 0-1-2-3
Associate data to edges
>>> G.add_edges_from([(1,2),(2,3)], weight=3)
>>> G.add_edges_from([(3,4),(1,4)], label='WN2898')
add_weighted_edges_from
MultiDiGraph.add_weighted_edges_from(ebunch, weight=’weight’, **attr)
Add all the edges in ebunch as weighted edges with specified weights.
Parameters ebunch : container of edges
Each edge given in the list or container will be added to the graph. The edges must be
given as 3-tuples (u,v,w) where w is a number.
weight : string, optional (default= �weight’)
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The attribute name for the edge weights to be added.
attr : keyword arguments, optional (default= no attributes)
Edge attributes to add/update for all edges.
See also:
add_edge add a single edge
add_edges_from add multiple edges
Notes
Adding the same edge twice for Graph/DiGraph simply updates the edge data. For MultiGraph/MultiDiGraph,
duplicate edges are stored.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_weighted_edges_from([(0,1,3.0),(1,2,7.5)])
remove_edge
MultiDiGraph.remove_edge(u, v, key=None)
Remove an edge between u and v.
Parameters u,v: nodes
Remove an edge between nodes u and v.
key : hashable identifier, optional (default=None)
Used to distinguish multiple edges between a pair of nodes. If None remove a single
(abritrary) edge between u and v.
Raises NetworkXError
If there is not an edge between u and v, or if there is no edge with the specified key.
See also:
remove_edges_from remove a collection of edges
Examples
>>>
>>>
>>>
>>>
>>>
G = nx.MultiDiGraph()
G.add_path([0,1,2,3])
G.remove_edge(0,1)
e = (1,2)
G.remove_edge(*e) # unpacks e from an edge tuple
For multiple edges
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>>> G = nx.MultiDiGraph()
>>> G.add_edges_from([(1,2),(1,2),(1,2)])
>>> G.remove_edge(1,2) # remove a single (arbitrary) edge
For edges with keys
>>>
>>>
>>>
>>>
G = nx.MultiDiGraph()
G.add_edge(1,2,key='first')
G.add_edge(1,2,key='second')
G.remove_edge(1,2,key='second')
remove_edges_from
MultiDiGraph.remove_edges_from(ebunch)
Remove all edges specified in ebunch.
Parameters ebunch: list or container of edge tuples
Each edge given in the list or container will be removed from the graph. The edges can
be:
• 2-tuples (u,v) All edges between u and v are removed.
• 3-tuples (u,v,key) The edge identified by key is removed.
• 4-tuples (u,v,key,data) where data is ignored.
See also:
remove_edge remove a single edge
Notes
Will fail silently if an edge in ebunch is not in the graph.
Examples
>>>
>>>
>>>
>>>
G = nx.MultiGraph() # or MultiDiGraph
G.add_path([0,1,2,3])
ebunch=[(1,2),(2,3)]
G.remove_edges_from(ebunch)
Removing multiple copies of edges
>>> G = nx.MultiGraph()
>>> G.add_edges_from([(1,2),(1,2),(1,2)])
>>> G.remove_edges_from([(1,2),(1,2)])
>>> G.edges()
[(1, 2)]
>>> G.remove_edges_from([(1,2),(1,2)]) # silently ignore extra copy
>>> G.edges() # now empty graph
[]
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add_star
MultiDiGraph.add_star(nodes, **attr)
Add a star.
The first node in nodes is the middle of the star. It is connected to all other nodes.
Parameters nodes : iterable container
A container of nodes.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in star.
See also:
add_path, add_cycle
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_star([0,1,2,3])
>>> G.add_star([10,11,12],weight=2)
add_path
MultiDiGraph.add_path(nodes, **attr)
Add a path.
Parameters nodes : iterable container
A container of nodes. A path will be constructed from the nodes (in order) and added
to the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in path.
See also:
add_star, add_cycle
Examples
>>> G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.add_path([10,11,12],weight=7)
add_cycle
MultiDiGraph.add_cycle(nodes, **attr)
Add a cycle.
Parameters nodes: iterable container
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A container of nodes. A cycle will be constructed from the nodes (in order) and added
to the graph.
attr : keyword arguments, optional (default= no attributes)
Attributes to add to every edge in cycle.
See also:
add_path, add_star
Examples
>>> G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_cycle([0,1,2,3])
>>> G.add_cycle([10,11,12],weight=7)
clear
MultiDiGraph.clear()
Remove all nodes and edges from the graph.
This also removes the name, and all graph, node, and edge attributes.
Examples
>>>
>>>
>>>
>>>
[]
>>>
[]
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.clear()
G.nodes()
G.edges()
Iterating over nodes and edges
MultiDiGraph.nodes([data])
MultiDiGraph.nodes_iter([data])
MultiDiGraph.__iter__()
MultiDiGraph.edges([nbunch, data, keys])
MultiDiGraph.edges_iter([nbunch, data, keys])
MultiDiGraph.out_edges([nbunch, keys, data])
MultiDiGraph.out_edges_iter([nbunch, data, keys])
MultiDiGraph.in_edges([nbunch, keys, data])
MultiDiGraph.in_edges_iter([nbunch, data, keys])
MultiDiGraph.get_edge_data(u, v[, key, default])
MultiDiGraph.neighbors(n)
MultiDiGraph.neighbors_iter(n)
MultiDiGraph.__getitem__(n)
MultiDiGraph.successors(n)
MultiDiGraph.successors_iter(n)
106
Return a list of the nodes in the graph.
Return an iterator over the nodes.
Iterate over the nodes.
Return a list of edges.
Return an iterator over the edges.
Return a list of the outgoing edges.
Return an iterator over the edges.
Return a list of the incoming edges.
Return an iterator over the incoming edges.
Return the attribute dictionary associated with edge (u,v).
Return a list of successor nodes of n.
Return an iterator over successor nodes of n.
Return a dict of neighbors of node n.
Return a list of successor nodes of n.
Return an iterator over successor nodes of n.
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Table 3.14 – continued from previous page
MultiDiGraph.predecessors(n)
Return a list of predecessor nodes of n.
MultiDiGraph.predecessors_iter(n)
Return an iterator over predecessor nodes of n.
MultiDiGraph.adjacency_list()
Return an adjacency list representation of the graph.
MultiDiGraph.adjacency_iter()
Return an iterator of (node, adjacency dict) tuples for all nodes.
MultiDiGraph.nbunch_iter([nbunch])
Return an iterator of nodes contained in nbunch that are also in the gr
nodes
MultiDiGraph.nodes(data=False)
Return a list of the nodes in the graph.
Parameters data : boolean, optional (default=False)
If False return a list of nodes. If True return a two-tuple of node and node data dictionary
Returns nlist : list
A list of nodes. If data=True a list of two-tuples containing (node, node data dictionary).
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> G.nodes()
[0, 1, 2]
>>> G.add_node(1, time='5pm')
>>> G.nodes(data=True)
[(0, {}), (1, {'time': '5pm'}), (2, {})]
nodes_iter
MultiDiGraph.nodes_iter(data=False)
Return an iterator over the nodes.
Parameters data : boolean, optional (default=False)
If False the iterator returns nodes. If True return a two-tuple of node and node data
dictionary
Returns niter : iterator
An iterator over nodes. If data=True the iterator gives two-tuples containing (node, node
data, dictionary)
Notes
If the node data is not required it is simpler and equivalent to use the expression �for n in G’.
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
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Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> [d for n,d in G.nodes_iter(data=True)]
[{}, {}, {}]
__iter__
MultiDiGraph.__iter__()
Iterate over the nodes. Use the expression �for n in G’.
Returns niter : iterator
An iterator over all nodes in the graph.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
edges
MultiDiGraph.edges(nbunch=None, data=False, keys=False)
Return a list of edges.
Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
Return two tuples (u,v) (False) or three-tuples (u,v,data) (True).
keys : bool, optional (default=False)
Return two tuples (u,v) (False) or three-tuples (u,v,key) (True).
Returns edge_list: list of edge tuples
Edges that are adjacent to any node in nbunch, or a list of all edges if nbunch is not
specified.
See also:
edges_iter return an iterator over the edges
Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.
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Examples
>>> G = nx.MultiGraph() # or MultiDiGraph
>>> G.add_path([0,1,2,3])
>>> G.edges()
[(0, 1), (1, 2), (2, 3)]
>>> G.edges(data=True) # default edge data is {} (empty dictionary)
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
>>> G.edges(keys=True) # default keys are integers
[(0, 1, 0), (1, 2, 0), (2, 3, 0)]
>>> G.edges(data=True,keys=True) # default keys are integers
[(0, 1, 0, {}), (1, 2, 0, {}), (2, 3, 0, {})]
>>> G.edges([0,3])
[(0, 1), (3, 2)]
>>> G.edges(0)
[(0, 1)]
edges_iter
MultiDiGraph.edges_iter(nbunch=None, data=False, keys=False)
Return an iterator over the edges.
Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
If True, return edge attribute dict with each edge.
keys : bool, optional (default=False)
If True, return edge keys with each edge.
Returns edge_iter : iterator
An iterator of (u,v), (u,v,d) or (u,v,key,d) tuples of edges.
See also:
edges return a list of edges
Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.
Examples
>>> G = nx.MultiDiGraph()
>>> G.add_path([0,1,2,3])
>>> [e for e in G.edges_iter()]
[(0, 1), (1, 2), (2, 3)]
>>> list(G.edges_iter(data=True)) # default data is {} (empty dict)
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
>>> list(G.edges_iter([0,2]))
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[(0, 1), (2, 3)]
>>> list(G.edges_iter(0))
[(0, 1)]
out_edges
MultiDiGraph.out_edges(nbunch=None, keys=False, data=False)
Return a list of the outgoing edges.
Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
If True, return edge attribute dict with each edge.
keys : bool, optional (default=False)
If True, return edge keys with each edge.
Returns out_edges : list
An listr of (u,v), (u,v,d) or (u,v,key,d) tuples of edges.
See also:
in_edges return a list of incoming edges
Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs edges() is the same as
out_edges().
out_edges_iter
MultiDiGraph.out_edges_iter(nbunch=None, data=False, keys=False)
Return an iterator over the edges.
Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data).
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
If True, return edge attribute dict with each edge.
keys : bool, optional (default=False)
If True, return edge keys with each edge.
Returns edge_iter : iterator
An iterator of (u,v), (u,v,d) or (u,v,key,d) tuples of edges.
See also:
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edges return a list of edges
Notes
Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.
Examples
>>> G = nx.MultiDiGraph()
>>> G.add_path([0,1,2,3])
>>> [e for e in G.edges_iter()]
[(0, 1), (1, 2), (2, 3)]
>>> list(G.edges_iter(data=True)) # default data is {} (empty dict)
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
>>> list(G.edges_iter([0,2]))
[(0, 1), (2, 3)]
>>> list(G.edges_iter(0))
[(0, 1)]
in_edges
MultiDiGraph.in_edges(nbunch=None, keys=False, data=False)
Return a list of the incoming edges.
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
If True, return edge attribute dict with each edge.
keys : bool, optional (default=False)
If True, return edge keys with each edge.
Returns in_edges : list
A list of (u,v), (u,v,d) or (u,v,key,d) tuples of edges.
See also:
out_edges return a list of outgoing edges
in_edges_iter
MultiDiGraph.in_edges_iter(nbunch=None, data=False, keys=False)
Return an iterator over the incoming edges.
Parameters nbunch : iterable container, optional (default= all nodes)
A container of nodes. The container will be iterated through once.
data : bool, optional (default=False)
If True, return edge attribute dict with each edge.
keys : bool, optional (default=False)
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If True, return edge keys with each edge.
Returns in_edge_iter : iterator
An iterator of (u,v), (u,v,d) or (u,v,key,d) tuples of edges.
See also:
edges_iter return an iterator of edges
get_edge_data
MultiDiGraph.get_edge_data(u, v, key=None, default=None)
Return the attribute dictionary associated with edge (u,v).
Parameters u,v : nodes
default: any Python object (default=None)
Value to return if the edge (u,v) is not found.
key : hashable identifier, optional (default=None)
Return data only for the edge with specified key.
Returns edge_dict : dictionary
The edge attribute dictionary.
Notes
It is faster to use G[u][v][key].
>>> G = nx.MultiGraph() # or MultiDiGraph
>>> G.add_edge(0,1,key='a',weight=7)
>>> G[0][1]['a'] # key='a'
{'weight': 7}
Warning: Assigning G[u][v][key] corrupts the graph data structure. But it is safe to assign attributes to that
dictionary,
>>> G[0][1]['a']['weight'] = 10
>>> G[0][1]['a']['weight']
10
>>> G[1][0]['a']['weight']
10
Examples
>>>
>>>
>>>
{0:
>>>
>>>
{0:
>>>
0
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G = nx.MultiGraph() # or MultiDiGraph
G.add_path([0,1,2,3])
G.get_edge_data(0,1)
{}}
e = (0,1)
G.get_edge_data(*e) # tuple form
{}}
G.get_edge_data('a','b',default=0) # edge not in graph, return 0
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neighbors
MultiDiGraph.neighbors(n)
Return a list of successor nodes of n.
neighbors() and successors() are the same function.
neighbors_iter
MultiDiGraph.neighbors_iter(n)
Return an iterator over successor nodes of n.
neighbors_iter() and successors_iter() are the same.
__getitem__
MultiDiGraph.__getitem__(n)
Return a dict of neighbors of node n. Use the expression �G[n]’.
Parameters n : node
A node in the graph.
Returns adj_dict : dictionary
The adjacency dictionary for nodes connected to n.
Notes
G[n] is similar to G.neighbors(n) but the internal data dictionary is returned instead of a list.
Assigning G[n] will corrupt the internal graph data structure. Use G[n] for reading data only.
Examples
>>>
>>>
>>>
{1:
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G[0]
{}}
successors
MultiDiGraph.successors(n)
Return a list of successor nodes of n.
neighbors() and successors() are the same function.
successors_iter
MultiDiGraph.successors_iter(n)
Return an iterator over successor nodes of n.
neighbors_iter() and successors_iter() are the same.
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predecessors
MultiDiGraph.predecessors(n)
Return a list of predecessor nodes of n.
predecessors_iter
MultiDiGraph.predecessors_iter(n)
Return an iterator over predecessor nodes of n.
adjacency_list
MultiDiGraph.adjacency_list()
Return an adjacency list representation of the graph.
The output adjacency list is in the order of G.nodes(). For directed graphs, only outgoing adjacencies are
included.
Returns adj_list : lists of lists
The adjacency structure of the graph as a list of lists.
See also:
adjacency_iter
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.adjacency_list() # in order given by G.nodes()
[[1], [0, 2], [1, 3], [2]]
adjacency_iter
MultiDiGraph.adjacency_iter()
Return an iterator of (node, adjacency dict) tuples for all nodes.
This is the fastest way to look at every edge. For directed graphs, only outgoing adjacencies are included.
Returns adj_iter : iterator
An iterator of (node, adjacency dictionary) for all nodes in the graph.
See also:
adjacency_list
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> [(n,nbrdict) for n,nbrdict in G.adjacency_iter()]
[(0, {1: {}}), (1, {0: {}, 2: {}}), (2, {1: {}, 3: {}}), (3, {2: {}})]
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nbunch_iter
MultiDiGraph.nbunch_iter(nbunch=None)
Return an iterator of nodes contained in nbunch that are also in the graph.
The nodes in nbunch are checked for membership in the graph and if not are silently ignored.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
Returns niter : iterator
An iterator over nodes in nbunch that are also in the graph. If nbunch is None, iterate
over all nodes in the graph.
Raises NetworkXError
If nbunch is not a node or or sequence of nodes. If a node in nbunch is not hashable.
See also:
Graph.__iter__
Notes
When nbunch is an iterator, the returned iterator yields values directly from nbunch, becoming exhausted when
nbunch is exhausted.
To test whether nbunch is a single node, one can use “if nbunch in self:”, even after processing with this routine.
If nbunch is not a node or a (possibly empty) sequence/iterator or None, a NetworkXError is raised. Also, if any
object in nbunch is not hashable, a NetworkXError is raised.
Information about graph structure
MultiDiGraph.has_node(n)
MultiDiGraph.__contains__(n)
MultiDiGraph.has_edge(u, v[, key])
MultiDiGraph.order()
MultiDiGraph.number_of_nodes()
MultiDiGraph.__len__()
MultiDiGraph.degree([nbunch, weight])
MultiDiGraph.degree_iter([nbunch, weight])
MultiDiGraph.in_degree([nbunch, weight])
MultiDiGraph.in_degree_iter([nbunch, weight])
MultiDiGraph.out_degree([nbunch, weight])
MultiDiGraph.out_degree_iter([nbunch, weight])
MultiDiGraph.size([weight])
MultiDiGraph.number_of_edges([u, v])
MultiDiGraph.nodes_with_selfloops()
MultiDiGraph.selfloop_edges([data, keys])
MultiDiGraph.number_of_selfloops()
3.2. Basic graph types
Return True if the graph contains the node n.
Return True if n is a node, False otherwise.
Return True if the graph has an edge between nodes u and v.
Return the number of nodes in the graph.
Return the number of nodes in the graph.
Return the number of nodes.
Return the degree of a node or nodes.
Return an iterator for (node, degree).
Return the in-degree of a node or nodes.
Return an iterator for (node, in-degree).
Return the out-degree of a node or nodes.
Return an iterator for (node, out-degree).
Return the number of edges.
Return the number of edges between two nodes.
Return a list of nodes with self loops.
Return a list of selfloop edges.
Return the number of selfloop edges.
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has_node
MultiDiGraph.has_node(n)
Return True if the graph contains the node n.
Parameters n : node
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> G.has_node(0)
True
It is more readable and simpler to use
>>> 0 in G
True
__contains__
MultiDiGraph.__contains__(n)
Return True if n is a node, False otherwise. Use the expression �n in G’.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> 1 in G
True
has_edge
MultiDiGraph.has_edge(u, v, key=None)
Return True if the graph has an edge between nodes u and v.
Parameters u,v : nodes
Nodes can be, for example, strings or numbers.
key : hashable identifier, optional (default=None)
If specified return True only if the edge with key is found.
Returns edge_ind : bool
True if edge is in the graph, False otherwise.
Examples
Can be called either using two nodes u,v, an edge tuple (u,v), or an edge tuple (u,v,key).
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>>> G = nx.MultiGraph()
# or MultiDiGraph
>>> G.add_path([0,1,2,3])
>>> G.has_edge(0,1) # using two nodes
True
>>> e = (0,1)
>>> G.has_edge(*e) # e is a 2-tuple (u,v)
True
>>> G.add_edge(0,1,key='a')
>>> G.has_edge(0,1,key='a') # specify key
True
>>> e=(0,1,'a')
>>> G.has_edge(*e) # e is a 3-tuple (u,v,'a')
True
The following syntax are equivalent:
>>> G.has_edge(0,1)
True
>>> 1 in G[0] # though this gives KeyError if 0 not in G
True
order
MultiDiGraph.order()
Return the number of nodes in the graph.
Returns nnodes : int
The number of nodes in the graph.
See also:
number_of_nodes, __len__
number_of_nodes
MultiDiGraph.number_of_nodes()
Return the number of nodes in the graph.
Returns nnodes : int
The number of nodes in the graph.
See also:
order, __len__
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2])
>>> len(G)
3
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__len__
MultiDiGraph.__len__()
Return the number of nodes. Use the expression �len(G)’.
Returns nnodes : int
The number of nodes in the graph.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> len(G)
4
degree
MultiDiGraph.degree(nbunch=None, weight=None)
Return the degree of a node or nodes.
The node degree is the number of edges adjacent to that node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd : dictionary, or number
A dictionary with nodes as keys and degree as values or a number if a single node is
specified.
Examples
>>>
>>>
>>>
1
>>>
{0:
>>>
[1,
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.degree(0)
G.degree([0,1])
1, 1: 2}
list(G.degree([0,1]).values())
2]
degree_iter
MultiDiGraph.degree_iter(nbunch=None, weight=None)
Return an iterator for (node, degree).
The node degree is the number of edges adjacent to the node.
Parameters nbunch : iterable container, optional (default=all nodes)
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A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights.
Returns nd_iter : an iterator
The iterator returns two-tuples of (node, degree).
See also:
degree
Examples
>>> G = nx.MultiDiGraph()
>>> G.add_path([0,1,2,3])
>>> list(G.degree_iter(0)) # node 0 with degree 1
[(0, 1)]
>>> list(G.degree_iter([0,1]))
[(0, 1), (1, 2)]
in_degree
MultiDiGraph.in_degree(nbunch=None, weight=None)
Return the in-degree of a node or nodes.
The node in-degree is the number of edges pointing in to the node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd : dictionary, or number
A dictionary with nodes as keys and in-degree as values or a number if a single node is
specified.
See also:
degree, out_degree, in_degree_iter
Examples
>>>
>>>
>>>
0
>>>
{0:
>>>
[0,
G = nx.DiGraph()
# or MultiDiGraph
G.add_path([0,1,2,3])
G.in_degree(0)
G.in_degree([0,1])
0, 1: 1}
list(G.in_degree([0,1]).values())
1]
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in_degree_iter
MultiDiGraph.in_degree_iter(nbunch=None, weight=None)
Return an iterator for (node, in-degree).
The node in-degree is the number of edges pointing in to the node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd_iter : an iterator
The iterator returns two-tuples of (node, in-degree).
See also:
degree, in_degree, out_degree, out_degree_iter
Examples
>>> G = nx.MultiDiGraph()
>>> G.add_path([0,1,2,3])
>>> list(G.in_degree_iter(0)) # node 0 with degree 0
[(0, 0)]
>>> list(G.in_degree_iter([0,1]))
[(0, 0), (1, 1)]
out_degree
MultiDiGraph.out_degree(nbunch=None, weight=None)
Return the out-degree of a node or nodes.
The node out-degree is the number of edges pointing out of the node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns nd : dictionary, or number
A dictionary with nodes as keys and out-degree as values or a number if a single node
is specified.
Examples
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>>>
>>>
>>>
1
>>>
{0:
>>>
[1,
G = nx.DiGraph()
# or MultiDiGraph
G.add_path([0,1,2,3])
G.out_degree(0)
G.out_degree([0,1])
1, 1: 1}
list(G.out_degree([0,1]).values())
1]
out_degree_iter
MultiDiGraph.out_degree_iter(nbunch=None, weight=None)
Return an iterator for (node, out-degree).
The node out-degree is the number of edges pointing out of the node.
Parameters nbunch : iterable container, optional (default=all nodes)
A container of nodes. The container will be iterated through once.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights.
Returns nd_iter : an iterator
The iterator returns two-tuples of (node, out-degree).
See also:
degree, in_degree, out_degree, in_degree_iter
Examples
>>> G = nx.MultiDiGraph()
>>> G.add_path([0,1,2,3])
>>> list(G.out_degree_iter(0)) # node 0 with degree 1
[(0, 1)]
>>> list(G.out_degree_iter([0,1]))
[(0, 1), (1, 1)]
size
MultiDiGraph.size(weight=None)
Return the number of edges.
Parameters weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1.
Returns nedges : int
The number of edges or sum of edge weights in the graph.
See also:
number_of_edges
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Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> G.size()
3
>>>
>>>
>>>
>>>
2
>>>
6.0
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge('a','b',weight=2)
G.add_edge('b','c',weight=4)
G.size()
G.size(weight='weight')
number_of_edges
MultiDiGraph.number_of_edges(u=None, v=None)
Return the number of edges between two nodes.
Parameters u,v : nodes, optional (default=all edges)
If u and v are specified, return the number of edges between u and v. Otherwise return
the total number of all edges.
Returns nedges : int
The number of edges in the graph. If nodes u and v are specified return the number of
edges between those nodes.
See also:
size
Examples
>>>
>>>
>>>
3
>>>
1
>>>
>>>
1
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_path([0,1,2,3])
G.number_of_edges()
G.number_of_edges(0,1)
e = (0,1)
G.number_of_edges(*e)
nodes_with_selfloops
MultiDiGraph.nodes_with_selfloops()
Return a list of nodes with self loops.
A node with a self loop has an edge with both ends adjacent to that node.
Returns nodelist : list
A list of nodes with self loops.
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See also:
selfloop_edges, number_of_selfloops
Examples
>>>
>>>
>>>
>>>
[1]
G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge(1,1)
G.add_edge(1,2)
G.nodes_with_selfloops()
selfloop_edges
MultiDiGraph.selfloop_edges(data=False, keys=False)
Return a list of selfloop edges.
A selfloop edge has the same node at both ends.
Parameters data : bool, optional (default=False)
Return selfloop edges as two tuples (u,v) (data=False) or three-tuples (u,v,data)
(data=True)
keys : bool, optional (default=False)
If True, return edge keys with each edge.
Returns edgelist : list of edge tuples
A list of all selfloop edges.
See also:
nodes_with_selfloops, number_of_selfloops
Examples
>>> G = nx.MultiGraph()
# or MultiDiGraph
>>> G.add_edge(1,1)
>>> G.add_edge(1,2)
>>> G.selfloop_edges()
[(1, 1)]
>>> G.selfloop_edges(data=True)
[(1, 1, {})]
>>> G.selfloop_edges(keys=True)
[(1, 1, 0)]
>>> G.selfloop_edges(keys=True, data=True)
[(1, 1, 0, {})]
number_of_selfloops
MultiDiGraph.number_of_selfloops()
Return the number of selfloop edges.
A selfloop edge has the same node at both ends.
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Returns nloops : int
The number of selfloops.
See also:
nodes_with_selfloops, selfloop_edges
Examples
>>>
>>>
>>>
>>>
1
G=nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
G.add_edge(1,1)
G.add_edge(1,2)
G.number_of_selfloops()
Making copies and subgraphs
MultiDiGraph.copy()
MultiDiGraph.to_undirected([reciprocal])
MultiDiGraph.to_directed()
MultiDiGraph.subgraph(nbunch)
MultiDiGraph.reverse([copy])
Return a copy of the graph.
Return an undirected representation of the digraph.
Return a directed copy of the graph.
Return the subgraph induced on nodes in nbunch.
Return the reverse of the graph.
copy
MultiDiGraph.copy()
Return a copy of the graph.
Returns G : Graph
A copy of the graph.
See also:
to_directed return a directed copy of the graph.
Notes
This makes a complete copy of the graph including all of the node or edge attributes.
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> H = G.copy()
to_undirected
MultiDiGraph.to_undirected(reciprocal=False)
Return an undirected representation of the digraph.
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Parameters reciprocal : bool (optional)
If True only keep edges that appear in both directions in the original digraph.
Returns G : MultiGraph
An undirected graph with the same name and nodes and with edge (u,v,data) if either
(u,v,data) or (v,u,data) is in the digraph. If both edges exist in digraph and their edge
data is different, only one edge is created with an arbitrary choice of which edge data to
use. You must check and correct for this manually if desired.
Notes
This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the
data and references.
This is in contrast to the similar D=DiGraph(G) which returns a shallow copy of the data.
See the Python copy module for
http://docs.python.org/library/copy.html.
more
information
on
shallow
and
deep
copies,
to_directed
MultiDiGraph.to_directed()
Return a directed copy of the graph.
Returns G : MultiDiGraph
A deepcopy of the graph.
Notes
If edges in both directions (u,v) and (v,u) exist in the graph, attributes for the new undirected edge will be a
combination of the attributes of the directed edges. The edge data is updated in the (arbitrary) order that the
edges are encountered. For more customized control of the edge attributes use add_edge().
This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the
data and references.
This is in contrast to the similar G=DiGraph(D) which returns a shallow copy of the data.
See the Python copy module for
http://docs.python.org/library/copy.html.
more
information
on
shallow
and
deep
copies,
Examples
>>> G = nx.Graph()
# or MultiGraph, etc
>>> G.add_path([0,1])
>>> H = G.to_directed()
>>> H.edges()
[(0, 1), (1, 0)]
If already directed, return a (deep) copy
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>>> G = nx.MultiDiGraph()
>>> G.add_path([0,1])
>>> H = G.to_directed()
>>> H.edges()
[(0, 1)]
subgraph
MultiDiGraph.subgraph(nbunch)
Return the subgraph induced on nodes in nbunch.
The induced subgraph of the graph contains the nodes in nbunch and the edges between those nodes.
Parameters nbunch : list, iterable
A container of nodes which will be iterated through once.
Returns G : Graph
A subgraph of the graph with the same edge attributes.
Notes
The graph, edge or node attributes just point to the original graph. So changes to the node or edge structure will
not be reflected in the original graph while changes to the attributes will.
To create a subgraph with its own copy of the edge/node attributes use: nx.Graph(G.subgraph(nbunch))
If edge attributes are containers, a deep copy can be obtained using: G.subgraph(nbunch).copy()
For an inplace reduction of a graph to a subgraph you can remove nodes: G.remove_nodes_from([ n in G if n
not in set(nbunch)])
Examples
>>> G = nx.Graph()
# or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_path([0,1,2,3])
>>> H = G.subgraph([0,1,2])
>>> H.edges()
[(0, 1), (1, 2)]
reverse
MultiDiGraph.reverse(copy=True)
Return the reverse of the graph.
The reverse is a graph with the same nodes and edges but with the directions of the edges reversed.
Parameters copy : bool optional (default=True)
If True, return a new DiGraph holding the reversed edges. If False, reverse the reverse
graph is created using the original graph (this changes the original graph).
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FOUR
ALGORITHMS
4.1 Approximation
4.1.1 Clique
Cliques.
max_clique(G)
clique_removal(G)
Find the Maximum Clique
Repeatedly remove cliques from the graph.
max_clique
max_clique(G)
Find the Maximum Clique
Finds the 𝑂(|𝑉 |/(𝑙𝑜𝑔|𝑉 |)2 ) apx of maximum clique/independent set in the worst case.
Parameters G : NetworkX graph
Undirected graph
Returns clique : set
The apx-maximum clique of the graph
Notes
A clique in an undirected graph G = (V, E) is a subset of the vertex set 𝐶 ⊆ 𝑉 , such that for every two vertices
in C, there exists an edge connecting the two. This is equivalent to saying that the subgraph induced by C is
complete (in some cases, the term clique may also refer to the subgraph).
A maximum clique is a clique of the largest possible size in a given graph. The clique number рќњ”(рќђє) of a graph
G is the number of vertices in a maximum clique in G. The intersection number of G is the smallest number of
cliques that together cover all edges of G.
http://en.wikipedia.org/wiki/Maximum_clique
References
[R158]
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clique_removal
clique_removal(G)
Repeatedly remove cliques from the graph.
Results in a 𝑂(|𝑉 |/(log |𝑉 |)2 ) approximation of maximum clique & independent set. Returns the largest independent set found, along with found maximal cliques.
Parameters G : NetworkX graph
Undirected graph
Returns max_ind_cliques : (set, list) tuple
Maximal independent set and list of maximal cliques (sets) in the graph.
References
[R157]
4.1.2 Clustering
average_clustering(G[, trials])
Estimates the average clustering coefficient of G.
average_clustering
average_clustering(G, trials=1000)
Estimates the average clustering coefficient of G.
The local clustering of each node in рќђє is the fraction of triangles that actually exist over all possible triangles in
its neighborhood. The average clustering coefficient of a graph рќђє is the mean of local clusterings.
This function finds an approximate average clustering coefficient for G by repeating рќ‘› times (defined in рќ‘Ўрќ‘џрќ‘–рќ‘Ћрќ‘™рќ‘ )
the following experiment: choose a node at random, choose two of its neighbors at random, and check if they
are connected. The approximate coefficient is the fraction of triangles found over the number of trials [R159].
Parameters G : NetworkX graph
trials : integer
Number of trials to perform (default 1000).
Returns c : float
Approximated average clustering coefficient.
References
[R159]
4.1.3 Dominating Set
A dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is joined to at least one
member of D by some edge. The domination number gamma(G) is the number of vertices in a smallest dominating
set for G. Given a graph G = (V, E) find a minimum weight dominating set V’.
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http://en.wikipedia.org/wiki/Dominating_set
An edge dominating set for a graph G = (V, E) is a subset D of E such that every edge not in D is adjacent to at least
one edge in D.
http://en.wikipedia.org/wiki/Edge_dominating_set
min_weighted_dominating_set(G[, weight])
min_edge_dominating_set(G)
Return minimum weight vertex dominating set.
Return minimum cardinality edge dominating set.
min_weighted_dominating_set
min_weighted_dominating_set(G, weight=None)
Return minimum weight vertex dominating set.
Parameters G : NetworkX graph
Undirected graph
weight : None or string, optional (default = None)
If None, every edge has weight/distance/weight 1. If a string, use this edge attribute as
the edge weight. Any edge attribute not present defaults to 1.
Returns min_weight_dominating_set : set
Returns a set of vertices whose weight sum is no more than log w(V) * OPT
Notes
This algorithm computes an approximate minimum weighted dominating set for the graph G. The upper-bound
on the size of the solution is log w(V) * OPT. Runtime of the algorithm is рќ‘‚(|рќђё|).
References
[R160]
min_edge_dominating_set
min_edge_dominating_set(G)
Return minimum cardinality edge dominating set.
Parameters G : NetworkX graph
Undirected graph
Returns min_edge_dominating_set : set
Returns a set of dominating edges whose size is no more than 2 * OPT.
Notes
The algorithm computes an approximate solution to the edge dominating set problem. The result is no more
than 2 * OPT in terms of size of the set. Runtime of the algorithm is рќ‘‚(|рќђё|).
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4.1.4 Independent Set
Independent Set
Independent set or stable set is a set of vertices in a graph, no two of which are adjacent. That is, it is a set I of vertices
such that for every two vertices in I, there is no edge connecting the two. Equivalently, each edge in the graph has at
most one endpoint in I. The size of an independent set is the number of vertices it contains.
A maximum independent set is a largest independent set for a given graph G and its size is denoted рќ›ј(G). The problem
of finding such a set is called the maximum independent set problem and is an NP-hard optimization problem. As
such, it is unlikely that there exists an efficient algorithm for finding a maximum independent set of a graph.
http://en.wikipedia.org/wiki/Independent_set_(graph_theory)
Independent set algorithm is based on the following paper:
𝑂(|𝑉 |/(𝑙𝑜𝑔|𝑉 |)2 ) apx of maximum clique/independent set.
Boppana, R., & HalldГіrsson, M. M. (1992). Approximating maximum independent sets by excluding subgraphs. BIT
Numerical Mathematics, 32(2), 180–196. Springer. doi:10.1007/BF01994876
maximum_independent_set(G)
Return an approximate maximum independent set.
maximum_independent_set
maximum_independent_set(G)
Return an approximate maximum independent set.
Parameters G : NetworkX graph
Undirected graph
Returns iset : Set
The apx-maximum independent set
Notes
Finds the 𝑂(|𝑉 |/(𝑙𝑜𝑔|𝑉 |)2 ) apx of independent set in the worst case.
References
[R161]
4.1.5 Matching
Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent edges; that is, no two edges share a
common vertex.
http://en.wikipedia.org/wiki/Matching_(graph_theory)
min_maximal_matching(G)
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min_maximal_matching
min_maximal_matching(G)
Returns the minimum maximal matching of G. That is, out of all maximal matchings of the graph G, the smallest
is returned.
Parameters G : NetworkX graph
Undirected graph
Returns min_maximal_matching : set
Returns a set of edges such that no two edges share a common endpoint and every edge
not in the set shares some common endpoint in the set. Cardinality will be 2*OPT in
the worst case.
Notes
The algorithm computes an approximate solution fo the minimum maximal cardinality matching problem. The
solution is no more than 2 * OPT in size. Runtime is рќ‘‚(|рќђё|).
References
[R162]
4.1.6 Ramsey
Ramsey numbers.
ramsey_R2(G)
Approximately computes the Ramsey number рќ‘…(2; рќ‘ , рќ‘Ў) for graph.
ramsey_R2
ramsey_R2(G)
Approximately computes the Ramsey number рќ‘…(2; рќ‘ , рќ‘Ў) for graph.
Parameters G : NetworkX graph
Undirected graph
Returns max_pair : (set, set) tuple
Maximum clique, Maximum independent set.
4.1.7 Vertex Cover
Given an undirected graph 𝐺 = (𝑉, 𝐸) and a function w assigning nonnegative weights to its vertices, find a minimum
weight subset of V such that each edge in E is incident to at least one vertex in the subset.
http://en.wikipedia.org/wiki/Vertex_cover
min_weighted_vertex_cover(G[, weight])
4.1. Approximation
2-OPT Local Ratio for Minimum Weighted Vertex Cover
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min_weighted_vertex_cover
min_weighted_vertex_cover(G, weight=None)
2-OPT Local Ratio for Minimum Weighted Vertex Cover
Find an approximate minimum weighted vertex cover of a graph.
Parameters G : NetworkX graph
Undirected graph
weight : None or string, optional (default = None)
If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the
edge weight. Any edge attribute not present defaults to 1.
Returns min_weighted_cover : set
Returns a set of vertices whose weight sum is no more than 2 * OPT.
Notes
Local-Ratio algorithm for computing an approximate vertex cover. Algorithm greedily reduces the costs over
edges and iteratively builds a cover. Worst-case runtime is рќ‘‚(|рќђё|).
References
[R163]
4.2 Assortativity
4.2.1 Assortativity
degree_assortativity_coefficient(G[, x, y, ...])
attribute_assortativity_coefficient(G, attribute)
numeric_assortativity_coefficient(G, attribute)
degree_pearson_correlation_coefficient(G[, ...])
Compute degree assortativity of graph.
Compute assortativity for node attributes.
Compute assortativity for numerical node attributes.
Compute degree assortativity of graph.
degree_assortativity_coefficient
degree_assortativity_coefficient(G, x=’out’, y=’in’, weight=None, nodes=None)
Compute degree assortativity of graph.
Assortativity measures the similarity of connections in the graph with respect to the node degree.
Parameters G : NetworkX graph
x: string (�in’,’out’)
The degree type for source node (directed graphs only).
y: string (�in’,’out’)
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The degree type for target node (directed graphs only).
weight: string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
nodes: list or iterable (optional)
Compute degree assortativity only for nodes in container. The default is all nodes.
Returns r : float
Assortativity of graph by degree.
See also:
attribute_assortativity_coefficient,
numeric_assortativity_coefficient,
neighbor_connectivity, degree_mixing_dict, degree_mixing_matrix
Notes
This computes Eq. (21) in Ref. [R167] , where e is the joint probability distribution (mixing matrix) of the
degrees. If G is directed than the matrix e is the joint probability of the user-specified degree type for the source
and target.
References
[R167], [R168]
Examples
>>> G=nx.path_graph(4)
>>> r=nx.degree_assortativity_coefficient(G)
>>> print("%3.1f"%r)
-0.5
attribute_assortativity_coefficient
attribute_assortativity_coefficient(G, attribute, nodes=None)
Compute assortativity for node attributes.
Assortativity measures the similarity of connections in the graph with respect to the given attribute.
Parameters G : NetworkX graph
attribute : string
Node attribute key
nodes: list or iterable (optional)
Compute attribute assortativity for nodes in container. The default is all nodes.
Returns r: float
Assortativity of graph for given attribute
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Notes
This computes Eq. (2) in Ref. [R164] , trace(M)-sum(M))/(1-sum(M), where M is the joint probability distribution (mixing matrix) of the specified attribute.
References
[R164]
Examples
>>>
>>>
>>>
>>>
>>>
1.0
G=nx.Graph()
G.add_nodes_from([0,1],color='red')
G.add_nodes_from([2,3],color='blue')
G.add_edges_from([(0,1),(2,3)])
print(nx.attribute_assortativity_coefficient(G,'color'))
numeric_assortativity_coefficient
numeric_assortativity_coefficient(G, attribute, nodes=None)
Compute assortativity for numerical node attributes.
Assortativity measures the similarity of connections in the graph with respect to the given numeric attribute.
Parameters G : NetworkX graph
attribute : string
Node attribute key
nodes: list or iterable (optional)
Compute numeric assortativity only for attributes of nodes in container. The default is
all nodes.
Returns r: float
Assortativity of graph for given attribute
Notes
This computes Eq. (21) in Ref. [R172] , for the mixing matrix of of the specified attribute.
References
[R172]
Examples
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>>>
>>>
>>>
>>>
>>>
1.0
G=nx.Graph()
G.add_nodes_from([0,1],size=2)
G.add_nodes_from([2,3],size=3)
G.add_edges_from([(0,1),(2,3)])
print(nx.numeric_assortativity_coefficient(G,'size'))
degree_pearson_correlation_coefficient
degree_pearson_correlation_coefficient(G, x=’out’, y=’in’, weight=None, nodes=None)
Compute degree assortativity of graph.
Assortativity measures the similarity of connections in the graph with respect to the node degree.
This is the same as degree_assortativity_coefficient but uses the potentially faster scipy.stats.pearsonr function.
Parameters G : NetworkX graph
x: string (�in’,’out’)
The degree type for source node (directed graphs only).
y: string (�in’,’out’)
The degree type for target node (directed graphs only).
weight: string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
nodes: list or iterable (optional)
Compute pearson correlation of degrees only for specified nodes. The default is all
nodes.
Returns r : float
Assortativity of graph by degree.
Notes
This calls scipy.stats.pearsonr.
References
[R169], [R170]
Examples
>>> G=nx.path_graph(4)
>>> r=nx.degree_pearson_correlation_coefficient(G)
>>> print("%3.1f"%r)
-0.5
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4.2.2 Average neighbor degree
average_neighbor_degree(G[, source, target, ...])
Returns the average degree of the neighborhood of each node.
average_neighbor_degree
average_neighbor_degree(G, source=’out’, target=’out’, nodes=None, weight=None)
Returns the average degree of the neighborhood of each node.
The average degree of a node рќ‘– is
�𝑛𝑛,𝑖 =
∑︁
1
�𝑗
|рќ‘Ѓ (рќ‘–)|
рќ‘—в€€рќ‘Ѓ (рќ‘–)
where 𝑁 (𝑖) are the neighbors of node 𝑖 and �𝑗 is the degree of node 𝑗 which belongs to 𝑁 (𝑖). For weighted
graphs, an analogous measure can be defined [R166],
𝑤
�𝑛𝑛,𝑖
=
1 ∑︁
𝑤𝑖𝑗 �𝑗
рќ‘ рќ‘–
рќ‘—в€€рќ‘Ѓ (рќ‘–)
where 𝑠𝑖 is the weighted degree of node 𝑖, 𝑤𝑖𝑗 is the weight of the edge that links 𝑖 and 𝑗 and 𝑁 (𝑖) are the
neighbors of node рќ‘–.
Parameters G : NetworkX graph
source : string (“in”|”out”)
Directed graphs only. Use “in”- or “out”-degree for source node.
target : string (“in”|”out”)
Directed graphs only. Use “in”- or “out”-degree for target node.
nodes : list or iterable, optional
Compute neighbor degree for specified nodes. The default is all nodes in the graph.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1.
Returns d: dict
A dictionary keyed by node with average neighbors degree value.
See also:
average_degree_connectivity
Notes
For directed graphs you can also specify in-degree or out-degree by passing keyword arguments.
References
[R166]
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Examples
>>> G=nx.path_graph(4)
>>> G.edge[0][1]['weight'] = 5
>>> G.edge[2][3]['weight'] = 3
>>>
{0:
>>>
{0:
nx.average_neighbor_degree(G)
2.0, 1: 1.5, 2: 1.5, 3: 2.0}
nx.average_neighbor_degree(G, weight='weight')
2.0, 1: 1.1666666666666667, 2: 1.25, 3: 2.0}
>>>
>>>
>>>
{0:
G=nx.DiGraph()
G.add_path([0,1,2,3])
nx.average_neighbor_degree(G, source='in', target='in')
1.0, 1: 1.0, 2: 1.0, 3: 0.0}
>>> nx.average_neighbor_degree(G, source='out', target='out')
{0: 1.0, 1: 1.0, 2: 0.0, 3: 0.0}
4.2.3 Average degree connectivity
average_degree_connectivity(G[, source, ...])
k_nearest_neighbors(G[, source, target, ...])
Compute the average degree connectivity of graph.
Compute the average degree connectivity of graph.
average_degree_connectivity
average_degree_connectivity(G, source=’in+out’, target=’in+out’, nodes=None, weight=None)
Compute the average degree connectivity of graph.
The average degree connectivity is the average nearest neighbor degree of nodes with degree k. For weighted
graphs, an analogous measure can be computed using the weighted average neighbors degree defined in [R165],
for a node рќ‘–, as:
𝑤
�𝑛𝑛,𝑖
=
1 ∑︁
𝑤𝑖𝑗 �𝑗
рќ‘ рќ‘–
рќ‘—в€€рќ‘Ѓ (рќ‘–)
where 𝑠𝑖 is the weighted degree of node 𝑖, 𝑤𝑖𝑗 is the weight of the edge that links 𝑖 and 𝑗, and 𝑁 (𝑖) are the
neighbors of node рќ‘–.
Parameters G : NetworkX graph
source : “in”|”out”|”in+out” (default:”in+out”)
Directed graphs only. Use “in”- or “out”-degree for source node.
target : “in”|”out”|”in+out” (default:”in+out”
Directed graphs only. Use “in”- or “out”-degree for target node.
nodes: list or iterable (optional)
Compute neighbor connectivity for these nodes. The default is all nodes.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1.
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Returns d: dict
A dictionary keyed by degree k with the value of average connectivity.
See also:
neighbors_average_degree
Notes
This algorithm is sometimes called “k nearest neighbors’ and is also available as �𝑛 𝑒𝑎𝑟𝑒𝑠𝑡𝑛 𝑒𝑖𝑔ℎ𝑏𝑜𝑟𝑠.
References
[R165]
Examples
>>>
>>>
>>>
{1:
>>>
{1:
G=nx.path_graph(4)
G.edge[1][2]['weight'] = 3
nx.k_nearest_neighbors(G)
2.0, 2: 1.5}
nx.k_nearest_neighbors(G, weight='weight')
2.0, 2: 1.75}
k_nearest_neighbors
k_nearest_neighbors(G, source=’in+out’, target=’in+out’, nodes=None, weight=None)
Compute the average degree connectivity of graph.
The average degree connectivity is the average nearest neighbor degree of nodes with degree k. For weighted
graphs, an analogous measure can be computed using the weighted average neighbors degree defined in [R171],
for a node рќ‘–, as:
𝑤
�𝑛𝑛,𝑖
=
1 ∑︁
𝑤𝑖𝑗 �𝑗
рќ‘ рќ‘–
рќ‘—в€€рќ‘Ѓ (рќ‘–)
where 𝑠𝑖 is the weighted degree of node 𝑖, 𝑤𝑖𝑗 is the weight of the edge that links 𝑖 and 𝑗, and 𝑁 (𝑖) are the
neighbors of node рќ‘–.
Parameters G : NetworkX graph
source : “in”|”out”|”in+out” (default:”in+out”)
Directed graphs only. Use “in”- or “out”-degree for source node.
target : “in”|”out”|”in+out” (default:”in+out”
Directed graphs only. Use “in”- or “out”-degree for target node.
nodes: list or iterable (optional)
Compute neighbor connectivity for these nodes. The default is all nodes.
weight : string or None, optional (default=None)
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The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1.
Returns d: dict
A dictionary keyed by degree k with the value of average connectivity.
See also:
neighbors_average_degree
Notes
This algorithm is sometimes called “k nearest neighbors’ and is also available as �𝑛 𝑒𝑎𝑟𝑒𝑠𝑡𝑛 𝑒𝑖𝑔ℎ𝑏𝑜𝑟𝑠.
References
[R171]
Examples
>>>
>>>
>>>
{1:
>>>
{1:
G=nx.path_graph(4)
G.edge[1][2]['weight'] = 3
nx.k_nearest_neighbors(G)
2.0, 2: 1.5}
nx.k_nearest_neighbors(G, weight='weight')
2.0, 2: 1.75}
4.2.4 Mixing
attribute_mixing_matrix(G, attribute[, ...])
degree_mixing_matrix(G[, x, y, weight, ...])
degree_mixing_dict(G[, x, y, weight, nodes, ...])
attribute_mixing_dict(G, attribute[, nodes, ...])
Return mixing matrix for attribute.
Return mixing matrix for attribute.
Return dictionary representation of mixing matrix for degree.
Return dictionary representation of mixing matrix for attribute.
attribute_mixing_matrix
attribute_mixing_matrix(G, attribute, nodes=None, mapping=None, normalized=True)
Return mixing matrix for attribute.
Parameters G : graph
NetworkX graph object.
attribute : string
Node attribute key.
nodes: list or iterable (optional)
Use only nodes in container to build the matrix. The default is all nodes.
mapping : dictionary, optional
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Mapping from node attribute to integer index in matrix. If not specified, an arbitrary
ordering will be used.
normalized : bool (default=False)
Return counts if False or probabilities if True.
Returns m: numpy array
Counts or joint probability of occurrence of attribute pairs.
degree_mixing_matrix
degree_mixing_matrix(G, x=’out’, y=’in’, weight=None, nodes=None, normalized=True)
Return mixing matrix for attribute.
Parameters G : graph
NetworkX graph object.
x: string (�in’,’out’)
The degree type for source node (directed graphs only).
y: string (�in’,’out’)
The degree type for target node (directed graphs only).
nodes: list or iterable (optional)
Build the matrix using only nodes in container. The default is all nodes.
weight: string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
normalized : bool (default=False)
Return counts if False or probabilities if True.
Returns m: numpy array
Counts, or joint probability, of occurrence of node degree.
degree_mixing_dict
degree_mixing_dict(G, x=’out’, y=’in’, weight=None, nodes=None, normalized=False)
Return dictionary representation of mixing matrix for degree.
Parameters G : graph
NetworkX graph object.
x: string (�in’,’out’)
The degree type for source node (directed graphs only).
y: string (�in’,’out’)
The degree type for target node (directed graphs only).
weight: string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
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normalized : bool (default=False)
Return counts if False or probabilities if True.
Returns d: dictionary
Counts or joint probability of occurrence of degree pairs.
attribute_mixing_dict
attribute_mixing_dict(G, attribute, nodes=None, normalized=False)
Return dictionary representation of mixing matrix for attribute.
Parameters G : graph
NetworkX graph object.
attribute : string
Node attribute key.
nodes: list or iterable (optional)
Unse nodes in container to build the dict. The default is all nodes.
normalized : bool (default=False)
Return counts if False or probabilities if True.
Returns d : dictionary
Counts or joint probability of occurrence of attribute pairs.
Examples
>>>
>>>
>>>
>>>
>>>
>>>
1
>>>
1
G=nx.Graph()
G.add_nodes_from([0,1],color='red')
G.add_nodes_from([2,3],color='blue')
G.add_edge(1,3)
d=nx.attribute_mixing_dict(G,'color')
print(d['red']['blue'])
print(d['blue']['red']) # d symmetric for undirected graphs
4.3 Bipartite
This module provides functions and operations for bipartite graphs. Bipartite graphs 𝐵 = (𝑈, 𝑉, 𝐸) have two node
sets 𝑈, 𝑉 and edges in 𝐸 that only connect nodes from opposite sets. It is common in the literature to use an spatial
analogy referring to the two node sets as top and bottom nodes.
The bipartite algorithms are not imported into the networkx namespace at the top level so the easiest way to use them
is with:
>>> import networkx as nx
>>> from networkx.algorithms import bipartite
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NetworkX does not have a custom bipartite graph class but the Graph() or DiGraph() classes can be used to represent
bipartite graphs. However, you have to keep track of which set each node belongs to, and make sure that there is no
edge between nodes of the same set. The convention used in NetworkX is to use a node attribute named “bipartite”
with values 0 or 1 to identify the sets each node belongs to.
For example:
>>>
>>>
>>>
>>>
B = nx.Graph()
B.add_nodes_from([1,2,3,4], bipartite=0) # Add the node attribute "bipartite"
B.add_nodes_from(['a','b','c'], bipartite=1)
B.add_edges_from([(1,'a'), (1,'b'), (2,'b'), (2,'c'), (3,'c'), (4,'a')])
Many algorithms of the bipartite module of NetworkX require, as an argument, a container with all the nodes that
belong to one set, in addition to the bipartite graph рќђµ. If рќђµ is connected, you can find the node sets using a twocoloring algorithm:
>>> nx.is_connected(B)
True
>>> bottom_nodes, top_nodes = bipartite.sets(B)
list(top_nodes) [1, 2, 3, 4] list(bottom_nodes) [’a’, �c’, �b’]
However, if the input graph is not connected, there are more than one possible colorations. Thus, the following result
is correct:
>>> B.remove_edge(2,'c')
>>> nx.is_connected(B)
False
>>> bottom_nodes, top_nodes = bipartite.sets(B)
list(top_nodes) [1, 2, 4, �c’] list(bottom_nodes) [’a’, 3, �b’]
Using the “bipartite” node attribute, you can easily get the two node sets:
>>> top_nodes = set(n for n,d in B.nodes(data=True) if d['bipartite']==0)
>>> bottom_nodes = set(B) - top_nodes
list(top_nodes) [1, 2, 3, 4] list(bottom_nodes) [’a’, �c’, �b’]
So you can easily use the bipartite algorithms that require, as an argument, a container with all nodes that belong to
one node set:
>>> print(round(bipartite.density(B, bottom_nodes),2))
0.42
>>> G = bipartite.projected_graph(B, top_nodes)
>>> G.edges()
[(1, 2), (1, 4)]
All bipartite graph generators in NetworkX build bipartite graphs with the “bipartite” node attribute. Thus, you can
use the same approach:
>>>
>>>
>>>
>>>
[0,
>>>
[5,
RB = nx.bipartite_random_graph(5, 7, 0.2)
RB_top = set(n for n,d in RB.nodes(data=True) if d['bipartite']==0)
RB_bottom = set(RB) - RB_top
list(RB_top)
1, 2, 3, 4]
list(RB_bottom)
6, 7, 8, 9, 10, 11]
For other bipartite graph generators see the bipartite section of Graph generators.
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4.3.1 Basic functions
is_bipartite(G)
is_bipartite_node_set(G, nodes)
sets(G)
color(G)
density(B, nodes)
degrees(B, nodes[, weight])
biadjacency_matrix(G, row_order[, ...])
Returns True if graph G is bipartite, False if not.
Returns True if nodes and G/nodes are a bipartition of G.
Returns bipartite node sets of graph G.
Returns a two-coloring of the graph.
Return density of bipartite graph B.
Return the degrees of the two node sets in the bipartite graph B.
Return the biadjacency matrix of the bipartite graph G.
is_bipartite
is_bipartite(G)
Returns True if graph G is bipartite, False if not.
Parameters G : NetworkX graph
See also:
color, is_bipartite_node_set
Examples
>>> from networkx.algorithms import bipartite
>>> G = nx.path_graph(4)
>>> print(bipartite.is_bipartite(G))
True
is_bipartite_node_set
is_bipartite_node_set(G, nodes)
Returns True if nodes and G/nodes are a bipartition of G.
Parameters G : NetworkX graph
nodes: list or container
Check if nodes are a one of a bipartite set.
Notes
For connected graphs the bipartite sets are unique. This function handles disconnected graphs.
Examples
>>> from networkx.algorithms import bipartite
>>> G = nx.path_graph(4)
>>> X = set([1,3])
>>> bipartite.is_bipartite_node_set(G,X)
True
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sets
sets(G)
Returns bipartite node sets of graph G.
Raises an exception if the graph is not bipartite.
Parameters G : NetworkX graph
Returns (X,Y) : two-tuple of sets
One set of nodes for each part of the bipartite graph.
See also:
color
Examples
>>>
>>>
>>>
>>>
[0,
>>>
[1,
from networkx.algorithms import bipartite
G = nx.path_graph(4)
X, Y = bipartite.sets(G)
list(X)
2]
list(Y)
3]
color
color(G)
Returns a two-coloring of the graph.
Raises an exception if the graph is not bipartite.
Parameters G : NetworkX graph
Returns color : dictionary
A dictionary keyed by node with a 1 or 0 as data for each node color.
Raises NetworkXError if the graph is not two-colorable.
Examples
>>>
>>>
>>>
>>>
{0:
from networkx.algorithms import bipartite
G = nx.path_graph(4)
c = bipartite.color(G)
print(c)
1, 1: 0, 2: 1, 3: 0}
You can use this to set a node attribute indicating the biparite set:
>>> nx.set_node_attributes(G, 'bipartite', c)
>>> print(G.node[0]['bipartite'])
1
>>> print(G.node[1]['bipartite'])
0
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density
density(B, nodes)
Return density of bipartite graph B.
Parameters G : NetworkX graph
nodes: list or container
Nodes in one set of the bipartite graph.
Returns d : float
The bipartite density
See also:
color
Examples
>>>
>>>
>>>
>>>
1.0
>>>
>>>
1.0
from networkx.algorithms import bipartite
G = nx.complete_bipartite_graph(3,2)
X=set([0,1,2])
bipartite.density(G,X)
Y=set([3,4])
bipartite.density(G,Y)
degrees
degrees(B, nodes, weight=None)
Return the degrees of the two node sets in the bipartite graph B.
Parameters G : NetworkX graph
nodes: list or container
Nodes in one set of the bipartite graph.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1. The degree is the sum of the edge weights adjacent to the node.
Returns (degX,degY) : tuple of dictionaries
The degrees of the two bipartite sets as dictionaries keyed by node.
See also:
color, density
Examples
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>>>
>>>
>>>
>>>
>>>
{0:
from networkx.algorithms import bipartite
G = nx.complete_bipartite_graph(3,2)
Y=set([3,4])
degX,degY=bipartite.degrees(G,Y)
degX
2, 1: 2, 2: 2}
biadjacency_matrix
biadjacency_matrix(G, row_order, column_order=None, weight=’weight’, dtype=None)
Return the biadjacency matrix of the bipartite graph G.
Let 𝐺 = (𝑈, 𝑉, 𝐸) be a bipartite graph with node sets 𝑈 = 𝑢1 , ..., 𝑢𝑟 and 𝑉 = 𝑣1 , ..., 𝑣𝑠. The biadjacency
matrix [1] is the 𝑟 x 𝑠matrix 𝐵 in which 𝑏𝑖,𝑗 = 1 if, and only if, (𝑢𝑖 , 𝑣𝑗 ) ∈ 𝐸. If the parameter 𝑤𝑒𝑖𝑔ℎ𝑡 is not
рќ‘Ѓ рќ‘њрќ‘›рќ‘’ and matches the name of an edge attribute, its value is used instead of 1.
Parameters G : graph
A NetworkX graph
row_order : list of nodes
The rows of the matrix are ordered according to the list of nodes.
column_order : list, optional
The columns of the matrix are ordered according to the list of nodes. If column_order
is None, then the ordering of columns is arbitrary.
weight : string or None, optional (default=’weight’)
The edge data key used to provide each value in the matrix. If None, then each edge has
weight 1.
dtype : NumPy data type, optional
A valid single NumPy data type used to initialize the array. This must be a simple type
such as int or numpy.float64 and not a compound data type (see to_numpy_recarray) If
None, then the NumPy default is used.
Returns B : numpy matrix
Biadjacency matrix representation of the bipartite graph G.
See also:
to_numpy_matrix, adjacency_matrix
Notes
No attempt is made to check that the input graph is bipartite.
For directed bipartite graphs only successors are considered as neighbors. To obtain an adjacency matrix with
ones (or weight values) for both predecessors and successors you have to generate two biadjacency matrices
where the rows of one of them are the columns of the other, and then add one to the transpose of the other.
References
[1] http://en.wikipedia.org/wiki/Adjacency_matrix#Adjacency_matrix_of_a_bipartite_graph
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4.3.2 Projections
One-mode (unipartite) projections of bipartite graphs.
projected_graph(B, nodes[, multigraph])
weighted_projected_graph(B, nodes[, ratio])
collaboration_weighted_projected_graph(B, nodes)
overlap_weighted_projected_graph(B, nodes[, ...])
generic_weighted_projected_graph(B, nodes[, ...])
Returns the projection of B onto one of its node sets.
Returns a weighted projection of B onto one of its node sets.
Newman’s weighted projection of B onto one of its node sets.
Overlap weighted projection of B onto one of its node sets.
Weighted projection of B with a user-specified weight function.
projected_graph
projected_graph(B, nodes, multigraph=False)
Returns the projection of B onto one of its node sets.
Returns the graph G that is the projection of the bipartite graph B onto the specified nodes. They retain their
attributes and are connected in G if they have a common neighbor in B.
Parameters B : NetworkX graph
The input graph should be bipartite.
nodes : list or iterable
Nodes to project onto (the “bottom” nodes).
multigraph: bool (default=False)
If True return a multigraph where the multiple edges represent multiple shared neighbors. They edge key in the multigraph is assigned to the label of the neighbor.
Returns Graph : NetworkX graph or multigraph
A graph that is the projection onto the given nodes.
See also:
is_bipartite,
is_bipartite_node_set,
collaboration_weighted_projected_graph,
generic_weighted_projected_graph
sets,
weighted_projected_graph,
overlap_weighted_projected_graph,
Notes
No attempt is made to verify that the input graph B is bipartite. Returns a simple graph that is the projection of
the bipartite graph B onto the set of nodes given in list nodes. If multigraph=True then a multigraph is returned
with an edge for every shared neighbor.
Directed graphs are allowed as input. The output will also then be a directed graph with edges if there is a
directed path between the nodes.
The graph and node properties are (shallow) copied to the projected graph.
Examples
>>> from networkx.algorithms import bipartite
>>> B = nx.path_graph(4)
>>> G = bipartite.projected_graph(B, [1,3])
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>>> print(G.nodes())
[1, 3]
>>> print(G.edges())
[(1, 3)]
If nodes рќ‘Ћ, and рќ‘Џ are connected through both nodes 1 and 2 then building a multigraph results in two edges in
the projection onto [𝑎,�b�]:
>>> B = nx.Graph()
>>> B.add_edges_from([('a', 1), ('b', 1), ('a', 2), ('b', 2)])
>>> G = bipartite.projected_graph(B, ['a', 'b'], multigraph=True)
>>> print([sorted((u,v)) for u,v in G.edges()])
[['a', 'b'], ['a', 'b']]
weighted_projected_graph
weighted_projected_graph(B, nodes, ratio=False)
Returns a weighted projection of B onto one of its node sets.
The weighted projected graph is the projection of the bipartite network B onto the specified nodes with weights
representing the number of shared neighbors or the ratio between actual shared neighbors and possible shared
neighbors if ratio=True [R182]. The nodes retain their attributes and are connected in the resulting graph if they
have an edge to a common node in the original graph.
Parameters B : NetworkX graph
The input graph should be bipartite.
nodes : list or iterable
Nodes to project onto (the “bottom” nodes).
ratio: Bool (default=False)
If True, edge weight is the ratio between actual shared neighbors and possible shared
neighbors. If False, edges weight is the number of shared neighbors.
Returns Graph : NetworkX graph
A graph that is the projection onto the given nodes.
See also:
is_bipartite, is_bipartite_node_set, sets, collaboration_weighted_projected_graph,
overlap_weighted_projected_graph,
generic_weighted_projected_graph,
projected_graph
Notes
No attempt is made to verify that the input graph B is bipartite. The graph and node properties are (shallow)
copied to the projected graph.
References
[R182]
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Examples
>>> from networkx.algorithms import bipartite
>>> B = nx.path_graph(4)
>>> G = bipartite.weighted_projected_graph(B, [1,3])
>>> print(G.nodes())
[1, 3]
>>> print(G.edges(data=True))
[(1, 3, {'weight': 1})]
>>> G = bipartite.weighted_projected_graph(B, [1,3], ratio=True)
>>> print(G.edges(data=True))
[(1, 3, {'weight': 0.5})]
collaboration_weighted_projected_graph
collaboration_weighted_projected_graph(B, nodes)
Newman’s weighted projection of B onto one of its node sets.
The collaboration weighted projection is the projection of the bipartite network B onto the specified nodes with
weights assigned using Newman’s collaboration model [R180]:
𝑤𝑣,𝑢 =
∑︁ 𝛿 𝑤 𝛿 �
𝑣 𝑤
�𝑤 − 1
�
where 𝑣 and 𝑢 are nodes from the same bipartite node set, and 𝑤 is a node of the opposite node set. The value
�𝑤 is the degree of node 𝑤 in the bipartite network and 𝛿𝑣𝑤 is 1 if node 𝑣 is linked to node 𝑤 in the original
bipartite graph or 0 otherwise.
The nodes retain their attributes and are connected in the resulting graph if have an edge to a common node in
the original bipartite graph.
Parameters B : NetworkX graph
The input graph should be bipartite.
nodes : list or iterable
Nodes to project onto (the “bottom” nodes).
Returns Graph : NetworkX graph
A graph that is the projection onto the given nodes.
See also:
is_bipartite,
is_bipartite_node_set,
overlap_weighted_projected_graph,
projected_graph
sets,
weighted_projected_graph,
generic_weighted_projected_graph,
Notes
No attempt is made to verify that the input graph B is bipartite. The graph and node properties are (shallow)
copied to the projected graph.
References
[R180]
4.3. Bipartite
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Examples
>>>
>>>
>>>
>>>
>>>
[0,
>>>
...
(0,
(0,
(2,
(2,
from networkx.algorithms import bipartite
B = nx.path_graph(5)
B.add_edge(1,5)
G = bipartite.collaboration_weighted_projected_graph(B, [0, 2, 4, 5])
print(G.nodes())
2, 4, 5]
for edge in G.edges(data=True): print(edge)
2,
5,
4,
5,
{'weight':
{'weight':
{'weight':
{'weight':
0.5})
0.5})
1.0})
0.5})
overlap_weighted_projected_graph
overlap_weighted_projected_graph(B, nodes, jaccard=True)
Overlap weighted projection of B onto one of its node sets.
The overlap weighted projection is the projection of the bipartite network B onto the specified nodes with
weights representing the Jaccard index between the neighborhoods of the two nodes in the original bipartite
network [R181]:
𝑤𝑣,𝑢 =
|𝑁 (𝑢) ∩ 𝑁 (𝑣)|
|рќ‘Ѓ (рќ‘ў) в€Є рќ‘Ѓ (рќ‘Ј)|
or if the parameter �jaccard’ is False, the fraction of common neighbors by minimum of both nodes degree in
the original bipartite graph [R181]:
𝑤𝑣,𝑢 =
|𝑁 (𝑢) ∩ 𝑁 (𝑣)|
рќ‘љрќ‘–рќ‘›(|рќ‘Ѓ (рќ‘ў)|, |рќ‘Ѓ (рќ‘Ј)|)
The nodes retain their attributes and are connected in the resulting graph if have an edge to a common node in
the original bipartite graph.
Parameters B : NetworkX graph
The input graph should be bipartite.
nodes : list or iterable
Nodes to project onto (the “bottom” nodes).
jaccard: Bool (default=True)
Returns Graph : NetworkX graph
A graph that is the projection onto the given nodes.
See also:
is_bipartite,
is_bipartite_node_set,
collaboration_weighted_projected_graph,
projected_graph
sets,
weighted_projected_graph,
generic_weighted_projected_graph,
Notes
No attempt is made to verify that the input graph B is bipartite. The graph and node properties are (shallow)
copied to the projected graph.
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References
[R181]
Examples
>>> from networkx.algorithms import bipartite
>>> B = nx.path_graph(5)
>>> G = bipartite.overlap_weighted_projected_graph(B, [0, 2, 4])
>>> print(G.nodes())
[0, 2, 4]
>>> print(G.edges(data=True))
[(0, 2, {'weight': 0.5}), (2, 4, {'weight': 0.5})]
>>> G = bipartite.overlap_weighted_projected_graph(B, [0, 2, 4], jaccard=False)
>>> print(G.edges(data=True))
[(0, 2, {'weight': 1.0}), (2, 4, {'weight': 1.0})]
generic_weighted_projected_graph
generic_weighted_projected_graph(B, nodes, weight_function=None)
Weighted projection of B with a user-specified weight function.
The bipartite network B is projected on to the specified nodes with weights computed by a user-specified function. This function must accept as a parameter the neighborhood sets of two nodes and return an integer or a
float.
The nodes retain their attributes and are connected in the resulting graph if they have an edge to a common node
in the original graph.
Parameters B : NetworkX graph
The input graph should be bipartite.
nodes : list or iterable
Nodes to project onto (the “bottom” nodes).
weight_function: function
This function must accept as parameters the same input graph that this function, and
two nodes; and return an integer or a float. The default function computes the number
of shared neighbors.
Returns Graph : NetworkX graph
A graph that is the projection onto the given nodes.
See also:
is_bipartite,
is_bipartite_node_set,
collaboration_weighted_projected_graph,
projected_graph
sets,
weighted_projected_graph,
overlap_weighted_projected_graph,
Notes
No attempt is made to verify that the input graph B is bipartite. The graph and node properties are (shallow)
copied to the projected graph.
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Examples
>>> from networkx.algorithms import bipartite
>>> # Define some custom weight functions
>>> def jaccard(G, u, v):
...
unbrs = set(G[u])
...
vnbrs = set(G[v])
...
return float(len(unbrs & vnbrs)) / len(unbrs | vnbrs)
...
>>> def my_weight(G, u, v, weight='weight'):
...
w = 0
...
for nbr in set(G[u]) & set(G[v]):
...
w += G.edge[u][nbr].get(weight, 1) + G.edge[v][nbr].get(weight, 1)
...
return w
...
>>> # A complete bipartite graph with 4 nodes and 4 edges
>>> B = nx.complete_bipartite_graph(2,2)
>>> # Add some arbitrary weight to the edges
>>> for i,(u,v) in enumerate(B.edges()):
...
B.edge[u][v]['weight'] = i + 1
...
>>> for edge in B.edges(data=True):
...
print(edge)
...
(0, 2, {'weight': 1})
(0, 3, {'weight': 2})
(1, 2, {'weight': 3})
(1, 3, {'weight': 4})
>>> # Without specifying a function, the weight is equal to # shared partners
>>> G = bipartite.generic_weighted_projected_graph(B, [0, 1])
>>> print(G.edges(data=True))
[(0, 1, {'weight': 2})]
>>> # To specify a custom weight function use the weight_function parameter
>>> G = bipartite.generic_weighted_projected_graph(B, [0, 1], weight_function=jaccard)
>>> print(G.edges(data=True))
[(0, 1, {'weight': 1.0})]
>>> G = bipartite.generic_weighted_projected_graph(B, [0, 1], weight_function=my_weight)
>>> print(G.edges(data=True))
[(0, 1, {'weight': 10})]
4.3.3 Spectral
Spectral bipartivity measure.
spectral_bipartivity(G[, nodes, weight])
Returns the spectral bipartivity.
spectral_bipartivity
spectral_bipartivity(G, nodes=None, weight=’weight’)
Returns the spectral bipartivity.
Parameters G : NetworkX graph
nodes : list or container optional(default is all nodes)
Nodes to return value of spectral bipartivity contribution.
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weight : string or None optional (default = �weight’)
Edge data key to use for edge weights. If None, weights set to 1.
Returns sb : float or dict
A single number if the keyword nodes is not specified, or a dictionary keyed by node
with the spectral bipartivity contribution of that node as the value.
See also:
color
Notes
This implementation uses Numpy (dense) matrices which are not efficient for storing large sparse graphs.
References
[R184]
Examples
>>> from networkx.algorithms import bipartite
>>> G = nx.path_graph(4)
>>> bipartite.spectral_bipartivity(G)
1.0
4.3.4 Clustering
clustering(G[, nodes, mode])
average_clustering(G[, nodes, mode])
latapy_clustering(G[, nodes, mode])
robins_alexander_clustering(G)
Compute a bipartite clustering coefficient for nodes.
Compute the average bipartite clustering coefficient.
Compute a bipartite clustering coefficient for nodes.
Compute the bipartite clustering of G.
clustering
clustering(G, nodes=None, mode=’dot’)
Compute a bipartite clustering coefficient for nodes.
The bipartie clustering coefficient is a measure of local density of connections defined as [R177]:
∑︀
рќ‘Јв€€рќ‘Ѓ (рќ‘Ѓ (рќ‘Ј)) рќ‘ђрќ‘ўрќ‘Ј
рќ‘ђрќ‘ў =
|рќ‘Ѓ (рќ‘Ѓ (рќ‘ў))|
where рќ‘Ѓ (рќ‘Ѓ (рќ‘ў)) are the second order neighbors of рќ‘ў in рќђє excluding рќ‘ў, and рќ‘ђрќ‘ўрќ‘Ј is the pairwise clustering
coefficient between nodes рќ‘ў and рќ‘Ј.
The mode selects the function for рќ‘ђрќ‘ўрќ‘Ј which can be:
рќ‘‘рќ‘њрќ‘Ў:
рќ‘ђрќ‘ўрќ‘Ј =
4.3. Bipartite
|𝑁 (𝑢) ∩ 𝑁 (𝑣)|
|рќ‘Ѓ (рќ‘ў) в€Є рќ‘Ѓ (рќ‘Ј)|
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рќ‘љрќ‘–рќ‘›:
рќ‘ђрќ‘ўрќ‘Ј =
|𝑁 (𝑢) ∩ 𝑁 (𝑣)|
рќ‘љрќ‘–рќ‘›(|рќ‘Ѓ (рќ‘ў)|, |рќ‘Ѓ (рќ‘Ј)|)
рќ‘ђрќ‘ўрќ‘Ј =
|𝑁 (𝑢) ∩ 𝑁 (𝑣)|
рќ‘љрќ‘Ћрќ‘Ґ(|рќ‘Ѓ (рќ‘ў)|, |рќ‘Ѓ (рќ‘Ј)|)
рќ‘љрќ‘Ћрќ‘Ґ:
Parameters G : graph
A bipartite graph
nodes : list or iterable (optional)
Compute bipartite clustering for these nodes. The default is all nodes in G.
mode : string
The pariwise bipartite clustering method to be used in the computation. It must be “dot”,
“max”, or “min”.
Returns clustering : dictionary
A dictionary keyed by node with the clustering coefficient value.
See also:
robins_alexander_clustering, square_clustering, average_clustering
References
[R177]
Examples
>>>
>>>
>>>
>>>
0.5
>>>
>>>
1.0
from networkx.algorithms import bipartite
G = nx.path_graph(4) # path graphs are bipartite
c = bipartite.clustering(G)
c[0]
c = bipartite.clustering(G,mode='min')
c[0]
average_clustering
average_clustering(G, nodes=None, mode=’dot’)
Compute the average bipartite clustering coefficient.
A clustering coefficient for the whole graph is the average,
рќђ¶=
1 ∑︁
рќ‘ђрќ‘Ј ,
рќ‘›
рќ‘Јв€€рќђє
where рќ‘› is the number of nodes in рќђє.
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Similar measures for the two bipartite sets can be defined [R176]
рќђ¶рќ‘‹ =
1 ∑︁
рќ‘ђрќ‘Ј ,
|рќ‘‹|
рќ‘Јв€€рќ‘‹
where рќ‘‹ is a bipartite set of рќђє.
Parameters G : graph
a bipartite graph
nodes : list or iterable, optional
A container of nodes to use in computing the average. The nodes should be either the
entire graph (the default) or one of the bipartite sets.
mode : string
The pariwise bipartite clustering method. It must be “dot”, “max”, or “min”
Returns clustering : float
The average bipartite clustering for the given set of nodes or the entire graph if no nodes
are specified.
See also:
clustering
Notes
The container of nodes passed to this function must contain all of the nodes in one of the bipartite sets (“top” or
“bottom”) in order to compute the correct average bipartite clustering coefficients.
References
[R176]
Examples
>>> from networkx.algorithms import bipartite
>>> G=nx.star_graph(3) # star graphs are bipartite
>>> bipartite.average_clustering(G)
0.75
>>> X,Y=bipartite.sets(G)
>>> bipartite.average_clustering(G,X)
0.0
>>> bipartite.average_clustering(G,Y)
1.0
latapy_clustering
latapy_clustering(G, nodes=None, mode=’dot’)
Compute a bipartite clustering coefficient for nodes.
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The bipartie clustering coefficient is a measure of local density of connections defined as [R178]:
∑︀
рќ‘Јв€€рќ‘Ѓ (рќ‘Ѓ (рќ‘Ј)) рќ‘ђрќ‘ўрќ‘Ј
рќ‘ђрќ‘ў =
|рќ‘Ѓ (рќ‘Ѓ (рќ‘ў))|
where рќ‘Ѓ (рќ‘Ѓ (рќ‘ў)) are the second order neighbors of рќ‘ў in рќђє excluding рќ‘ў, and рќ‘ђрќ‘ўрќ‘Ј is the pairwise clustering
coefficient between nodes рќ‘ў and рќ‘Ј.
The mode selects the function for рќ‘ђрќ‘ўрќ‘Ј which can be:
рќ‘‘рќ‘њрќ‘Ў:
рќ‘ђрќ‘ўрќ‘Ј =
|𝑁 (𝑢) ∩ 𝑁 (𝑣)|
|рќ‘Ѓ (рќ‘ў) в€Є рќ‘Ѓ (рќ‘Ј)|
рќ‘љрќ‘–рќ‘›:
рќ‘ђрќ‘ўрќ‘Ј =
|𝑁 (𝑢) ∩ 𝑁 (𝑣)|
рќ‘љрќ‘–рќ‘›(|рќ‘Ѓ (рќ‘ў)|, |рќ‘Ѓ (рќ‘Ј)|)
рќ‘ђрќ‘ўрќ‘Ј =
|𝑁 (𝑢) ∩ 𝑁 (𝑣)|
рќ‘љрќ‘Ћрќ‘Ґ(|рќ‘Ѓ (рќ‘ў)|, |рќ‘Ѓ (рќ‘Ј)|)
рќ‘љрќ‘Ћрќ‘Ґ:
Parameters G : graph
A bipartite graph
nodes : list or iterable (optional)
Compute bipartite clustering for these nodes. The default is all nodes in G.
mode : string
The pariwise bipartite clustering method to be used in the computation. It must be “dot”,
“max”, or “min”.
Returns clustering : dictionary
A dictionary keyed by node with the clustering coefficient value.
See also:
robins_alexander_clustering, square_clustering, average_clustering
References
[R178]
Examples
>>>
>>>
>>>
>>>
0.5
>>>
>>>
1.0
156
from networkx.algorithms import bipartite
G = nx.path_graph(4) # path graphs are bipartite
c = bipartite.clustering(G)
c[0]
c = bipartite.clustering(G,mode='min')
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robins_alexander_clustering
robins_alexander_clustering(G)
Compute the bipartite clustering of G.
Robins and Alexander [R179] defined bipartite clustering coefficient as four times the number of four cycles рќђ¶4
divided by the number of three paths рќђї3 in a bipartite graph:
рќђ¶рќђ¶4 =
4 * рќђ¶4
рќђї3
Parameters G : graph
a bipartite graph
Returns clustering : float
The Robins and Alexander bipartite clustering for the input graph.
See also:
latapy_clustering, square_clustering
References
[R179]
Examples
>>> from networkx.algorithms import bipartite
>>> G = nx.davis_southern_women_graph()
>>> print(round(bipartite.robins_alexander_clustering(G), 3))
0.468
4.3.5 Redundancy
Node redundancy for bipartite graphs.
node_redundancy(G[, nodes])
Compute bipartite node redundancy coefficient.
node_redundancy
node_redundancy(G, nodes=None)
Compute bipartite node redundancy coefficient.
The redundancy coefficient of a node рќ‘Ј is the fraction of pairs of neighbors of рќ‘Ј that are both linked to other
nodes. In a one-mode projection these nodes would be linked together even if рќ‘Ј were not there.
рќ‘џрќ‘ђ(рќ‘Ј) =
|{{𝑢, 𝑤} ⊆ 𝑁 (𝑣), ∃𝑣 ′ ̸= 𝑣, (𝑣 ′ , 𝑢) ∈ 𝐸 and (𝑣 ′ , 𝑤) ∈ 𝐸}|
|рќ‘Ѓ (рќ‘Ј)|(|рќ‘Ѓ (рќ‘Ј)|в€’1)
2
where рќ‘Ѓ (рќ‘Ј) are the neighbors of рќ‘Ј in рќђє.
Parameters G : graph
A bipartite graph
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nodes : list or iterable (optional)
Compute redundancy for these nodes. The default is all nodes in G.
Returns redundancy : dictionary
A dictionary keyed by node with the node redundancy value.
References
[R183]
Examples
>>>
>>>
>>>
>>>
1.0
from networkx.algorithms import bipartite
G = nx.cycle_graph(4)
rc = bipartite.node_redundancy(G)
rc[0]
Compute the average redundancy for the graph:
>>> sum(rc.values())/len(G)
1.0
Compute the average redundancy for a set of nodes:
>>> nodes = [0, 2]
>>> sum(rc[n] for n in nodes)/len(nodes)
1.0
4.3.6 Centrality
closeness_centrality(G, nodes[, normalized])
degree_centrality(G, nodes)
betweenness_centrality(G, nodes)
Compute the closeness centrality for nodes in a bipartite network.
Compute the degree centrality for nodes in a bipartite network.
Compute betweenness centrality for nodes in a bipartite network.
closeness_centrality
closeness_centrality(G, nodes, normalized=True)
Compute the closeness centrality for nodes in a bipartite network.
The closeness of a node is the distance to all other nodes in the graph or in the case that the graph is not connected
to all other nodes in the connected component containing that node.
Parameters G : graph
A bipartite network
nodes : list or container
Container with all nodes in one bipartite node set.
normalized : bool, optional
If True (default) normalize by connected component size.
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Returns closeness : dictionary
Dictionary keyed by node with bipartite closeness centrality as the value.
See also:
betweenness_centrality, degree_centrality, sets, is_bipartite
Notes
The nodes input parameter must conatin all nodes in one bipartite node set, but the dictionary returned contains
all nodes from both node sets.
Closeness centrality is normalized by the minimum distance possible. In the bipartite case the minimum distance
for a node in one bipartite node set is 1 from all nodes in the other node set and 2 from all other nodes in its
own set [R174]. Thus the closeness centrality for node 𝑣 in the two bipartite sets 𝑈 with 𝑛 nodes and 𝑉 with 𝑚
nodes is
рќ‘љ + 2(рќ‘› в€’ 1)
, forрќ‘Ј в€€ рќ‘€,
рќ‘‘
рќ‘› + 2(рќ‘љ в€’ 1)
рќ‘ђрќ‘Ј =
, for𝑣 ∈ 𝑉,
рќ‘‘
рќ‘ђрќ‘Ј =
where рќ‘‘ is the sum of the distances from рќ‘Ј to all other nodes.
Higher values of closeness indicate higher centrality.
As in the unipartite case, setting normalized=True causes the values to normalized further to n-1 / size(G)-1
where n is the number of nodes in the connected part of graph containing the node. If the graph is not completely
connected, this algorithm computes the closeness centrality for each connected part separately.
References
[R174]
degree_centrality
degree_centrality(G, nodes)
Compute the degree centrality for nodes in a bipartite network.
The degree centrality for a node рќ‘Ј is the fraction of nodes connected to it.
Parameters G : graph
A bipartite network
nodes : list or container
Container with all nodes in one bipartite node set.
Returns centrality : dictionary
Dictionary keyed by node with bipartite degree centrality as the value.
See also:
betweenness_centrality, closeness_centrality, sets, is_bipartite
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Notes
The nodes input parameter must conatin all nodes in one bipartite node set, but the dictionary returned contains
all nodes from both bipartite node sets.
For unipartite networks, the degree centrality values are normalized by dividing by the maximum possible degree
(which is рќ‘› в€’ 1 where рќ‘› is the number of nodes in G).
In the bipartite case, the maximum possible degree of a node in a bipartite node set is the number of nodes in
the opposite node set [R175]. The degree centrality for a node 𝑣 in the bipartite sets 𝑈 with 𝑛 nodes and 𝑉 with
рќ‘љ nodes is
рќ‘‘рќ‘’рќ‘”(рќ‘Ј)
, forрќ‘Ј в€€ рќ‘€,
рќ‘љ
рќ‘‘рќ‘’рќ‘”(рќ‘Ј)
рќ‘‘рќ‘Ј =
, for𝑣 ∈ 𝑉,
рќ‘›
рќ‘‘рќ‘Ј =
where рќ‘‘рќ‘’рќ‘”(рќ‘Ј) is the degree of node рќ‘Ј.
References
[R175]
betweenness_centrality
betweenness_centrality(G, nodes)
Compute betweenness centrality for nodes in a bipartite network.
Betweenness centrality of a node рќ‘Ј is the sum of the fraction of all-pairs shortest paths that pass through рќ‘Ј.
Values of betweenness are normalized by the maximum possible value which for bipartite graphs is limited by
the relative size of the two node sets [R173].
Let 𝑛 be the number of nodes in the node set 𝑈 and 𝑚 be the number of nodes in the node set 𝑉 , then nodes in
рќ‘€ are normalized by dividing by
1 2
[рќ‘љ (рќ‘ + 1)2 + рќ‘љ(рќ‘ + 1)(2рќ‘Ў в€’ рќ‘ в€’ 1) в€’ рќ‘Ў(2рќ‘ в€’ рќ‘Ў + 3)],
2
where
рќ‘ = (рќ‘› в€’ 1) Г· рќ‘љ, рќ‘Ў = (рќ‘› в€’ 1)
mod рќ‘љ,
and nodes in 𝑉 are normalized by dividing by
1 2
[рќ‘› (рќ‘ќ + 1)2 + рќ‘›(рќ‘ќ + 1)(2рќ‘џ в€’ рќ‘ќ в€’ 1) в€’ рќ‘џ(2рќ‘ќ в€’ рќ‘џ + 3)],
2
where,
рќ‘ќ = (рќ‘љ в€’ 1) Г· рќ‘›, рќ‘џ = (рќ‘љ в€’ 1)
mod рќ‘›.
Parameters G : graph
A bipartite graph
nodes : list or container
Container with all nodes in one bipartite node set.
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Returns betweenness : dictionary
Dictionary keyed by node with bipartite betweenness centrality as the value.
See also:
degree_centrality, closeness_centrality, sets, is_bipartite
Notes
The nodes input parameter must contain all nodes in one bipartite node set, but the dictionary returned contains
all nodes from both node sets.
References
[R173]
4.4 Blockmodeling
Functions for creating network blockmodels from node partitions.
Created by Drew Conway <[email protected]> Copyright (c) 2010. All rights reserved.
blockmodel(G, partitions[, multigraph])
Returns a reduced graph constructed using the generalized block modeling technique.
4.4.1 blockmodel
blockmodel(G, partitions, multigraph=False)
Returns a reduced graph constructed using the generalized block modeling technique.
The blockmodel technique collapses nodes into blocks based on a given partitioning of the node set. Each
partition of nodes (block) is represented as a single node in the reduced graph.
Edges between nodes in the block graph are added according to the edges in the original graph. If the parameter
multigraph is False (the default) a single edge is added with a weight equal to the sum of the edge weights
between nodes in the original graph The default is a weight of 1 if weights are not specified. If the parameter
multigraph is True then multiple edges are added each with the edge data from the original graph.
Parameters G : graph
A networkx Graph or DiGraph
partitions : list of lists, or list of sets
The partition of the nodes. Must be non-overlapping.
multigraph : bool, optional
If True return a MultiGraph with the edge data of the original graph applied to each
corresponding edge in the new graph. If False return a Graph with the sum of the edge
weights, or a count of the edges if the original graph is unweighted.
Returns blockmodel : a Networkx graph object
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References
[R185]
Examples
>>> G=nx.path_graph(6)
>>> partition=[[0,1],[2,3],[4,5]]
>>> M=nx.blockmodel(G,partition)
4.5 Boundary
Routines to find the boundary of a set of nodes.
Edge boundaries are edges that have only one end in the set of nodes.
Node boundaries are nodes outside the set of nodes that have an edge to a node in the set.
edge_boundary(G, nbunch1[, nbunch2])
node_boundary(G, nbunch1[, nbunch2])
Return the edge boundary.
Return the node boundary.
4.5.1 edge_boundary
edge_boundary(G, nbunch1, nbunch2=None)
Return the edge boundary.
Edge boundaries are edges that have only one end in the given set of nodes.
Parameters G : graph
A networkx graph
nbunch1 : list, container
Interior node set
nbunch2 : list, container
Exterior node set. If None then it is set to all of the nodes in G not in nbunch1.
Returns elist : list
List of edges
Notes
Nodes in nbunch1 and nbunch2 that are not in G are ignored.
nbunch1 and nbunch2 are usually meant to be disjoint, but in the interest of speed and generality, that is not
required here.
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4.5.2 node_boundary
node_boundary(G, nbunch1, nbunch2=None)
Return the node boundary.
The node boundary is all nodes in the edge boundary of a given set of nodes that are in the set.
Parameters G : graph
A networkx graph
nbunch1 : list, container
Interior node set
nbunch2 : list, container
Exterior node set. If None then it is set to all of the nodes in G not in nbunch1.
Returns nlist : list
List of nodes.
Notes
Nodes in nbunch1 and nbunch2 that are not in G are ignored.
nbunch1 and nbunch2 are usually meant to be disjoint, but in the interest of speed and generality, that is not
required here.
4.6 Centrality
4.6.1 Degree
degree_centrality(G)
in_degree_centrality(G)
out_degree_centrality(G)
Compute the degree centrality for nodes.
Compute the in-degree centrality for nodes.
Compute the out-degree centrality for nodes.
degree_centrality
degree_centrality(G)
Compute the degree centrality for nodes.
The degree centrality for a node v is the fraction of nodes it is connected to.
Parameters G : graph
A networkx graph
Returns nodes : dictionary
Dictionary of nodes with degree centrality as the value.
See also:
betweenness_centrality, load_centrality, eigenvector_centrality
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Notes
The degree centrality values are normalized by dividing by the maximum possible degree in a simple graph n-1
where n is the number of nodes in G.
For multigraphs or graphs with self loops the maximum degree might be higher than n-1 and values of degree
centrality greater than 1 are possible.
in_degree_centrality
in_degree_centrality(G)
Compute the in-degree centrality for nodes.
The in-degree centrality for a node v is the fraction of nodes its incoming edges are connected to.
Parameters G : graph
A NetworkX graph
Returns nodes : dictionary
Dictionary of nodes with in-degree centrality as values.
Raises NetworkXError
If the graph is undirected.
See also:
degree_centrality, out_degree_centrality
Notes
The degree centrality values are normalized by dividing by the maximum possible degree in a simple graph n-1
where n is the number of nodes in G.
For multigraphs or graphs with self loops the maximum degree might be higher than n-1 and values of degree
centrality greater than 1 are possible.
out_degree_centrality
out_degree_centrality(G)
Compute the out-degree centrality for nodes.
The out-degree centrality for a node v is the fraction of nodes its outgoing edges are connected to.
Parameters G : graph
A NetworkX graph
Returns nodes : dictionary
Dictionary of nodes with out-degree centrality as values.
Raises NetworkXError
If the graph is undirected.
See also:
degree_centrality, in_degree_centrality
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Notes
The degree centrality values are normalized by dividing by the maximum possible degree in a simple graph n-1
where n is the number of nodes in G.
For multigraphs or graphs with self loops the maximum degree might be higher than n-1 and values of degree
centrality greater than 1 are possible.
4.6.2 Closeness
closeness_centrality(G[, u, distance, ...])
Compute closeness centrality for nodes.
closeness_centrality
closeness_centrality(G, u=None, distance=None, normalized=True)
Compute closeness centrality for nodes.
Closeness centrality [R191] of a node рќ‘ў is the reciprocal of the sum of the shortest path distances from рќ‘ў to
all рќ‘› в€’ 1 other nodes. Since the sum of distances depends on the number of nodes in the graph, closeness is
normalized by the sum of minimum possible distances рќ‘› в€’ 1.
рќ‘›в€’1
𝐶(𝑢) = ∑︀𝑛−1
,
рќ‘Ј=1 рќ‘‘(рќ‘Ј, рќ‘ў)
where рќ‘‘(рќ‘Ј, рќ‘ў) is the shortest-path distance between рќ‘Ј and рќ‘ў, and рќ‘› is the number of nodes in the graph.
Notice that higher values of closeness indicate higher centrality.
Parameters G : graph
A NetworkX graph
u : node, optional
Return only the value for node u
distance : edge attribute key, optional (default=None)
Use the specified edge attribute as the edge distance in shortest path calculations
normalized : bool, optional
If True (default) normalize by the number of nodes in the connected part of the graph.
Returns nodes : dictionary
Dictionary of nodes with closeness centrality as the value.
See also:
betweenness_centrality,
degree_centrality
load_centrality,
eigenvector_centrality,
Notes
The closeness centrality is normalized to (рќ‘› в€’ 1)/(|рќђє| в€’ 1) where рќ‘› is the number of nodes in the connected part
of graph containing the node. If the graph is not completely connected, this algorithm computes the closeness
centrality for each connected part separately.
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If the �distance’ keyword is set to an edge attribute key then the shortest-path length will be computed using
Dijkstra’s algorithm with that edge attribute as the edge weight.
References
[R191]
4.6.3 Betweenness
betweenness_centrality(G[, k, normalized, ...])
edge_betweenness_centrality(G[, normalized, ...])
Compute the shortest-path betweenness centrality for nodes.
Compute betweenness centrality for edges.
betweenness_centrality
betweenness_centrality(G, k=None, normalized=True, weight=None, endpoints=False, seed=None)
Compute the shortest-path betweenness centrality for nodes.
Betweenness centrality of a node рќ‘Ј is the sum of the fraction of all-pairs shortest paths that pass through рќ‘Ј:
рќ‘ђрќђµ (рќ‘Ј) =
∑︁ 𝜎(𝑠, 𝑡|𝑣)
рќњЋ(рќ‘ , рќ‘Ў)
𝑠,𝑡∈𝑉
where 𝑉 is the set of nodes, 𝜎(𝑠, 𝑡) is the number of shortest (𝑠, 𝑡)-paths, and 𝜎(𝑠, 𝑡|𝑣) is the number of those
paths passing through some node рќ‘Ј other than рќ‘ , рќ‘Ў. If рќ‘ = рќ‘Ў, рќњЋ(рќ‘ , рќ‘Ў) = 1, and if рќ‘Ј в€€ рќ‘ , рќ‘Ў, рќњЋ(рќ‘ , рќ‘Ў|рќ‘Ј) = 0 [R188].
Parameters G : graph
A NetworkX graph
k : int, optional (default=None)
If k is not None use k node samples to estimate betweenness. The value of k <= n where
n is the number of nodes in the graph. Higher values give better approximation.
normalized : bool, optional
If True the betweenness values are normalized by 2/((рќ‘› в€’ 1)(рќ‘› в€’ 2)) for graphs, and
1/((рќ‘› в€’ 1)(рќ‘› в€’ 2)) for directed graphs where рќ‘› is the number of nodes in G.
weight : None or string, optional
If None, all edge weights are considered equal. Otherwise holds the name of the edge
attribute used as weight.
endpoints : bool, optional
If True include the endpoints in the shortest path counts.
Returns nodes : dictionary
Dictionary of nodes with betweenness centrality as the value.
See also:
edge_betweenness_centrality, load_centrality
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Notes
The algorithm is from Ulrik Brandes [R187]. See [R190] for the original first published version and [R188] for
details on algorithms for variations and related metrics.
For approximate betweenness calculations set k=#samples to use k nodes (“pivots”) to estimate the betweenness
values. For an estimate of the number of pivots needed see [R189].
For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite
number of equal length paths between pairs of nodes.
References
[R187], [R188], [R189], [R190]
edge_betweenness_centrality
edge_betweenness_centrality(G, normalized=True, weight=None)
Compute betweenness centrality for edges.
Betweenness centrality of an edge рќ‘’ is the sum of the fraction of all-pairs shortest paths that pass through рќ‘’:
рќ‘ђрќђµ (рќ‘Ј) =
∑︁ 𝜎(𝑠, 𝑡|𝑒)
рќњЋ(рќ‘ , рќ‘Ў)
𝑠,𝑡∈𝑉
where 𝑉 is the set of nodes,�sigma(s, t)� is the number of shortest (𝑠, 𝑡)-paths, and 𝜎(𝑠, 𝑡|𝑒) is the number of
those paths passing through edge рќ‘’ [R206].
Parameters G : graph
A NetworkX graph
normalized : bool, optional
If True the betweenness values are normalized by 2/(рќ‘›(рќ‘›в€’1)) for graphs, and 1/(рќ‘›(рќ‘›в€’
1)) for directed graphs where рќ‘› is the number of nodes in G.
weight : None or string, optional
If None, all edge weights are considered equal. Otherwise holds the name of the edge
attribute used as weight.
Returns edges : dictionary
Dictionary of edges with betweenness centrality as the value.
See also:
betweenness_centrality, edge_load
Notes
The algorithm is from Ulrik Brandes [R205].
For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite
number of equal length paths between pairs of nodes.
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References
[R205], [R206]
4.6.4 Current Flow Closeness
current_flow_closeness_centrality(G[, ...])
Compute current-flow closeness centrality for nodes.
current_flow_closeness_centrality
current_flow_closeness_centrality(G, weight=’weight’, dtype=<type �float’>, solver=’lu’)
Compute current-flow closeness centrality for nodes.
Current-flow closeness centrality is variant of closeness centrality based on effective resistance between nodes
in a network. This metric is also known as information centrality.
Parameters G : graph
A NetworkX graph
dtype: data type (float)
Default data type for internal matrices. Set to np.float32 for lower memory consumption.
solver: string (default=’lu’)
Type of linear solver to use for computing the flow matrix. Options are “full” (uses most
memory), “lu” (recommended), and “cg” (uses least memory).
Returns nodes : dictionary
Dictionary of nodes with current flow closeness centrality as the value.
See also:
closeness_centrality
Notes
The algorithm is from Brandes [R202].
See also [R203] for the original definition of information centrality.
References
[R202], [R203]
4.6.5 Current-Flow Betweenness
current_flow_betweenness_centrality(G[, ...])
edge_current_flow_betweenness_centrality(G)
approximate_current_flow_betweenness_centrality(G)
168
Compute current-flow betweenness centrality for nodes.
Compute current-flow betweenness centrality for edges.
Compute the approximate current-flow betweenness centr
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current_flow_betweenness_centrality
current_flow_betweenness_centrality(G, normalized=True, weight=’weight’, dtype=<type
�float’>, solver=’full’)
Compute current-flow betweenness centrality for nodes.
Current-flow betweenness centrality uses an electrical current model for information spreading in contrast to
betweenness centrality which uses shortest paths.
Current-flow betweenness centrality is also known as random-walk betweenness centrality [R201].
Parameters G : graph
A NetworkX graph
normalized : bool, optional (default=True)
If True the betweenness values are normalized by 2/[(n-1)(n-2)] where n is the number
of nodes in G.
weight : string or None, optional (default=’weight’)
Key for edge data used as the edge weight. If None, then use 1 as each edge weight.
dtype: data type (float)
Default data type for internal matrices. Set to np.float32 for lower memory consumption.
solver: string (default=’lu’)
Type of linear solver to use for computing the flow matrix. Options are “full” (uses most
memory), “lu” (recommended), and “cg” (uses least memory).
Returns nodes : dictionary
Dictionary of nodes with betweenness centrality as the value.
See also:
approximate_current_flow_betweenness_centrality,
betweenness_centrality,
edge_betweenness_centrality, edge_current_flow_betweenness_centrality
Notes
Current-flow betweenness can be computed in рќ‘‚(рќђј(рќ‘› в€’ 1) + рќ‘љрќ‘› log рќ‘›) time [R200], where рќђј(рќ‘› в€’ 1) is the
time needed to в€љ
compute the inverse Laplacian. For a full matrix this is рќ‘‚(рќ‘›3 ) but using sparse methods you can
achieve 𝑂(𝑛𝑚 �) where � is the Laplacian matrix condition number.
The space required is 𝑂(𝑛𝑤) where 𝑤 is the width of the sparse Laplacian matrix. Worse case is 𝑤 = 𝑛 for
рќ‘‚(рќ‘›2 ).
If the edges have a �weight’ attribute they will be used as weights in this algorithm. Unspecified weights are set
to 1.
References
[R200], [R201]
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edge_current_flow_betweenness_centrality
edge_current_flow_betweenness_centrality(G,
normalized=True,
weight=’weight’,
dtype=<type �float’>, solver=’full’)
Compute current-flow betweenness centrality for edges.
Current-flow betweenness centrality uses an electrical current model for information spreading in contrast to
betweenness centrality which uses shortest paths.
Current-flow betweenness centrality is also known as random-walk betweenness centrality [R208].
Parameters G : graph
A NetworkX graph
normalized : bool, optional (default=True)
If True the betweenness values are normalized by 2/[(n-1)(n-2)] where n is the number
of nodes in G.
weight : string or None, optional (default=’weight’)
Key for edge data used as the edge weight. If None, then use 1 as each edge weight.
dtype: data type (float)
Default data type for internal matrices. Set to np.float32 for lower memory consumption.
solver: string (default=’lu’)
Type of linear solver to use for computing the flow matrix. Options are “full” (uses most
memory), “lu” (recommended), and “cg” (uses least memory).
Returns nodes : dictionary
Dictionary of edge tuples with betweenness centrality as the value.
See also:
betweenness_centrality, edge_betweenness_centrality, current_flow_betweenness_centrality
Notes
Current-flow betweenness can be computed in рќ‘‚(рќђј(рќ‘› в€’ 1) + рќ‘љрќ‘› log рќ‘›) time [R207], where рќђј(рќ‘› в€’ 1) is the
time needed to в€љ
compute the inverse Laplacian. For a full matrix this is рќ‘‚(рќ‘›3 ) but using sparse methods you can
achieve 𝑂(𝑛𝑚 �) where � is the Laplacian matrix condition number.
The space required is 𝑂(𝑛𝑤)𝑤ℎ𝑒𝑟𝑒�𝑤 is the width of the sparse Laplacian matrix. Worse case is 𝑤 = 𝑛 for
рќ‘‚(рќ‘›2 ).
If the edges have a �weight’ attribute they will be used as weights in this algorithm. Unspecified weights are set
to 1.
References
[R207], [R208]
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approximate_current_flow_betweenness_centrality
approximate_current_flow_betweenness_centrality(G, normalized=True, weight=’weight’,
dtype=<type �float’>, solver=’full’,
epsilon=0.5, kmax=10000)
Compute the approximate current-flow betweenness centrality for nodes.
Approximates the current-flow betweenness centrality within absolute error of epsilon with high probability
[R186].
Parameters G : graph
A NetworkX graph
normalized : bool, optional (default=True)
If True the betweenness values are normalized by 2/[(n-1)(n-2)] where n is the number
of nodes in G.
weight : string or None, optional (default=’weight’)
Key for edge data used as the edge weight. If None, then use 1 as each edge weight.
dtype: data type (float)
Default data type for internal matrices. Set to np.float32 for lower memory consumption.
solver: string (default=’lu’)
Type of linear solver to use for computing the flow matrix. Options are “full” (uses most
memory), “lu” (recommended), and “cg” (uses least memory).
epsilon: float
Absolute error tolerance.
kmax: int
Maximum number of sample node pairs to use for approximation.
Returns nodes : dictionary
Dictionary of nodes with betweenness centrality as the value.
See also:
current_flow_betweenness_centrality
Notes
в€љ
The running time is 𝑂((1/𝜖2 )𝑚 � log 𝑛) and the space required is 𝑂(𝑚) for n nodes and m edges.
If the edges have a �weight’ attribute they will be used as weights in this algorithm. Unspecified weights are set
to 1.
References
[R186]
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4.6.6 Eigenvector
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eigenvector_centrality(G[, max_iter, tol, ...])
eigenvector_centrality_numpy(G[, weight])
katz_centrality(G[, alpha, beta, max_iter, ...])
katz_centrality_numpy(G[, alpha, beta, ...])
Compute the eigenvector centrality for the graph G.
Compute the eigenvector centrality for the graph G.
Compute the Katz centrality for the nodes of the graph G.
Compute the Katz centrality for the graph G.
eigenvector_centrality
eigenvector_centrality(G, max_iter=100, tol=1e-06, nstart=None, weight=’weight’)
Compute the eigenvector centrality for the graph G.
Eigenvector centrality computes the centrality for a node based on the centrality of its neighbors. The eigenvector centrality for node рќ‘– is
Ax = рќњ†x
where 𝐴 is the adjacency matrix of the graph G with eigenvalue 𝜆. By virtue of the Perron–Frobenius theorem,
there is a unique and positive solution if рќњ† is the largest eigenvalue associated with the eigenvector of the
adjacency matrix рќђґ ([R210]).
Parameters G : graph
A networkx graph
max_iter : interger, optional
Maximum number of iterations in power method.
tol : float, optional
Error tolerance used to check convergence in power method iteration.
nstart : dictionary, optional
Starting value of eigenvector iteration for each node.
weight : None or string, optional
If None, all edge weights are considered equal. Otherwise holds the name of the edge
attribute used as weight.
Returns nodes : dictionary
Dictionary of nodes with eigenvector centrality as the value.
See also:
eigenvector_centrality_numpy, pagerank, hits
Notes
The measure was introduced by [R209].
The eigenvector calculation is done by the power iteration method and has no guarantee of convergence. The
iteration will stop after max_iter iterations or an error tolerance of number_of_nodes(G)*tol has been
reached.
For directed graphs this is “left” eigenvector centrality which corresponds to the in-edges in the graph. For
out-edges eigenvector centrality first reverse the graph with G.reverse().
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Examples
>>>
>>>
>>>
['0
G = nx.path_graph(4)
centrality = nx.eigenvector_centrality(G)
print(['%s %0.2f'%(node,centrality[node]) for node in centrality])
0.37', '1 0.60', '2 0.60', '3 0.37']
eigenvector_centrality_numpy
eigenvector_centrality_numpy(G, weight=’weight’)
Compute the eigenvector centrality for the graph G.
Eigenvector centrality computes the centrality for a node based on the centrality of its neighbors. The eigenvector centrality for node рќ‘– is
Ax = рќњ†x
where 𝐴 is the adjacency matrix of the graph G with eigenvalue 𝜆. By virtue of the Perron–Frobenius theorem,
there is a unique and positive solution if рќњ† is the largest eigenvalue associated with the eigenvector of the
adjacency matrix рќђґ ([R212]).
Parameters G : graph
A networkx graph
weight : None or string, optional
The name of the edge attribute used as weight. If None, all edge weights are considered
equal.
Returns nodes : dictionary
Dictionary of nodes with eigenvector centrality as the value.
See also:
eigenvector_centrality, pagerank, hits
Notes
The measure was introduced by [R211].
This algorithm uses the SciPy sparse eigenvalue solver (ARPACK) to find the largest eigenvalue/eigenvector
pair.
For directed graphs this is “left” eigenvector centrality which corresponds to the in-edges in the graph. For
out-edges eigenvector centrality first reverse the graph with G.reverse().
Examples
>>>
>>>
>>>
['0
174
G = nx.path_graph(4)
centrality = nx.eigenvector_centrality_numpy(G)
print(['%s %0.2f'%(node,centrality[node]) for node in centrality])
0.37', '1 0.60', '2 0.60', '3 0.37']
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katz_centrality
katz_centrality(G, alpha=0.1, beta=1.0, max_iter=1000, tol=1e-06, nstart=None, normalized=True,
weight=’weight’)
Compute the Katz centrality for the nodes of the graph G.
Katz centrality computes the centrality for a node based on the centrality of its neighbors. It is a generalization
of the eigenvector centrality. The Katz centrality for node рќ‘– is
∑︁
рќ‘Ґрќ‘– = рќ›ј
рќђґрќ‘–рќ‘— рќ‘Ґрќ‘— + рќ›Ѕ,
рќ‘—
where рќђґ is the adjacency matrix of the graph G with eigenvalues рќњ†.
The parameter рќ›Ѕ controls the initial centrality and
рќ›ј<
1
рќњ†рќ‘љрќ‘Ћрќ‘Ґ
.
Katz centrality computes the relative influence of a node within a network by measuring the number of the
immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under
consideration through these immediate neighbors.
Extra weight can be provided to immediate neighbors through the parameter рќ›Ѕ. Connections made with distant
neighbors are, however, penalized by an attenuation factor рќ›ј which should be strictly less than the inverse largest
eigenvalue of the adjacency matrix in order for the Katz centrality to be computed correctly. More information
is provided in [R214] .
Parameters G : graph
A NetworkX graph
alpha : float
Attenuation factor
beta : scalar or dictionary, optional (default=1.0)
Weight attributed to the immediate neighborhood. If not a scalar, the dictionary must
have an value for every node.
max_iter : integer, optional (default=1000)
Maximum number of iterations in power method.
tol : float, optional (default=1.0e-6)
Error tolerance used to check convergence in power method iteration.
nstart : dictionary, optional
Starting value of Katz iteration for each node.
normalized : bool, optional (default=True)
If True normalize the resulting values.
weight : None or string, optional
If None, all edge weights are considered equal. Otherwise holds the name of the edge
attribute used as weight.
Returns nodes : dictionary
Dictionary of nodes with Katz centrality as the value.
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Raises NetworkXError
If the parameter рќ‘Џрќ‘’рќ‘Ўрќ‘Ћ is not a scalar but lacks a value for at least one node
See also:
katz_centrality_numpy, eigenvector_centrality, eigenvector_centrality_numpy,
pagerank, hits
Notes
Katz centrality was introduced by [R215].
This algorithm it uses the power method to find the eigenvector corresponding to the largest eigenvalue of the
adjacency matrix of G. The constant alpha should be strictly less than the inverse of largest eigenvalue of the
adjacency matrix for the algorithm to converge. The iteration will stop after max_iter iterations or an error
tolerance of number_of_nodes(G)*tol has been reached.
When рќ›ј = 1/рќњ†рќ‘љрќ‘Ћрќ‘Ґ and рќ›Ѕ = 0, Katz centrality is the same as eigenvector centrality.
For directed graphs this finds “left” eigenvectors which corresponds to the in-edges in the graph. For out-edges
Katz centrality first reverse the graph with G.reverse().
References
[R214], [R215]
Examples
>>> import math
>>> G = nx.path_graph(4)
>>> phi = (1+math.sqrt(5))/2.0 # largest eigenvalue of adj matrix
>>> centrality = nx.katz_centrality(G,1/phi-0.01)
>>> for n,c in sorted(centrality.items()):
...
print("%d %0.2f"%(n,c))
0 0.37
1 0.60
2 0.60
3 0.37
katz_centrality_numpy
katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True, weight=’weight’)
Compute the Katz centrality for the graph G.
Katz centrality computes the centrality for a node based on the centrality of its neighbors. It is a generalization
of the eigenvector centrality. The Katz centrality for node рќ‘– is
∑︁
рќ‘Ґрќ‘– = рќ›ј
рќђґрќ‘–рќ‘— рќ‘Ґрќ‘— + рќ›Ѕ,
рќ‘—
where рќђґ is the adjacency matrix of the graph G with eigenvalues рќњ†.
The parameter рќ›Ѕ controls the initial centrality and
рќ›ј<
176
1
.
рќњ†рќ‘љрќ‘Ћрќ‘Ґ
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Katz centrality computes the relative influence of a node within a network by measuring the number of the
immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under
consideration through these immediate neighbors.
Extra weight can be provided to immediate neighbors through the parameter рќ›Ѕ. Connections made with distant
neighbors are, however, penalized by an attenuation factor рќ›ј which should be strictly less than the inverse largest
eigenvalue of the adjacency matrix in order for the Katz centrality to be computed correctly. More information
is provided in [R216] .
Parameters G : graph
A NetworkX graph
alpha : float
Attenuation factor
beta : scalar or dictionary, optional (default=1.0)
Weight attributed to the immediate neighborhood. If not a scalar the dictionary must
have an value for every node.
normalized : bool
If True normalize the resulting values.
weight : None or string, optional
If None, all edge weights are considered equal. Otherwise holds the name of the edge
attribute used as weight.
Returns nodes : dictionary
Dictionary of nodes with Katz centrality as the value.
Raises NetworkXError
If the parameter рќ‘Џрќ‘’рќ‘Ўрќ‘Ћ is not a scalar but lacks a value for at least one node
See also:
katz_centrality,
pagerank, hits
eigenvector_centrality_numpy,
eigenvector_centrality,
Notes
Katz centrality was introduced by [R217].
This algorithm uses a direct linear solver to solve the above equation. The constant alpha should be strictly less
than the inverse of largest eigenvalue of the adjacency matrix for there to be a solution. When рќ›ј = 1/рќњ†рќ‘љрќ‘Ћрќ‘Ґ and
рќ›Ѕ = 0, Katz centrality is the same as eigenvector centrality.
For directed graphs this finds “left” eigenvectors which corresponds to the in-edges in the graph. For out-edges
Katz centrality first reverse the graph with G.reverse().
References
[R216], [R217]
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Examples
>>> import math
>>> G = nx.path_graph(4)
>>> phi = (1+math.sqrt(5))/2.0 # largest eigenvalue of adj matrix
>>> centrality = nx.katz_centrality_numpy(G,1/phi)
>>> for n,c in sorted(centrality.items()):
...
print("%d %0.2f"%(n,c))
0 0.37
1 0.60
2 0.60
3 0.37
4.6.7 Communicability
communicability(G)
communicability_exp(G)
communicability_centrality(G)
communicability_centrality_exp(G)
communicability_betweenness_centrality(G[, ...])
estrada_index(G)
Return communicability between all pairs of nodes in G.
Return communicability between all pairs of nodes in G.
Return communicability centrality for each node in G.
Return the communicability centrality for each node of G
Return communicability betweenness for all pairs of nodes in G.
Return the Estrada index of a the graph G.
communicability
communicability(G)
Return communicability between all pairs of nodes in G.
The communicability between pairs of nodes in G is the sum of closed walks of different lengths starting at node
u and ending at node v.
Parameters G: graph
Returns comm: dictionary of dictionaries
Dictionary of dictionaries keyed by nodes with communicability as the value.
Raises NetworkXError
If the graph is not undirected and simple.
See also:
communicability_centrality_exp Communicability centrality for each node of G using matrix exponential.
communicability_centrality Communicability centrality for each node in G using spectral decomposition.
communicability Communicability between pairs of nodes in G.
Notes
This algorithm uses a spectral decomposition of the adjacency matrix. Let G=(V,E) be a simple undirected
graph. Using the connection between the powers of the adjacency matrix and the number of walks in the graph,
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the communicability between nodes рќ‘ў and рќ‘Ј based on the graph spectrum is [R192]
рќђ¶(рќ‘ў, рќ‘Ј) =
рќ‘›
∑︁
рќњ‘рќ‘— (рќ‘ў)рќњ‘рќ‘— (рќ‘Ј)рќ‘’рќњ†рќ‘— ,
рќ‘—=1
where рќњ‘рќ‘— (рќ‘ў) is the рќ‘ўth element of the рќ‘—th orthonormal eigenvector of the adjacency matrix associated with the
eigenvalue рќњ†рќ‘— .
References
[R192]
Examples
>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> c = nx.communicability(G)
communicability_exp
communicability_exp(G)
Return communicability between all pairs of nodes in G.
Communicability between pair of node (u,v) of node in G is the sum of closed walks of different lengths starting
at node u and ending at node v.
Parameters G: graph
Returns comm: dictionary of dictionaries
Dictionary of dictionaries keyed by nodes with communicability as the value.
Raises NetworkXError
If the graph is not undirected and simple.
See also:
communicability_centrality_exp Communicability centrality for each node of G using matrix exponential.
communicability_centrality Communicability centrality for each node in G using spectral decomposition.
communicability_exp Communicability between all pairs of nodes in G using spectral decomposition.
Notes
This algorithm uses matrix exponentiation of the adjacency matrix.
Let G=(V,E) be a simple undirected graph. Using the connection between the powers of the adjacency matrix
and the number of walks in the graph, the communicability between nodes u and v is [R199],
рќђ¶(рќ‘ў, рќ‘Ј) = (рќ‘’рќђґ )рќ‘ўрќ‘Ј ,
where рќђґ is the adjacency matrix of G.
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References
[R199]
Examples
>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> c = nx.communicability_exp(G)
communicability_centrality
communicability_centrality(G)
Return communicability centrality for each node in G.
Communicability centrality, also called subgraph centrality, of a node рќ‘› is the sum of closed walks of all lengths
starting and ending at node рќ‘›.
Parameters G: graph
Returns nodes: dictionary
Dictionary of nodes with communicability centrality as the value.
Raises NetworkXError
If the graph is not undirected and simple.
See also:
communicability Communicability between all pairs of nodes in G.
communicability_centrality Communicability centrality for each node of G.
Notes
This version of the algorithm computes eigenvalues and eigenvectors of the adjacency matrix.
Communicability centrality of a node рќ‘ў in G can be found using a spectral decomposition of the adjacency
matrix [R195] [R196],
𝑆𝐶(𝑢) =
рќ‘Ѓ
∑︁
(рќ‘Јрќ‘—рќ‘ў )2 рќ‘’рќњ†рќ‘— ,
рќ‘—=1
where рќ‘Јрќ‘— is an eigenvector of the adjacency matrix рќђґ of G corresponding corresponding to the eigenvalue рќњ†рќ‘— .
References
[R195], [R196]
Examples
>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> sc = nx.communicability_centrality(G)
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communicability_centrality_exp
communicability_centrality_exp(G)
Return the communicability centrality for each node of G
Communicability centrality, also called subgraph centrality, of a node рќ‘› is the sum of closed walks of all lengths
starting and ending at node рќ‘›.
Parameters G: graph
Returns nodes:dictionary
Dictionary of nodes with communicability centrality as the value.
Raises NetworkXError
If the graph is not undirected and simple.
See also:
communicability Communicability between all pairs of nodes in G.
communicability_centrality Communicability centrality for each node of G.
Notes
This version of the algorithm exponentiates the adjacency matrix. The communicability centrality of a node рќ‘ў
in G can be found using the matrix exponential of the adjacency matrix of G [R197] [R198],
𝑆𝐶(𝑢) = (𝑒𝐴 )𝑢𝑢 .
References
[R197], [R198]
Examples
>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> sc = nx.communicability_centrality_exp(G)
communicability_betweenness_centrality
communicability_betweenness_centrality(G, normalized=True)
Return communicability betweenness for all pairs of nodes in G.
Communicability betweenness measure makes use of the number of walks connecting every pair of nodes as the
basis of a betweenness centrality measure.
Parameters G: graph
Returns nodes:dictionary
Dictionary of nodes with communicability betweenness as the value.
Raises NetworkXError
If the graph is not undirected and simple.
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See also:
communicability Communicability between all pairs of nodes in G.
communicability_centrality Communicability centrality for each node of G using matrix exponential.
communicability_centrality_exp Communicability centrality for each node in G using spectral decomposition.
Notes
Let 𝐺 = (𝑉, 𝐸) be a simple undirected graph with 𝑛 nodes and 𝑚 edges, and 𝐴 denote the adjacency matrix of
рќђє.
Let 𝐺(𝑟) = (𝑉, 𝐸(𝑟)) be the graph resulting from removing all edges connected to node 𝑟 but not the node
itself.
The adjacency matrix for рќђє(рќ‘џ) is рќђґ + рќђё(рќ‘џ), where рќђё(рќ‘џ) has nonzeros only in row and column рќ‘џ.
The communicability betweenness of a node рќ‘џ is [R194]
рќњ”рќ‘џ =
1 ∑︁ ∑︁ 𝐺𝑝𝑟𝑞
, рќ‘ќ Мё= рќ‘ћ, рќ‘ћ Мё= рќ‘џ,
рќђ¶ рќ‘ќ рќ‘ћ рќђєрќ‘ќрќ‘ћ
рќђґ+рќђё(рќ‘џ)
)рќ‘ќрќ‘ћ is the number of walks involving node r, рќђєрќ‘ќрќ‘ћ = (рќ‘’рќђґ )рќ‘ќрќ‘ћ is the number of
where рќђєрќ‘ќрќ‘џрќ‘ћ = (рќ‘’рќђґ
рќ‘ќрќ‘ћ в€’ (рќ‘’
closed walks starting at node рќ‘ќ and ending at node рќ‘ћ, and рќђ¶ = (рќ‘› в€’ 1)2 в€’ (рќ‘› в€’ 1) is a normalization factor
equal to the number of terms in the sum.
The resulting рќњ”рќ‘џ takes values between zero and one. The lower bound cannot be attained for a connected graph,
and the upper bound is attained in the star graph.
References
[R194]
Examples
>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> cbc = nx.communicability_betweenness_centrality(G)
estrada_index
estrada_index(G)
Return the Estrada index of a the graph G.
Parameters G: graph
Returns estrada index: float
Raises NetworkXError
If the graph is not undirected and simple.
See also:
estrada_index_exp
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Notes
Let 𝐺 = (𝑉, 𝐸) be a simple undirected graph with 𝑛 nodes and let 𝜆1 ≤ 𝜆2 ≤ · · · 𝜆𝑛 be a non-increasing
ordering of the eigenvalues of its adjacency matrix рќђґ. The Estrada index is
рќђёрќђё(рќђє) =
рќ‘›
∑︁
рќ‘’рќњ†рќ‘— .
рќ‘—=1
References
[R213]
Examples
>>> G=nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> ei=nx.estrada_index(G)
4.6.8 Load
load_centrality(G[, v, cutoff, normalized, ...])
edge_load(G[, nodes, cutoff])
Compute load centrality for nodes.
Compute edge load.
load_centrality
load_centrality(G, v=None, cutoff=None, normalized=True, weight=None)
Compute load centrality for nodes.
The load centrality of a node is the fraction of all shortest paths that pass through that node.
Parameters G : graph
A networkx graph
normalized : bool, optional
If True the betweenness values are normalized by b=b/(n-1)(n-2) where n is the number
of nodes in G.
weight : None or string, optional
If None, edge weights are ignored. Otherwise holds the name of the edge attribute used
as weight.
cutoff : bool, optional
If specified, only consider paths of length <= cutoff.
Returns nodes : dictionary
Dictionary of nodes with centrality as the value.
See also:
betweenness_centrality
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Notes
Load centrality is slightly different than betweenness. It was originally introduced by [R219]. For this load
algorithm see [R218].
References
[R218], [R219]
edge_load
edge_load(G, nodes=None, cutoff=False)
Compute edge load.
WARNING:
This module is for demonstration and testing purposes.
4.6.9 Dispersion
dispersion(G[, u, v, normalized, alpha, b, c])
Calculate dispersion between рќ‘ў and рќ‘Ј in рќђє.
dispersion
dispersion(G, u=None, v=None, normalized=True, alpha=1.0, b=0.0, c=0.0)
Calculate dispersion between рќ‘ў and рќ‘Ј in рќђє.
A link between two actors (рќ‘ў and рќ‘Ј) has a high dispersion when their mutual ties (рќ‘ and рќ‘Ў) are not well connected
with each other.
Parameters G : graph
A NetworkX graph.
u : node, optional
The source for the dispersion score (e.g. ego node of the network).
v : node, optional
The target of the dispersion score if specified.
normalized : bool
If True (default) normalize by the embededness of the nodes (u and v).
Returns nodes : dictionary
If u (v) is specified, returns a dictionary of nodes with dispersion score for all “target”
(“source”) nodes. If neither u nor v is specified, returns a dictionary of dictionaries for
all nodes �u’ in the graph with a dispersion score for each node �v’.
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Notes
This implementation follows Lars Backstrom and Jon Kleinberg [R204]. Typical usage would be to run dispersion on the ego network рќђєрќ‘ў if рќ‘ў were specified. Running dispersion() with neither рќ‘ў nor рќ‘Ј specified can
take some time to complete.
References
[R204]
4.7 Chordal
Algorithms for chordal graphs.
A graph is chordal if every cycle of length at least 4 has a chord (an edge joining two nodes not adjacent in the cycle).
http://en.wikipedia.org/wiki/Chordal_graph
is_chordal(G)
chordal_graph_cliques(G)
chordal_graph_treewidth(G)
find_induced_nodes(G, s, t[, treewidth_bound])
Checks whether G is a chordal graph.
Returns the set of maximal cliques of a chordal graph.
Returns the treewidth of the chordal graph G.
Returns the set of induced nodes in the path from s to t.
4.7.1 is_chordal
is_chordal(G)
Checks whether G is a chordal graph.
A graph is chordal if every cycle of length at least 4 has a chord (an edge joining two nodes not adjacent in the
cycle).
Parameters G : graph
A NetworkX graph.
Returns chordal : bool
True if G is a chordal graph and False otherwise.
Raises NetworkXError
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. If the input
graph is an instance of one of these classes, a NetworkXError is raised.
Notes
The routine tries to go through every node following maximum cardinality search. It returns False when it finds
that the separator for any node is not a clique. Based on the algorithms in [R222].
References
[R222]
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Examples
>>> import networkx as nx
>>> e=[(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)]
>>> G=nx.Graph(e)
>>> nx.is_chordal(G)
True
4.7.2 chordal_graph_cliques
chordal_graph_cliques(G)
Returns the set of maximal cliques of a chordal graph.
The algorithm breaks the graph in connected components and performs a maximum cardinality search in each
component to get the cliques.
Parameters G : graph
A NetworkX graph
Returns cliques : A set containing the maximal cliques in G.
Raises NetworkXError
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. If the input
graph is an instance of one of these classes, a NetworkXError is raised. The algorithm
can only be applied to chordal graphs. If the input graph is found to be non-chordal, a
NetworkXError is raised.
Examples
>>>
>>>
>>>
>>>
>>>
import networkx as nx
e= [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
G = nx.Graph(e)
G.add_node(9)
setlist = nx.chordal_graph_cliques(G)
4.7.3 chordal_graph_treewidth
chordal_graph_treewidth(G)
Returns the treewidth of the chordal graph G.
Parameters G : graph
A NetworkX graph
Returns treewidth : int
The size of the largest clique in the graph minus one.
Raises NetworkXError
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. If the input
graph is an instance of one of these classes, a NetworkXError is raised. The algorithm
can only be applied to chordal graphs. If the input graph is found to be non-chordal, a
NetworkXError is raised.
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References
[R220]
Examples
>>>
>>>
>>>
>>>
>>>
3
import networkx as nx
e = [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
G = nx.Graph(e)
G.add_node(9)
nx.chordal_graph_treewidth(G)
4.7.4 find_induced_nodes
find_induced_nodes(G, s, t, treewidth_bound=9223372036854775807)
Returns the set of induced nodes in the path from s to t.
Parameters G : graph
A chordal NetworkX graph
s : node
Source node to look for induced nodes
t : node
Destination node to look for induced nodes
treewith_bound: float
Maximum treewidth acceptable for the graph H. The search for induced nodes will end
as soon as the treewidth_bound is exceeded.
Returns I : Set of nodes
The set of induced nodes in the path from s to t in G
Raises NetworkXError
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. If the input
graph is an instance of one of these classes, a NetworkXError is raised. The algorithm
can only be applied to chordal graphs. If the input graph is found to be non-chordal, a
NetworkXError is raised.
Notes
G must be a chordal graph and (s,t) an edge that is not in G.
If a treewidth_bound is provided, the search for induced nodes will end as soon as the treewidth_bound is
exceeded.
The algorithm is inspired by Algorithm 4 in [R221]. A formal definition of induced node can also be found on
that reference.
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References
[R221]
Examples
>>>
>>>
>>>
>>>
>>>
[1,
import networkx as nx
G=nx.Graph()
G = nx.generators.classic.path_graph(10)
I = nx.find_induced_nodes(G,1,9,2)
list(I)
2, 3, 4, 5, 6, 7, 8, 9]
4.8 Clique
Find and manipulate cliques of graphs.
Note that finding the largest clique of a graph has been shown to be an NP-complete problem; the algorithms here
could take a long time to run.
http://en.wikipedia.org/wiki/Clique_problem
enumerate_all_cliques(G)
find_cliques(G)
make_max_clique_graph(G[, create_using, name])
make_clique_bipartite(G[, fpos, ...])
graph_clique_number(G[, cliques])
graph_number_of_cliques(G[, cliques])
node_clique_number(G[, nodes, cliques])
number_of_cliques(G[, nodes, cliques])
cliques_containing_node(G[, nodes, cliques])
Returns all cliques in an undirected graph.
Search for all maximal cliques in a graph.
Create the maximal clique graph of a graph.
Create a bipartite clique graph from a graph G.
Return the clique number (size of the largest clique) for G.
Returns the number of maximal cliques in G.
Returns the size of the largest maximal clique containing each given node.
Returns the number of maximal cliques for each node.
Returns a list of cliques containing the given node.
4.8.1 enumerate_all_cliques
enumerate_all_cliques(G)
Returns all cliques in an undirected graph.
This method returns cliques of size (cardinality) k = 1, 2, 3, ..., maxDegree - 1.
Where maxDegree is the maximal degree of any node in the graph.
Parameters G: undirected graph
Returns generator of lists: generator of list for each clique.
Notes
To obtain a list of all cliques, use list(enumerate_all_cliques(G)).
Based on the algorithm published by Zhang et al. (2005) [R223] and adapted to output all cliques discovered.
This algorithm is not applicable on directed graphs.
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This algorithm ignores self-loops and parallel edges as clique is not conventionally defined with such edges.
There are often many cliques in graphs. This algorithm however, hopefully, does not run out of memory since it
only keeps candidate sublists in memory and continuously removes exhausted sublists.
References
[R223]
4.8.2 find_cliques
find_cliques(G)
Search for all maximal cliques in a graph.
Maximal cliques are the largest complete subgraph containing a given node. The largest maximal clique is
sometimes called the maximum clique.
Returns generator of lists: genetor of member list for each maximal clique
See also:
find_cliques_recursive, A
Notes
To obtain a list of cliques, use list(find_cliques(G)).
Based on the algorithm published by Bron & Kerbosch (1973) [R224] as adapted by Tomita, Tanaka and Takahashi (2006) [R225] and discussed in Cazals and Karande (2008) [R226]. The method essentially unrolls the
recursion used in the references to avoid issues of recursion stack depth.
This algorithm is not suitable for directed graphs.
This algorithm ignores self-loops and parallel edges as clique is not conventionally defined with such edges.
There are often many cliques in graphs. This algorithm can run out of memory for large graphs.
References
[R224], [R225], [R226]
4.8.3 make_max_clique_graph
make_max_clique_graph(G, create_using=None, name=None)
Create the maximal clique graph of a graph.
Finds the maximal cliques and treats these as nodes. The nodes are connected if they have common members in
the original graph. Theory has done a lot with clique graphs, but I haven’t seen much on maximal clique graphs.
Notes
This should be the same as make_clique_bipartite followed by project_up, but it saves all the intermediate steps.
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4.8.4 make_clique_bipartite
make_clique_bipartite(G, fpos=None, create_using=None, name=None)
Create a bipartite clique graph from a graph G.
Nodes of G are retained as the “bottom nodes” of B and cliques of G become “top nodes” of B. Edges are
present if a bottom node belongs to the clique represented by the top node.
Returns a Graph with additional attribute dict B.node_type which is keyed by nodes to “Bottom” or “Top”
appropriately.
if fpos is not None, a second additional attribute dict B.pos is created to hold the position tuple of each node for
viewing the bipartite graph.
4.8.5 graph_clique_number
graph_clique_number(G, cliques=None)
Return the clique number (size of the largest clique) for G.
An optional list of cliques can be input if already computed.
4.8.6 graph_number_of_cliques
graph_number_of_cliques(G, cliques=None)
Returns the number of maximal cliques in G.
An optional list of cliques can be input if already computed.
4.8.7 node_clique_number
node_clique_number(G, nodes=None, cliques=None)
Returns the size of the largest maximal clique containing each given node.
Returns a single or list depending on input nodes. Optional list of cliques can be input if already computed.
4.8.8 number_of_cliques
number_of_cliques(G, nodes=None, cliques=None)
Returns the number of maximal cliques for each node.
Returns a single or list depending on input nodes. Optional list of cliques can be input if already computed.
4.8.9 cliques_containing_node
cliques_containing_node(G, nodes=None, cliques=None)
Returns a list of cliques containing the given node.
Returns a single list or list of lists depending on input nodes. Optional list of cliques can be input if already
computed.
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4.9 Clustering
Algorithms to characterize the number of triangles in a graph.
triangles(G[, nodes])
transitivity(G)
clustering(G[, nodes, weight])
average_clustering(G[, nodes, weight, ...])
square_clustering(G[, nodes])
Compute the number of triangles.
Compute graph transitivity, the fraction of all possible triangles present in G.
Compute the clustering coefficient for nodes.
Compute the average clustering coefficient for the graph G.
Compute the squares clustering coefficient for nodes.
4.9.1 triangles
triangles(G, nodes=None)
Compute the number of triangles.
Finds the number of triangles that include a node as one vertex.
Parameters G : graph
A networkx graph
nodes : container of nodes, optional (default= all nodes in G)
Compute triangles for nodes in this container.
Returns out : dictionary
Number of triangles keyed by node label.
Notes
When computing triangles for the entire graph each triangle is counted three times, once at each node. Self
loops are ignored.
Examples
>>>
>>>
6
>>>
{0:
>>>
[6,
G=nx.complete_graph(5)
print(nx.triangles(G,0))
print(nx.triangles(G))
6, 1: 6, 2: 6, 3: 6, 4: 6}
print(list(nx.triangles(G,(0,1)).values()))
6]
4.9.2 transitivity
transitivity(G)
Compute graph transitivity, the fraction of all possible triangles present in G.
Possible triangles are identified by the number of “triads” (two edges with a shared vertex).
The transitivity is
𝑇 =3
4.9. Clustering
#рќ‘Ўрќ‘џрќ‘–рќ‘Ћрќ‘›рќ‘”рќ‘™рќ‘’рќ‘ .
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Parameters G : graph
Returns out : float
Transitivity
Examples
>>> G = nx.complete_graph(5)
>>> print(nx.transitivity(G))
1.0
4.9.3 clustering
clustering(G, nodes=None, weight=None)
Compute the clustering coefficient for nodes.
For unweighted graphs, the clustering of a node рќ‘ў is the fraction of possible triangles through that node that
exist,
рќ‘ђрќ‘ў =
2𝑇 (𝑢)
,
рќ‘‘рќ‘’рќ‘”(рќ‘ў)(рќ‘‘рќ‘’рќ‘”(рќ‘ў) в€’ 1)
where 𝑇 (𝑢) is the number of triangles through node 𝑢 and 𝑑𝑒𝑔(𝑢) is the degree of 𝑢.
For weighted graphs, the clustering is defined as the geometric average of the subgraph edge weights [R229],
рќ‘ђрќ‘ў =
∑︁
1
(𝑤
ˆ𝑢𝑣 𝑤
ˆ𝑢𝑤 𝑤
ˆ𝑣𝑤 )1/3 .
рќ‘‘рќ‘’рќ‘”(рќ‘ў)(рќ‘‘рќ‘’рќ‘”(рќ‘ў) в€’ 1)) рќ‘ўрќ‘Ј
The edge weights 𝑤
ˆ𝑢𝑣 are normalized by the maximum weight in the network 𝑤
ˆ𝑢𝑣 = 𝑤𝑢𝑣 / max(𝑤).
The value of рќ‘ђрќ‘ў is assigned to 0 if рќ‘‘рќ‘’рќ‘”(рќ‘ў) < 2.
Parameters G : graph
nodes : container of nodes, optional (default=all nodes in G)
Compute clustering for nodes in this container.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1.
Returns out : float, or dictionary
Clustering coefficient at specified nodes
Notes
Self loops are ignored.
References
[R229]
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Examples
>>>
>>>
1.0
>>>
{0:
G=nx.complete_graph(5)
print(nx.clustering(G,0))
print(nx.clustering(G))
1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
4.9.4 average_clustering
average_clustering(G, nodes=None, weight=None, count_zeros=True)
Compute the average clustering coefficient for the graph G.
The clustering coefficient for the graph is the average,
рќђ¶=
1 ∑︁
рќ‘ђрќ‘Ј ,
рќ‘›
рќ‘Јв€€рќђє
where рќ‘› is the number of nodes in рќђє.
Parameters G : graph
nodes : container of nodes, optional (default=all nodes in G)
Compute average clustering for nodes in this container.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight. If None, then each
edge has weight 1.
count_zeros : bool
If False include only the nodes with nonzero clustering in the average.
Returns avg : float
Average clustering
Notes
This is a space saving routine; it might be faster to use the clustering function to get a list and then take the
average.
Self loops are ignored.
References
[R227], [R228]
Examples
>>> G=nx.complete_graph(5)
>>> print(nx.average_clustering(G))
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4.9.5 square_clustering
square_clustering(G, nodes=None)
Compute the squares clustering coefficient for nodes.
For each node return the fraction of possible squares that exist at the node [R230]
∑︀�𝑣 ∑︀�𝑣
𝐶4 (𝑣) = ∑︀�𝑣 ∑︀𝑢=1
�𝑣
рќ‘ў=1
𝑤=𝑢+1 𝑞𝑣 (𝑢, 𝑤)
𝑤=𝑢+1 [𝑎𝑣 (𝑢, 𝑤)
+ 𝑞𝑣 (𝑢, 𝑤)]
,
where 𝑞𝑣 (𝑢, 𝑤) are the number of common neighbors of 𝑢 and 𝑤 other than 𝑣 (ie squares), and 𝑎𝑣 (𝑢, 𝑤) =
(�𝑢 − (1 + 𝑞𝑣 (𝑢, 𝑤) + 𝜃𝑢𝑣 ))(�𝑤 − (1 + 𝑞𝑣 (𝑢, 𝑤) + 𝜃𝑢𝑤 )), where 𝜃𝑢𝑤 = 1 if 𝑢 and 𝑤 are connected and 0
otherwise.
Parameters G : graph
nodes : container of nodes, optional (default=all nodes in G)
Compute clustering for nodes in this container.
Returns c4 : dictionary
A dictionary keyed by node with the square clustering coefficient value.
Notes
While рќђ¶3 (рќ‘Ј) (triangle clustering) gives the probability that two neighbors of node v are connected with each
other, рќђ¶4 (рќ‘Ј) is the probability that two neighbors of node v share a common neighbor different from v. This
algorithm can be applied to both bipartite and unipartite networks.
References
[R230]
Examples
>>>
>>>
1.0
>>>
{0:
G=nx.complete_graph(5)
print(nx.square_clustering(G,0))
print(nx.square_clustering(G))
1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
4.10 Coloring
greedy_color(G[, strategy, interchange])
Color a graph using various strategies of greedy graph coloring.
4.10.1 greedy_color
greedy_color(G, strategy=<function strategy_largest_first at 0x23092a8>, interchange=False)
Color a graph using various strategies of greedy graph coloring. The strategies are described in [R231].
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Attempts to color a graph using as few colors as possible, where no neighbours of a node can have same color
as the node itself.
Parameters G : NetworkX graph
strategy : function(G, colors)
A function that provides the coloring strategy, by returning nodes in the ordering they
should be colored. G is the graph, and colors is a dict of the currently assigned colors,
keyed by nodes.
You can pass your own ordering function, or use one of the built in:
• strategy_largest_first
• strategy_random_sequential
• strategy_smallest_last
• strategy_independent_set
• strategy_connected_sequential_bfs
• strategy_connected_sequential_dfs
• strategy_connected_sequential (alias of strategy_connected_sequential_bfs)
• strategy_saturation_largest_first (also known as DSATUR)
interchange: bool
Will use the color interchange algorithm described by [R232] if set to true.
Note that saturation largest first and independent set do not work with interchange.
Furthermore, if you use interchange with your own strategy function, you cannot rely
on the values in the colors argument.
Returns A dictionary with keys representing nodes and values representing
corresponding coloring.
References
[R231], [R232]
Examples
>>> G = nx.cycle_graph(4)
>>> d = nx.coloring.greedy_color(G, strategy=nx.coloring.strategy_largest_first)
>>> d in [{0: 0, 1: 1, 2: 0, 3: 1}, {0: 1, 1: 0, 2: 1, 3: 0}]
True
4.11 Communities
4.11.1 K-Clique
k_clique_communities(G, k[, cliques])
4.11. Communities
Find k-clique communities in graph using the percolation method.
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k_clique_communities
k_clique_communities(G, k, cliques=None)
Find k-clique communities in graph using the percolation method.
A k-clique community is the union of all cliques of size k that can be reached through adjacent (sharing k-1
nodes) k-cliques.
Parameters G : NetworkX graph
k : int
Size of smallest clique
cliques: list or generator
Precomputed cliques (use networkx.find_cliques(G))
Returns Yields sets of nodes, one for each k-clique community.
References
[R233]
Examples
>>>
>>>
>>>
>>>
>>>
[0,
>>>
[]
G = nx.complete_graph(5)
K5 = nx.convert_node_labels_to_integers(G,first_label=2)
G.add_edges_from(K5.edges())
c = list(nx.k_clique_communities(G, 4))
list(c[0])
1, 2, 3, 4, 5, 6]
list(nx.k_clique_communities(G, 6))
4.12 Components
4.12.1 Connectivity
Connected components.
is_connected(G)
number_connected_components(G)
connected_components(G)
connected_component_subgraphs(G[, copy])
node_connected_component(G, n)
Return True if the graph is connected, false otherwise.
Return the number of connected components.
Generate connected components.
Generate connected components as subgraphs.
Return the nodes in the component of graph containing node n.
is_connected
is_connected(G)
Return True if the graph is connected, false otherwise.
Parameters G : NetworkX Graph
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An undirected graph.
Returns connected : bool
True if the graph is connected, false otherwise.
See also:
connected_components
Notes
For undirected graphs only.
Examples
>>> G = nx.path_graph(4)
>>> print(nx.is_connected(G))
True
number_connected_components
number_connected_components(G)
Return the number of connected components.
Parameters G : NetworkX graph
An undirected graph.
Returns n : integer
Number of connected components
See also:
connected_components
Notes
For undirected graphs only.
connected_components
connected_components(G)
Generate connected components.
Parameters G : NetworkX graph
An undirected graph
Returns comp : generator of lists
A list of nodes for each component of G.
See also:
strongly_connected_components
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Notes
For undirected graphs only.
Examples
Generate a sorted list of connected components, largest first.
>>> G = nx.path_graph(4)
>>> G.add_path([10, 11, 12])
>>> sorted(nx.connected_components(G), key = len, reverse=True)
[[0, 1, 2, 3], [10, 11, 12]]
connected_component_subgraphs
connected_component_subgraphs(G, copy=True)
Generate connected components as subgraphs.
Parameters G : NetworkX graph
An undirected graph.
copy: bool (default=True)
If True make a copy of the graph attributes
Returns comp : generator
A generator of graphs, one for each connected component of G.
See also:
connected_components
Notes
For undirected graphs only. Graph, node, and edge attributes are copied to the subgraphs by default.
Examples
>>> G = nx.path_graph(4)
>>> G.add_edge(5,6)
>>> graphs = list(nx.connected_component_subgraphs(G))
node_connected_component
node_connected_component(G, n)
Return the nodes in the component of graph containing node n.
Parameters G : NetworkX Graph
An undirected graph.
n : node label
A node in G
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Returns comp : lists
A list of nodes in component of G containing node n.
See also:
connected_components
Notes
For undirected graphs only.
4.12.2 Strong connectivity
Strongly connected components.
is_strongly_connected(G)
number_strongly_connected_components(G)
strongly_connected_components(G)
strongly_connected_component_subgraphs(G[, copy])
strongly_connected_components_recursive(G)
kosaraju_strongly_connected_components(G[, ...])
condensation(G[, scc])
Test directed graph for strong connectivity.
Return number of strongly connected components in graph.
Generate nodes in strongly connected components of graph.
Generate strongly connected components as subgraphs.
Generate nodes in strongly connected components of graph.
Generate nodes in strongly connected components of graph.
Returns the condensation of G.
is_strongly_connected
is_strongly_connected(G)
Test directed graph for strong connectivity.
Parameters G : NetworkX Graph
A directed graph.
Returns connected : bool
True if the graph is strongly connected, False otherwise.
See also:
strongly_connected_components
Notes
For directed graphs only.
number_strongly_connected_components
number_strongly_connected_components(G)
Return number of strongly connected components in graph.
Parameters G : NetworkX graph
A directed graph.
Returns n : integer
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Number of strongly connected components
See also:
connected_components
Notes
For directed graphs only.
strongly_connected_components
strongly_connected_components(G)
Generate nodes in strongly connected components of graph.
Parameters G : NetworkX Graph
An directed graph.
Returns comp : generator of lists
A list of nodes for each strongly connected component of G.
Raises NetworkXNotImplemented: If G is undirected.
See also:
connected_components, weakly_connected_components
Notes
Uses Tarjan’s algorithm with Nuutila’s modifications. Nonrecursive version of algorithm.
References
[R239], [R240]
strongly_connected_component_subgraphs
strongly_connected_component_subgraphs(G, copy=True)
Generate strongly connected components as subgraphs.
Parameters G : NetworkX Graph
A graph.
Returns comp : generator of lists
A list of graphs, one for each strongly connected component of G.
copy : boolean
if copy is True, Graph, node, and edge attributes are copied to the subgraphs.
See also:
connected_component_subgraphs
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strongly_connected_components_recursive
strongly_connected_components_recursive(G)
Generate nodes in strongly connected components of graph.
Recursive version of algorithm.
Parameters G : NetworkX Graph
An directed graph.
Returns comp : generator of lists
A list of nodes for each component of G. The list is ordered from largest connected
component to smallest.
Raises NetworkXNotImplemented : If G is undirected
See also:
connected_components
Notes
Uses Tarjan’s algorithm with Nuutila’s modifications.
References
[R241], [R242]
kosaraju_strongly_connected_components
kosaraju_strongly_connected_components(G, source=None)
Generate nodes in strongly connected components of graph.
Parameters G : NetworkX Graph
An directed graph.
Returns comp : generator of lists
A list of nodes for each component of G.
Raises NetworkXNotImplemented: If G is undirected.
See also:
connected_components
Notes
Uses Kosaraju’s algorithm.
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condensation
condensation(G, scc=None)
Returns the condensation of G.
The condensation of G is the graph with each of the strongly connected components contracted into a single
node.
Parameters G : NetworkX DiGraph
A directed graph.
scc: list or generator (optional, default=None)
Strongly connected components.
If provided, the elements in рќ‘ рќ‘ђрќ‘ђ must
partition the nodes in рќђє.
If not provided, it will be calculated as
scc=nx.strongly_connected_components(G).
Returns C : NetworkX DiGraph
The condensation graph C of G. The node labels are integers corresponding to the index
of the component in the list of strongly connected components of G. C has a graph
attribute named �mapping’ with a dictionary mapping the original nodes to the nodes in
C to which they belong. Each node in C also has a node attribute �members’ with the
list of original nodes in G that form the SCC that the node in C represents.
Raises NetworkXNotImplemented: If G is not directed
Notes
After contracting all strongly connected components to a single node, the resulting graph is a directed acyclic
graph.
4.12.3 Weak connectivity
Weakly connected components.
is_weakly_connected(G)
number_weakly_connected_components(G)
weakly_connected_components(G)
weakly_connected_component_subgraphs(G[, copy])
Test directed graph for weak connectivity.
Return the number of connected components in G.
Generate weakly connected components of G.
Generate weakly connected components as subgraphs.
is_weakly_connected
is_weakly_connected(G)
Test directed graph for weak connectivity.
A directed graph is weakly connected if, and only if, the graph is connected when the direction of the edge
between nodes is ignored.
Parameters G : NetworkX Graph
A directed graph.
Returns connected : bool
True if the graph is weakly connected, False otherwise.
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See also:
is_strongly_connected, is_semiconnected, is_connected
Notes
For directed graphs only.
number_weakly_connected_components
number_weakly_connected_components(G)
Return the number of connected components in G. For directed graphs only.
weakly_connected_components
weakly_connected_components(G)
Generate weakly connected components of G.
weakly_connected_component_subgraphs
weakly_connected_component_subgraphs(G, copy=True)
Generate weakly connected components as subgraphs.
Parameters G : NetworkX Graph
A directed graph.
copy : bool
If copy is True, graph, node, and edge attributes are copied to the subgraphs.
4.12.4 Attracting components
Attracting components.
is_attracting_component(G)
number_attracting_components(G)
attracting_components(G)
attracting_component_subgraphs(G[, copy])
Returns True if рќђє consists of a single attracting component.
Returns the number of attracting components in рќђє.
Generates a list of attracting components in рќђє.
Generates a list of attracting component subgraphs from рќђє.
is_attracting_component
is_attracting_component(G)
Returns True if рќђє consists of a single attracting component.
Parameters G : DiGraph, MultiDiGraph
The graph to be analyzed.
Returns attracting : bool
True if рќђє has a single attracting component. Otherwise, False.
See also:
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attracting_components, number_attracting_components, attracting_component_subgraphs
number_attracting_components
number_attracting_components(G)
Returns the number of attracting components in рќђє.
Parameters G : DiGraph, MultiDiGraph
The graph to be analyzed.
Returns n : int
The number of attracting components in G.
See also:
attracting_components, is_attracting_component, attracting_component_subgraphs
attracting_components
attracting_components(G)
Generates a list of attracting components in рќђє.
An attracting component in a directed graph рќђє is a strongly connected component with the property that a
random walker on the graph will never leave the component, once it enters the component.
The nodes in attracting components can also be thought of as recurrent nodes. If a random walker enters the
attractor containing the node, then the node will be visited infinitely often.
Parameters G : DiGraph, MultiDiGraph
The graph to be analyzed.
Returns attractors : generator of list
The list of attracting components, sorted from largest attracting component to smallest
attracting component.
See also:
number_attracting_components,
attracting_component_subgraphs
is_attracting_component,
attracting_component_subgraphs
attracting_component_subgraphs(G, copy=True)
Generates a list of attracting component subgraphs from рќђє.
Parameters G : DiGraph, MultiDiGraph
The graph to be analyzed.
Returns subgraphs : list
A list of node-induced subgraphs of the attracting components of рќђє.
copy : bool
If copy is True, graph, node, and edge attributes are copied to the subgraphs.
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See also:
attracting_components, number_attracting_components, is_attracting_component
4.12.5 Biconnected components
Biconnected components and articulation points.
is_biconnected(G)
biconnected_components(G)
biconnected_component_edges(G)
biconnected_component_subgraphs(G[, copy])
articulation_points(G)
Return True if the graph is biconnected, False otherwise.
Return a generator of sets of nodes, one set for each biconnected
Return a generator of lists of edges, one list for each biconnected compon
Return a generator of graphs, one graph for each biconnected component
Return a generator of articulation points, or cut vertices, of a graph.
is_biconnected
is_biconnected(G)
Return True if the graph is biconnected, False otherwise.
A graph is biconnected if, and only if, it cannot be disconnected by removing only one node (and all edges
incident on that node). If removing a node increases the number of disconnected components in the graph, that
node is called an articulation point, or cut vertex. A biconnected graph has no articulation points.
Parameters G : NetworkX Graph
An undirected graph.
Returns biconnected : bool
True if the graph is biconnected, False otherwise.
Raises NetworkXNotImplemented :
If the input graph is not undirected.
See also:
biconnected_components, articulation_points,
biconnected_component_subgraphs
biconnected_component_edges,
Notes
The algorithm to find articulation points and biconnected components is implemented using a non-recursive
depth-first-search (DFS) that keeps track of the highest level that back edges reach in the DFS tree. A node рќ‘›
is an articulation point if, and only if, there exists a subtree rooted at рќ‘› such that there is no back edge from
any successor of рќ‘› that links to a predecessor of рќ‘› in the DFS tree. By keeping track of all the edges traversed
by the DFS we can obtain the biconnected components because all edges of a bicomponent will be traversed
consecutively between articulation points.
References
[R238]
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Examples
>>> G=nx.path_graph(4)
>>> print(nx.is_biconnected(G))
False
>>> G.add_edge(0,3)
>>> print(nx.is_biconnected(G))
True
biconnected_components
biconnected_components(G)
Return a generator of sets of nodes, one set for each biconnected component of the graph
Biconnected components are maximal subgraphs such that the removal of a node (and all edges incident on that
node) will not disconnect the subgraph. Note that nodes may be part of more than one biconnected component.
Those nodes are articulation points, or cut vertices. The removal of articulation points will increase the number
of connected components of the graph.
Notice that by convention a dyad is considered a biconnected component.
Parameters G : NetworkX Graph
An undirected graph.
Returns nodes : generator
Generator of sets of nodes, one set for each biconnected component.
Raises NetworkXNotImplemented :
If the input graph is not undirected.
See also:
is_biconnected,
articulation_points,
biconnected_component_subgraphs
biconnected_component_edges,
Notes
The algorithm to find articulation points and biconnected components is implemented using a non-recursive
depth-first-search (DFS) that keeps track of the highest level that back edges reach in the DFS tree. A node рќ‘›
is an articulation point if, and only if, there exists a subtree rooted at рќ‘› such that there is no back edge from
any successor of рќ‘› that links to a predecessor of рќ‘› in the DFS tree. By keeping track of all the edges traversed
by the DFS we can obtain the biconnected components because all edges of a bicomponent will be traversed
consecutively between articulation points.
References
[R237]
Examples
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>>> G = nx.barbell_graph(4,2)
>>> print(nx.is_biconnected(G))
False
>>> components = nx.biconnected_components(G)
>>> G.add_edge(2,8)
>>> print(nx.is_biconnected(G))
True
>>> components = nx.biconnected_components(G)
biconnected_component_edges
biconnected_component_edges(G)
Return a generator of lists of edges, one list for each biconnected component of the input graph.
Biconnected components are maximal subgraphs such that the removal of a node (and all edges incident on that
node) will not disconnect the subgraph. Note that nodes may be part of more than one biconnected component. Those nodes are articulation points, or cut vertices. However, each edge belongs to one, and only one,
biconnected component.
Notice that by convention a dyad is considered a biconnected component.
Parameters G : NetworkX Graph
An undirected graph.
Returns edges : generator of lists
Generator of lists of edges, one list for each bicomponent.
Raises NetworkXNotImplemented :
If the input graph is not undirected.
See also:
is_biconnected,
biconnected_components,
biconnected_component_subgraphs
articulation_points,
Notes
The algorithm to find articulation points and biconnected components is implemented using a non-recursive
depth-first-search (DFS) that keeps track of the highest level that back edges reach in the DFS tree. A node рќ‘›
is an articulation point if, and only if, there exists a subtree rooted at рќ‘› such that there is no back edge from
any successor of рќ‘› that links to a predecessor of рќ‘› in the DFS tree. By keeping track of all the edges traversed
by the DFS we can obtain the biconnected components because all edges of a bicomponent will be traversed
consecutively between articulation points.
References
[R235]
Examples
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>>> G = nx.barbell_graph(4,2)
>>> print(nx.is_biconnected(G))
False
>>> components = nx.biconnected_component_edges(G)
>>> G.add_edge(2,8)
>>> print(nx.is_biconnected(G))
True
>>> components = nx.biconnected_component_edges(G)
biconnected_component_subgraphs
biconnected_component_subgraphs(G, copy=True)
Return a generator of graphs, one graph for each biconnected component of the input graph.
Biconnected components are maximal subgraphs such that the removal of a node (and all edges incident on that
node) will not disconnect the subgraph. Note that nodes may be part of more than one biconnected component.
Those nodes are articulation points, or cut vertices. The removal of articulation points will increase the number
of connected components of the graph.
Notice that by convention a dyad is considered a biconnected component.
Parameters G : NetworkX Graph
An undirected graph.
Returns graphs : generator
Generator of graphs, one graph for each biconnected component.
Raises NetworkXNotImplemented :
If the input graph is not undirected.
See also:
is_biconnected,
articulation_points,
biconnected_components
biconnected_component_edges,
Notes
The algorithm to find articulation points and biconnected components is implemented using a non-recursive
depth-first-search (DFS) that keeps track of the highest level that back edges reach in the DFS tree. A node рќ‘›
is an articulation point if, and only if, there exists a subtree rooted at рќ‘› such that there is no back edge from
any successor of рќ‘› that links to a predecessor of рќ‘› in the DFS tree. By keeping track of all the edges traversed
by the DFS we can obtain the biconnected components because all edges of a bicomponent will be traversed
consecutively between articulation points.
Graph, node, and edge attributes are copied to the subgraphs.
References
[R236]
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Examples
>>> G = nx.barbell_graph(4,2)
>>> print(nx.is_biconnected(G))
False
>>> subgraphs = list(nx.biconnected_component_subgraphs(G))
articulation_points
articulation_points(G)
Return a generator of articulation points, or cut vertices, of a graph.
An articulation point or cut vertex is any node whose removal (along with all its incident edges) increases the
number of connected components of a graph. An undirected connected graph without articulation points is
biconnected. Articulation points belong to more than one biconnected component of a graph.
Notice that by convention a dyad is considered a biconnected component.
Parameters G : NetworkX Graph
An undirected graph.
Returns articulation points : generator
generator of nodes
Raises NetworkXNotImplemented :
If the input graph is not undirected.
See also:
is_biconnected,
biconnected_components,
biconnected_component_subgraphs
biconnected_component_edges,
Notes
The algorithm to find articulation points and biconnected components is implemented using a non-recursive
depth-first-search (DFS) that keeps track of the highest level that back edges reach in the DFS tree. A node рќ‘›
is an articulation point if, and only if, there exists a subtree rooted at рќ‘› such that there is no back edge from
any successor of рќ‘› that links to a predecessor of рќ‘› in the DFS tree. By keeping track of all the edges traversed
by the DFS we can obtain the biconnected components because all edges of a bicomponent will be traversed
consecutively between articulation points.
References
[R234]
Examples
>>> G = nx.barbell_graph(4,2)
>>> print(nx.is_biconnected(G))
False
>>> list(nx.articulation_points(G))
[6, 5, 4, 3]
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>>> G.add_edge(2,8)
>>> print(nx.is_biconnected(G))
True
>>> list(nx.articulation_points(G))
[]
4.12.6 Semiconnectedness
Semiconnectedness.
is_semiconnected(G)
Return True if the graph is semiconnected, False otherwise.
is_semiconnected
is_semiconnected(G)
Return True if the graph is semiconnected, False otherwise.
A graph is semiconnected if, and only if, for any pair of nodes, either one is reachable from the other, or they
are mutually reachable.
Parameters G : NetworkX graph
A directed graph.
Returns semiconnected : bool
True if the graph is semiconnected, False otherwise.
Raises NetworkXNotImplemented :
If the input graph is not directed.
NetworkXPointlessConcept :
If the graph is empty.
See also:
is_strongly_connected, is_weakly_connected
Examples
>>> G=nx.path_graph(4,create_using=nx.DiGraph())
>>> print(nx.is_semiconnected(G))
True
>>> G=nx.DiGraph([(1, 2), (3, 2)])
>>> print(nx.is_semiconnected(G))
False
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Connectivity and cut algorithms
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4.13.1 Flow-based Connectivity
Flow based connectivity algorithms
average_node_connectivity(G[, flow_func])
all_pairs_node_connectivity(G[, nbunch, ...])
edge_connectivity(G[, s, t, flow_func])
local_edge_connectivity(G, u, v[, ...])
local_node_connectivity(G, s, t[, ...])
node_connectivity(G[, s, t, flow_func])
Returns the average connectivity of a graph G.
Compute node connectivity between all pairs of nodes of G.
Returns the edge connectivity of the graph or digraph G.
Returns local edge connectivity for nodes s and t in G.
Computes local node connectivity for nodes s and t.
Returns node connectivity for a graph or digraph G.
average_node_connectivity
average_node_connectivity(G, flow_func=None)
Returns the average connectivity of a graph G.
The average connectivity рќњ…
ВЇ of a graph G is the average of local node connectivity over all pairs of nodes of G
[R243] .
∑︀
рќ‘ў,рќ‘Ј рќњ…рќђє (рќ‘ў, рќ‘Ј)
(пёЂрќ‘›)пёЂ
рќњ…
ВЇ (рќђє) =
2
Parameters G : NetworkX graph
Undirected graph
flow_func : function
A function for computing the maximum flow among a pair of nodes. The function has to
accept at least three parameters: a Digraph, a source node, and a target node. And return
a residual network that follows NetworkX conventions (see maximum_flow() for details). If flow_func is None, the default maximum flow function (edmonds_karp())
is used. See local_node_connectivity() for details. The choice of the default
function may change from version to version and should not be relied on. Default value:
None.
Returns K : float
Average node connectivity
See also:
local_node_connectivity(),
node_connectivity(),
edge_connectivity(),
maximum_flow(), edmonds_karp(), preflow_push(), shortest_augmenting_path()
References
[R243]
all_pairs_node_connectivity
all_pairs_node_connectivity(G, nbunch=None, flow_func=None)
Compute node connectivity between all pairs of nodes of G.
Parameters G : NetworkX graph
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Undirected graph
nbunch: container
Container of nodes. If provided node connectivity will be computed only over pairs of
nodes in nbunch.
flow_func : function
A function for computing the maximum flow among a pair of nodes. The function has to
accept at least three parameters: a Digraph, a source node, and a target node. And return
a residual network that follows NetworkX conventions (see maximum_flow() for details). If flow_func is None, the default maximum flow function (edmonds_karp())
is used. See below for details. The choice of the default function may change from
version to version and should not be relied on. Default value: None.
Returns all_pairs : dict
A dictionary with node connectivity between all pairs of nodes in G, or in nbunch if
provided.
See also:
local_node_connectivity(), edge_connectivity(), local_edge_connectivity(),
maximum_flow(), edmonds_karp(), preflow_push(), shortest_augmenting_path()
edge_connectivity
edge_connectivity(G, s=None, t=None, flow_func=None)
Returns the edge connectivity of the graph or digraph G.
The edge connectivity is equal to the minimum number of edges that must be removed to disconnect G or render
it trivial. If source and target nodes are provided, this function returns the local edge connectivity: the minimum
number of edges that must be removed to break all paths from source to target in G.
Parameters G : NetworkX graph
Undirected or directed graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes. The function has to
accept at least three parameters: a Digraph, a source node, and a target node. And return
a residual network that follows NetworkX conventions (see maximum_flow() for details). If flow_func is None, the default maximum flow function (edmonds_karp())
is used. See below for details. The choice of the default function may change from
version to version and should not be relied on. Default value: None.
Returns K : integer
Edge connectivity for G, or local edge connectivity if source and target were provided
See also:
local_edge_connectivity(), local_node_connectivity(), node_connectivity(),
maximum_flow(), edmonds_karp(), preflow_push(), shortest_augmenting_path()
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Notes
This is a flow based implementation of global edge connectivity. For undirected graphs the algorithm works
by finding a �small’ dominating set of nodes of G (see algorithm 7 in [R244] ) and computing local maximum
flow (see local_edge_connectivity()) between an arbitrary node in the dominating set and the rest of
nodes in it. This is an implementation of algorithm 6 in [R244] . For directed graphs, the algorithm does n calls
to the maximum flow function. This is an implementation of algorithm 8 in [R244] .
References
[R244]
Examples
>>> # Platonic icosahedral graph is 5-edge-connected
>>> G = nx.icosahedral_graph()
>>> nx.edge_connectivity(G)
5
You can use alternative flow algorithms for the underlying maximum flow computation. In dense networks the algorithm shortest_augmenting_path() will usually perform better than the default
edmonds_karp(), which is faster for sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> nx.edge_connectivity(G, flow_func=shortest_augmenting_path)
5
If you specify a pair of nodes (source and target) as parameters, this function returns the value of local edge
connectivity.
>>> nx.edge_connectivity(G, 3, 7)
5
If you need to perform several local computations among different pairs of nodes on the same graph,
it is recommended that you reuse the data structures used in the maximum flow computations. See
local_edge_connectivity() for details.
local_edge_connectivity
local_edge_connectivity(G, u, v, flow_func=None, auxiliary=None, residual=None, cutoff=None)
Returns local edge connectivity for nodes s and t in G.
Local edge connectivity for two nodes s and t is the minimum number of edges that must be removed to disconnect them.
This is a flow based implementation of edge connectivity. We compute the maximum flow on an auxiliary
digraph build from the original network (see below for details). This is equal to the local edge connectivity
because the value of a maximum s-t-flow is equal to the capacity of a minimum s-t-cut (Ford and Fulkerson
theorem) [R245] .
Parameters G : NetworkX graph
Undirected or directed graph
s : node
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Source node
t : node
Target node
flow_func : function
A function for computing the maximum flow among a pair of nodes. The function has to
accept at least three parameters: a Digraph, a source node, and a target node. And return
a residual network that follows NetworkX conventions (see maximum_flow() for details). If flow_func is None, the default maximum flow function (edmonds_karp())
is used. See below for details. The choice of the default function may change from
version to version and should not be relied on. Default value: None.
auxiliary : NetworkX DiGraph
Auxiliary digraph for computing flow based edge connectivity. If provided it will be
reused instead of recreated. Default value: None.
residual : NetworkX DiGraph
Residual network to compute maximum flow. If provided it will be reused instead of
recreated. Default value: None.
cutoff : integer, float
If specified, the maximum flow algorithm will terminate when the flow value reaches
or exceeds the cutoff. This is only for the algorithms that support the cutoff parameter: edmonds_karp() and shortest_augmenting_path(). Other algorithms
will ignore this parameter. Default value: None.
Returns K : integer
local edge connectivity for nodes s and t.
See also:
edge_connectivity(),
local_node_connectivity(),
node_connectivity(),
maximum_flow(), edmonds_karp(), preflow_push(), shortest_augmenting_path()
Notes
This is a flow based implementation of edge connectivity. We compute the maximum flow using, by default, the
edmonds_karp() algorithm on an auxiliary digraph build from the original input graph:
If the input graph is undirected, we replace each edge (𝑢,�v�) with two reciprocal arcs (𝑢, 𝑣) and (𝑣, 𝑢) and
then we set the attribute �capacity’ for each arc to 1. If the input graph is directed we simply add the �capacity’
attribute. This is an implementation of algorithm 1 in [R245].
The maximum flow in the auxiliary network is equal to the local edge connectivity because the value of a
maximum s-t-flow is equal to the capacity of a minimum s-t-cut (Ford and Fulkerson theorem).
References
[R245]
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Examples
This function is not imported in the base NetworkX namespace, so you have to explicitly import it from the
connectivity package:
>>> from networkx.algorithms.connectivity import local_edge_connectivity
We use in this example the platonic icosahedral graph, which has edge connectivity 5.
>>> G = nx.icosahedral_graph()
>>> local_edge_connectivity(G, 0, 6)
5
If you need to compute local connectivity on several pairs of nodes in the same graph, it is recommended that
you reuse the data structures that NetworkX uses in the computation: the auxiliary digraph for edge connectivity,
and the residual network for the underlying maximum flow computation.
Example of how to compute local edge connectivity among all pairs of nodes of the platonic icosahedral graph
reusing the data structures.
>>> import itertools
>>> # You also have to explicitly import the function for
>>> # building the auxiliary digraph from the connectivity package
>>> from networkx.algorithms.connectivity import (
...
build_auxiliary_edge_connectivity)
>>> H = build_auxiliary_edge_connectivity(G)
>>> # And the function for building the residual network from the
>>> # flow package
>>> from networkx.algorithms.flow import build_residual_network
>>> # Note that the auxiliary digraph has an edge attribute named capacity
>>> R = build_residual_network(H, 'capacity')
>>> result = dict.fromkeys(G, dict())
>>> # Reuse the auxiliary digraph and the residual network by passing them
>>> # as parameters
>>> for u, v in itertools.combinations(G, 2):
...
k = local_edge_connectivity(G, u, v, auxiliary=H, residual=R)
...
result[u][v] = k
>>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))
True
You can also use alternative flow algorithms for computing edge connectivity. For instance, in dense
networks the algorithm shortest_augmenting_path() will usually perform better than the default
edmonds_karp() which is faster for sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> local_edge_connectivity(G, 0, 6, flow_func=shortest_augmenting_path)
5
local_node_connectivity
local_node_connectivity(G, s, t, flow_func=None, auxiliary=None, residual=None, cutoff=None)
Computes local node connectivity for nodes s and t.
Local node connectivity for two non adjacent nodes s and t is the minimum number of nodes that must be
removed (along with their incident edges) to disconnect them.
This is a flow based implementation of node connectivity. We compute the maximum flow on an auxiliary
digraph build from the original input graph (see below for details).
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Parameters G : NetworkX graph
Undirected graph
s : node
Source node
t : node
Target node
flow_func : function
A function for computing the maximum flow among a pair of nodes. The function has to
accept at least three parameters: a Digraph, a source node, and a target node. And return
a residual network that follows NetworkX conventions (see maximum_flow() for details). If flow_func is None, the default maximum flow function (edmonds_karp())
is used. See below for details. The choice of the default function may change from
version to version and should not be relied on. Default value: None.
auxiliary : NetworkX DiGraph
Auxiliary digraph to compute flow based node connectivity. It has to have a graph
attribute called mapping with a dictionary mapping node names in G and in the auxiliary
digraph. If provided it will be reused instead of recreated. Default value: None.
residual : NetworkX DiGraph
Residual network to compute maximum flow. If provided it will be reused instead of
recreated. Default value: None.
cutoff : integer, float
If specified, the maximum flow algorithm will terminate when the flow value reaches
or exceeds the cutoff. This is only for the algorithms that support the cutoff parameter: edmonds_karp() and shortest_augmenting_path(). Other algorithms
will ignore this parameter. Default value: None.
Returns K : integer
local node connectivity for nodes s and t
See also:
local_edge_connectivity(),
node_connectivity(),
minimum_node_cut(),
maximum_flow(), edmonds_karp(), preflow_push(), shortest_augmenting_path()
Notes
This is a flow based implementation of node connectivity. We compute the maximum flow using, by default,
the edmonds_karp() algorithm (see: maximum_flow()) on an auxiliary digraph build from the original
input graph:
For an undirected graph G having рќ‘› nodes and рќ‘љ edges we derive a directed graph H with 2рќ‘› nodes and 2рќ‘љ + рќ‘›
arcs by replacing each original node рќ‘Ј with two nodes рќ‘Јрќђґ , рќ‘Јрќђµ linked by an (internal) arc in H. Then for each
edge (рќ‘ў, рќ‘Ј) in G we add two arcs (рќ‘ўрќђµ , рќ‘Јрќђґ ) and (рќ‘Јрќђµ , рќ‘ўрќђґ ) in H. Finally we set the attribute capacity = 1 for each
arc in H [R247] .
For a directed graph G having рќ‘› nodes and рќ‘љ arcs we derive a directed graph H with 2рќ‘› nodes and рќ‘љ + рќ‘› arcs
by replacing each original node рќ‘Ј with two nodes рќ‘Јрќђґ , рќ‘Јрќђµ linked by an (internal) arc (рќ‘Јрќђґ , рќ‘Јрќђµ ) in H. Then for
each arc (рќ‘ў, рќ‘Ј) in G we add one arc (рќ‘ўрќђµ , рќ‘Јрќђґ ) in H. Finally we set the attribute capacity = 1 for each arc in H.
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This is equal to the local node connectivity because the value of a maximum s-t-flow is equal to the capacity of
a minimum s-t-cut.
References
[R247]
Examples
This function is not imported in the base NetworkX namespace, so you have to explicitly import it from the
connectivity package:
>>> from networkx.algorithms.connectivity import local_node_connectivity
We use in this example the platonic icosahedral graph, which has node connectivity 5.
>>> G = nx.icosahedral_graph()
>>> local_node_connectivity(G, 0, 6)
5
If you need to compute local connectivity on several pairs of nodes in the same graph, it is recommended that
you reuse the data structures that NetworkX uses in the computation: the auxiliary digraph for node connectivity,
and the residual network for the underlying maximum flow computation.
Example of how to compute local node connectivity among all pairs of nodes of the platonic icosahedral graph
reusing the data structures.
>>> import itertools
>>> # You also have to explicitly import the function for
>>> # building the auxiliary digraph from the connectivity package
>>> from networkx.algorithms.connectivity import (
...
build_auxiliary_node_connectivity)
>>> H = build_auxiliary_node_connectivity(G)
>>> # And the function for building the residual network from the
>>> # flow package
>>> from networkx.algorithms.flow import build_residual_network
>>> # Note that the auxiliary digraph has an edge attribute named capacity
>>> R = build_residual_network(H, 'capacity')
>>> result = dict.fromkeys(G, dict())
>>> # Reuse the auxiliary digraph and the residual network by passing them
>>> # as parameters
>>> for u, v in itertools.combinations(G, 2):
...
k = local_node_connectivity(G, u, v, auxiliary=H, residual=R)
...
result[u][v] = k
>>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))
True
You can also use alternative flow algorithms for computing node connectivity. For instance, in dense
networks the algorithm shortest_augmenting_path() will usually perform better than the default
edmonds_karp() which is faster for sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> local_node_connectivity(G, 0, 6, flow_func=shortest_augmenting_path)
5
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node_connectivity
node_connectivity(G, s=None, t=None, flow_func=None)
Returns node connectivity for a graph or digraph G.
Node connectivity is equal to the minimum number of nodes that must be removed to disconnect G or render it
trivial. If source and target nodes are provided, this function returns the local node connectivity: the minimum
number of nodes that must be removed to break all paths from source to target in G.
Parameters G : NetworkX graph
Undirected graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes. The function has to
accept at least three parameters: a Digraph, a source node, and a target node. And return
a residual network that follows NetworkX conventions (see maximum_flow() for details). If flow_func is None, the default maximum flow function (edmonds_karp())
is used. See below for details. The choice of the default function may change from
version to version and should not be relied on. Default value: None.
Returns K : integer
Node connectivity of G, or local node connectivity if source and target are provided.
See also:
local_node_connectivity(),
edge_connectivity(),
edmonds_karp(), preflow_push(), shortest_augmenting_path()
maximum_flow(),
Notes
This is a flow based implementation of node connectivity. The algorithm works by solving рќ‘‚((рќ‘› в€’ рќ›ї в€’ 1 + рќ›ї(рќ›ї в€’
1)/2) maximum flow problems on an auxiliary digraph. Where рќ›ї is the minimum degree of G. For details about
the auxiliary digraph and the computation of local node connectivity see local_node_connectivity().
This implementation is based on algorithm 11 in [R248].
References
[R248]
Examples
>>> # Platonic icosahedral graph is 5-node-connected
>>> G = nx.icosahedral_graph()
>>> nx.node_connectivity(G)
5
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You can use alternative flow algorithms for the underlying maximum flow computation. In dense networks the algorithm shortest_augmenting_path() will usually perform better than the default
edmonds_karp(), which is faster for sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> nx.node_connectivity(G, flow_func=shortest_augmenting_path)
5
If you specify a pair of nodes (source and target) as parameters, this function returns the value of local node
connectivity.
>>> nx.node_connectivity(G, 3, 7)
5
If you need to perform several local computations among different pairs of nodes on the same graph,
it is recommended that you reuse the data structures used in the maximum flow computations. See
local_node_connectivity() for details.
4.13.2 Flow-based Minimum Cuts
Flow based cut algorithms
minimum_edge_cut(G[, s, t, flow_func])
minimum_node_cut(G[, s, t, flow_func])
minimum_st_edge_cut(G, s, t[, flow_func, ...])
minimum_st_node_cut(G, s, t[, flow_func, ...])
Returns a set of edges of minimum cardinality that disconnects G.
Returns a set of nodes of minimum cardinality that disconnects G.
Returns the edges of the cut-set of a minimum (s, t)-cut.
Returns a set of nodes of minimum cardinality that disconnect source from targ
minimum_edge_cut
minimum_edge_cut(G, s=None, t=None, flow_func=None)
Returns a set of edges of minimum cardinality that disconnects G.
If source and target nodes are provided, this function returns the set of edges of minimum cardinality that, if
removed, would break all paths among source and target in G. If not, it returns a set of edges of minimum
cardinality that disconnects G.
Parameters G : NetworkX graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes. The function has to
accept at least three parameters: a Digraph, a source node, and a target node. And return
a residual network that follows NetworkX conventions (see maximum_flow() for details). If flow_func is None, the default maximum flow function (edmonds_karp())
is used. See below for details. The choice of the default function may change from
version to version and should not be relied on. Default value: None.
Returns cutset : set
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Set of edges that, if removed, would disconnect G. If source and target nodes are provided, the set contians the edges that if removed, would destroy all paths between source
and target.
See also:
minimum_st_edge_cut(),
minimum_node_cut(),
node_connectivity(), edge_connectivity(), maximum_flow(),
preflow_push(), shortest_augmenting_path()
stoer_wagner(),
edmonds_karp(),
Notes
This is a flow based implementation of minimum edge cut. For undirected graphs the algorithm works by finding
a �small’ dominating set of nodes of G (see algorithm 7 in [R249]) and computing the maximum flow between
an arbitrary node in the dominating set and the rest of nodes in it. This is an implementation of algorithm 6 in
[R249]. For directed graphs, the algorithm does n calls to the max flow function. It is an implementation of
algorithm 8 in [R249].
References
[R249]
Examples
>>> # Platonic icosahedral graph has edge connectivity 5
>>> G = nx.icosahedral_graph()
>>> len(nx.minimum_edge_cut(G))
5
You can use alternative flow algorithms for the underlying maximum flow computation. In dense networks the algorithm shortest_augmenting_path() will usually perform better than the default
edmonds_karp(), which is faster for sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> len(nx.minimum_edge_cut(G, flow_func=shortest_augmenting_path))
5
If you specify a pair of nodes (source and target) as parameters, this function returns the value of local edge
connectivity.
>>> nx.edge_connectivity(G, 3, 7)
5
If you need to perform several local computations among different pairs of nodes on the same graph,
it is recommended that you reuse the data structures used in the maximum flow computations. See
local_edge_connectivity() for details.
minimum_node_cut
minimum_node_cut(G, s=None, t=None, flow_func=None)
Returns a set of nodes of minimum cardinality that disconnects G.
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If source and target nodes are provided, this function returns the set of nodes of minimum cardinality that, if
removed, would destroy all paths among source and target in G. If not, it returns a set of nodes of minimum
cardinality that disconnects G.
Parameters G : NetworkX graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes. The function has to
accept at least three parameters: a Digraph, a source node, and a target node. And return
a residual network that follows NetworkX conventions (see maximum_flow() for details). If flow_func is None, the default maximum flow function (edmonds_karp())
is used. See below for details. The choice of the default function may change from
version to version and should not be relied on. Default value: None.
Returns cutset : set
Set of nodes that, if removed, would disconnect G. If source and target nodes are provided, the set contians the nodes that if removed, would destroy all paths between source
and target.
See also:
minimum_st_node_cut(), minimum_cut(), minimum_edge_cut(),
node_connectivity(), edge_connectivity(), maximum_flow(),
preflow_push(), shortest_augmenting_path()
stoer_wagner(),
edmonds_karp(),
Notes
This is a flow based implementation of minimum node cut. The algorithm is based in solving a number of
maximum flow computations to determine the capacity of the minimum cut on an auxiliary directed network that
corresponds to the minimum node cut of G. It handles both directed and undirected graphs. This implementation
is based on algorithm 11 in [R250].
References
[R250]
Examples
>>>
>>>
>>>
>>>
5
# Platonic icosahedral graph has node connectivity 5
G = nx.icosahedral_graph()
node_cut = nx.minimum_node_cut(G)
len(node_cut)
You can use alternative flow algorithms for the underlying maximum flow computation. In dense networks the algorithm shortest_augmenting_path() will usually perform better than the default
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edmonds_karp(), which is faster for sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> node_cut == nx.minimum_node_cut(G, flow_func=shortest_augmenting_path)
True
If you specify a pair of nodes (source and target) as parameters, this function returns a local st node cut.
>>> len(nx.minimum_node_cut(G, 3, 7))
5
If you need to perform several local st cuts among different pairs of nodes on the same graph, it is recommended
that you reuse the data structures used in the maximum flow computations. See minimum_st_node_cut()
for details.
minimum_st_edge_cut
minimum_st_edge_cut(G, s, t, flow_func=None, auxiliary=None, residual=None)
Returns the edges of the cut-set of a minimum (s, t)-cut.
This function returns the set of edges of minimum cardinality that, if removed, would destroy all paths among
source and target in G. Edge weights are not considered
Parameters G : NetworkX graph
Edges of the graph are expected to have an attribute called �capacity’. If this attribute is
not present, the edge is considered to have infinite capacity.
s : node
Source node for the flow.
t : node
Sink node for the flow.
auxiliary : NetworkX DiGraph
Auxiliary digraph to compute flow based node connectivity. It has to have a graph
attribute called mapping with a dictionary mapping node names in G and in the auxiliary
digraph. If provided it will be reused instead of recreated. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes. The function has to
accept at least three parameters: a Digraph, a source node, and a target node. And return
a residual network that follows NetworkX conventions (see maximum_flow() for details). If flow_func is None, the default maximum flow function (edmonds_karp())
is used. See node_connectivity() for details. The choice of the default function
may change from version to version and should not be relied on. Default value: None.
residual : NetworkX DiGraph
Residual network to compute maximum flow. If provided it will be reused instead of
recreated. Default value: None.
Returns cutset : set
Set of edges that, if removed from the graph, will disconnect it.
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See also:
minimum_cut(),
minimum_node_cut(),
minimum_edge_cut(),
node_connectivity(), edge_connectivity(), maximum_flow(),
preflow_push(), shortest_augmenting_path()
stoer_wagner(),
edmonds_karp(),
Examples
This function is not imported in the base NetworkX namespace, so you have to explicitly import it from the
connectivity package:
>>> from networkx.algorithms.connectivity import minimum_st_edge_cut
We use in this example the platonic icosahedral graph, which has edge connectivity 5.
>>> G = nx.icosahedral_graph()
>>> len(minimum_st_edge_cut(G, 0, 6))
5
If you need to compute local edge cuts on several pairs of nodes in the same graph, it is recommended that you
reuse the data structures that NetworkX uses in the computation: the auxiliary digraph for edge connectivity,
and the residual network for the underlying maximum flow computation.
Example of how to compute local edge cuts among all pairs of nodes of the platonic icosahedral graph reusing
the data structures.
>>> import itertools
>>> # You also have to explicitly import the function for
>>> # building the auxiliary digraph from the connectivity package
>>> from networkx.algorithms.connectivity import (
...
build_auxiliary_edge_connectivity)
>>> H = build_auxiliary_edge_connectivity(G)
>>> # And the function for building the residual network from the
>>> # flow package
>>> from networkx.algorithms.flow import build_residual_network
>>> # Note that the auxiliary digraph has an edge attribute named capacity
>>> R = build_residual_network(H, 'capacity')
>>> result = dict.fromkeys(G, dict())
>>> # Reuse the auxiliary digraph and the residual network by passing them
>>> # as parameters
>>> for u, v in itertools.combinations(G, 2):
...
k = len(minimum_st_edge_cut(G, u, v, auxiliary=H, residual=R))
...
result[u][v] = k
>>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))
True
You can also use alternative flow algorithms for computing edge cuts. For instance, in dense networks the algorithm shortest_augmenting_path() will usually perform better than the default edmonds_karp()
which is faster for sparse networks with highly skewed degree distributions. Alternative flow functions have to
be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> len(minimum_st_edge_cut(G, 0, 6, flow_func=shortest_augmenting_path))
5
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minimum_st_node_cut
minimum_st_node_cut(G, s, t, flow_func=None, auxiliary=None, residual=None)
Returns a set of nodes of minimum cardinality that disconnect source from target in G.
This function returns the set of nodes of minimum cardinality that, if removed, would destroy all paths among
source and target in G.
Parameters G : NetworkX graph
s : node
Source node.
t : node
Target node.
flow_func : function
A function for computing the maximum flow among a pair of nodes. The function has to
accept at least three parameters: a Digraph, a source node, and a target node. And return
a residual network that follows NetworkX conventions (see maximum_flow() for details). If flow_func is None, the default maximum flow function (edmonds_karp())
is used. See below for details. The choice of the default function may change from
version to version and should not be relied on. Default value: None.
auxiliary : NetworkX DiGraph
Auxiliary digraph to compute flow based node connectivity. It has to have a graph
attribute called mapping with a dictionary mapping node names in G and in the auxiliary
digraph. If provided it will be reused instead of recreated. Default value: None.
residual : NetworkX DiGraph
Residual network to compute maximum flow. If provided it will be reused instead of
recreated. Default value: None.
Returns cutset : set
Set of nodes that, if removed, would destroy all paths between source and target in G.
See also:
minimum_node_cut(), minimum_edge_cut(), stoer_wagner(), node_connectivity(),
edge_connectivity(),
maximum_flow(),
edmonds_karp(),
preflow_push(),
shortest_augmenting_path()
Notes
This is a flow based implementation of minimum node cut. The algorithm is based in solving a number of
maximum flow computations to determine the capacity of the minimum cut on an auxiliary directed network that
corresponds to the minimum node cut of G. It handles both directed and undirected graphs. This implementation
is based on algorithm 11 in [R252].
References
[R252]
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Examples
This function is not imported in the base NetworkX namespace, so you have to explicitly import it from the
connectivity package:
>>> from networkx.algorithms.connectivity import minimum_st_node_cut
We use in this example the platonic icosahedral graph, which has node connectivity 5.
>>> G = nx.icosahedral_graph()
>>> len(minimum_st_node_cut(G, 0, 6))
5
If you need to compute local st cuts between several pairs of nodes in the same graph, it is recommended that
you reuse the data structures that NetworkX uses in the computation: the auxiliary digraph for node connectivity
and node cuts, and the residual network for the underlying maximum flow computation.
Example of how to compute local st node cuts reusing the data structures:
>>>
>>>
>>>
...
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
5
# You also have to explicitly import the function for
# building the auxiliary digraph from the connectivity package
from networkx.algorithms.connectivity import (
build_auxiliary_node_connectivity)
H = build_auxiliary_node_connectivity(G)
# And the function for building the residual network from the
# flow package
from networkx.algorithms.flow import build_residual_network
# Note that the auxiliary digraph has an edge attribute named capacity
R = build_residual_network(H, 'capacity')
# Reuse the auxiliary digraph and the residual network by passing them
# as parameters
len(minimum_st_node_cut(G, 0, 6, auxiliary=H, residual=R))
You can also use alternative flow algorithms for computing minimum st node cuts. For instance, in dense
networks the algorithm shortest_augmenting_path() will usually perform better than the default
edmonds_karp() which is faster for sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> len(minimum_st_node_cut(G, 0, 6, flow_func=shortest_augmenting_path))
5
4.13.3 Stoer-Wagner minimum cut
Stoer-Wagner minimum cut algorithm.
stoer_wagner(G[, weight, heap])
Returns the weighted minimum edge cut using the Stoer-Wagner algorithm.
stoer_wagner
stoer_wagner(G, weight=’weight’, heap=<class �networkx.utils.heaps.BinaryHeap’>)
Returns the weighted minimum edge cut using the Stoer-Wagner algorithm.
Determine the minimum edge cut of a connected graph using the Stoer-Wagner algorithm. In weighted cases,
all weights must be nonnegative.
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The running time of the algorithm depends on the type of heaps used:
Type of heap
Binary heap
Fibonacci heap
Pairing heap
Running time
рќ‘‚(рќ‘›(рќ‘љ + рќ‘›) log рќ‘›)
рќ‘‚(рќ‘›рќ‘љ
+ рќ‘›2 log рќ‘›)
в€љ
2 log log рќ‘›
рќ‘‚(2
рќ‘›рќ‘љ + рќ‘›2 log рќ‘›)
Parameters G : NetworkX graph
Edges of the graph are expected to have an attribute named by the weight parameter
below. If this attribute is not present, the edge is considered to have unit weight.
weight : string
Name of the weight attribute of the edges. If the attribute is not present, unit weight is
assumed. Default value: �weight’.
heap : class
Type of heap to be used in the algorithm. It should be a subclass of MinHeap or
implement a compatible interface.
If a stock heap implementation is to be used, BinaryHeap is recommeded over
PairingHeap for Python implementations without optimized attribute accesses (e.g.,
CPython) despite a slower asymptotic running time. For Python implementations with
optimized attribute accesses (e.g., PyPy), PairingHeap provides better performance.
Default value: BinaryHeap.
Returns cut_value : integer or float
The sum of weights of edges in a minimum cut.
partition : pair of node lists
A partitioning of the nodes that defines a minimum cut.
Raises NetworkXNotImplemented
If the graph is directed or a multigraph.
NetworkXError
If the graph has less than two nodes, is not connected or has a negative-weighted edge.
Examples
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
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226
G = nx.Graph()
G.add_edge('x','a', weight=3)
G.add_edge('x','b', weight=1)
G.add_edge('a','c', weight=3)
G.add_edge('b','c', weight=5)
G.add_edge('b','d', weight=4)
G.add_edge('d','e', weight=2)
G.add_edge('c','y', weight=2)
G.add_edge('e','y', weight=3)
cut_value, partition = nx.stoer_wagner(G)
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4.13.4 Utils for flow-based connectivity
Utilities for connectivity package
build_auxiliary_edge_connectivity(G)
build_auxiliary_node_connectivity(G)
Auxiliary digraph for computing flow based edge connectivity
Creates a directed graph D from an undirected graph G to compute flow based
build_auxiliary_edge_connectivity
build_auxiliary_edge_connectivity(G)
Auxiliary digraph for computing flow based edge connectivity
If the input graph is undirected, we replace each edge (𝑢,�v�) with two reciprocal arcs (𝑢, 𝑣) and (𝑣, 𝑢) and
then we set the attribute �capacity’ for each arc to 1. If the input graph is directed we simply add the �capacity’
attribute. Part of algorithm 1 in [R254] .
References
[R254]
build_auxiliary_node_connectivity
build_auxiliary_node_connectivity(G)
Creates a directed graph D from an undirected graph G to compute flow based node connectivity.
For an undirected graph G having рќ‘› nodes and рќ‘љ edges we derive a directed graph D with 2рќ‘› nodes and 2рќ‘љ + рќ‘›
arcs by replacing each original node рќ‘Ј with two nodes рќ‘Јрќђґ, рќ‘Јрќђµ linked by an (internal) arc in D. Then for each
edge (рќ‘ў, рќ‘Ј) in G we add two arcs (рќ‘ўрќђµ, рќ‘Јрќђґ) and (рќ‘Јрќђµ, рќ‘ўрќђґ) in D. Finally we set the attribute capacity = 1 for each
arc in D [R255].
For a directed graph having рќ‘› nodes and рќ‘љ arcs we derive a directed graph D with 2рќ‘› nodes and рќ‘љ + рќ‘› arcs by
replacing each original node рќ‘Ј with two nodes рќ‘Јрќђґ, рќ‘Јрќђµ linked by an (internal) arc (рќ‘Јрќђґ, рќ‘Јрќђµ) in D. Then for each
arc (рќ‘ў, рќ‘Ј) in G we add one arc (рќ‘ўрќђµ, рќ‘Јрќђґ) in D. Finally we set the attribute capacity = 1 for each arc in D.
A dictionary with a mapping between nodes in the original graph and the auxiliary digraph is stored as a graph
attribute: H.graph[’mapping’].
References
[R255]
4.14 Cores
Find the k-cores of a graph.
The k-core is found by recursively pruning nodes with degrees less than k.
See the following reference for details:
An O(m) Algorithm for Cores Decomposition of Networks Vladimir Batagelj and Matjaz Zaversnik, 2003.
http://arxiv.org/abs/cs.DS/0310049
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core_number(G)
k_core(G[, k, core_number])
k_shell(G[, k, core_number])
k_crust(G[, k, core_number])
k_corona(G, k[, core_number])
Return the core number for each vertex.
Return the k-core of G.
Return the k-shell of G.
Return the k-crust of G.
Return the k-corona of G.
4.14.1 core_number
core_number(G)
Return the core number for each vertex.
A k-core is a maximal subgraph that contains nodes of degree k or more.
The core number of a node is the largest value k of a k-core containing that node.
Parameters G : NetworkX graph
A graph or directed graph
Returns core_number : dictionary
A dictionary keyed by node to the core number.
Raises NetworkXError
The k-core is not defined for graphs with self loops or parallel edges.
Notes
Not implemented for graphs with parallel edges or self loops.
For directed graphs the node degree is defined to be the in-degree + out-degree.
References
[R256]
4.14.2 k_core
k_core(G, k=None, core_number=None)
Return the k-core of G.
A k-core is a maximal subgraph that contains nodes of degree k or more.
Parameters G : NetworkX graph
A graph or directed graph
k : int, optional
The order of the core. If not specified return the main core.
core_number : dictionary, optional
Precomputed core numbers for the graph G.
Returns G : NetworkX graph
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The k-core subgraph
Raises NetworkXError
The k-core is not defined for graphs with self loops or parallel edges.
See also:
core_number
Notes
The main core is the core with the largest degree.
Not implemented for graphs with parallel edges or self loops.
For directed graphs the node degree is defined to be the in-degree + out-degree.
Graph, node, and edge attributes are copied to the subgraph.
References
[R257]
4.14.3 k_shell
k_shell(G, k=None, core_number=None)
Return the k-shell of G.
The k-shell is the subgraph of nodes in the k-core but not in the (k+1)-core.
Parameters G : NetworkX graph
A graph or directed graph.
k : int, optional
The order of the shell. If not specified return the main shell.
core_number : dictionary, optional
Precomputed core numbers for the graph G.
Returns G : NetworkX graph
The k-shell subgraph
Raises NetworkXError
The k-shell is not defined for graphs with self loops or parallel edges.
See also:
core_number, k_corona, ---------Shai Carmi, Shlomo Havlin, Scott Kirkpatrick, Yuval Shavitt, and Eran Shir, PNAS July 3, 2007 vol. 104 no.
27 11150-11154
http //www.pnas.org/content/104/27/11150.full
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Notes
This is similar to k_corona but in that case only neighbors in the k-core are considered.
Not implemented for graphs with parallel edges or self loops.
For directed graphs the node degree is defined to be the in-degree + out-degree.
Graph, node, and edge attributes are copied to the subgraph.
4.14.4 k_crust
k_crust(G, k=None, core_number=None)
Return the k-crust of G.
The k-crust is the graph G with the k-core removed.
Parameters G : NetworkX graph
A graph or directed graph.
k : int, optional
The order of the shell. If not specified return the main crust.
core_number : dictionary, optional
Precomputed core numbers for the graph G.
Returns G : NetworkX graph
The k-crust subgraph
Raises NetworkXError
The k-crust is not defined for graphs with self loops or parallel edges.
See also:
core_number
Notes
This definition of k-crust is different than the definition in [R259]. The k-crust in [R259] is equivalent to the
k+1 crust of this algorithm.
Not implemented for graphs with parallel edges or self loops.
For directed graphs the node degree is defined to be the in-degree + out-degree.
Graph, node, and edge attributes are copied to the subgraph.
References
[R259]
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4.14.5 k_corona
k_corona(G, k, core_number=None)
Return the k-corona of G.
The k-corona is the subgraph of nodes in the k-core which have exactly k neighbours in the k-core.
Parameters G : NetworkX graph
A graph or directed graph
k : int
The order of the corona.
core_number : dictionary, optional
Precomputed core numbers for the graph G.
Returns G : NetworkX graph
The k-corona subgraph
Raises NetworkXError
The k-cornoa is not defined for graphs with self loops or parallel edges.
See also:
core_number
Notes
Not implemented for graphs with parallel edges or self loops.
For directed graphs the node degree is defined to be the in-degree + out-degree.
Graph, node, and edge attributes are copied to the subgraph.
References
[R258]
4.15 Cycles
cycle_basis(G[, root])
simple_cycles(G)
find_cycle(G[, source, orientation])
Returns a list of cycles which form a basis for cycles of G.
Find simple cycles (elementary circuits) of a directed graph.
Returns the edges of a cycle found via a directed, depth-first traversal.
4.15.1 cycle_basis
cycle_basis(G, root=None)
Returns a list of cycles which form a basis for cycles of G.
A basis for cycles of a network is a minimal collection of cycles such that any cycle in the network can be written
as a sum of cycles in the basis. Here summation of cycles is defined as “exclusive or” of the edges. Cycle bases
are useful, e.g. when deriving equations for electric circuits using Kirchhoff’s Laws.
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Parameters G : NetworkX Graph
root : node, optional
Specify starting node for basis.
Returns A list of cycle lists. Each cycle list is a list of nodes
which forms a cycle (loop) in G.
See also:
simple_cycles
Notes
This is adapted from algorithm CACM 491 [R260].
References
[R260]
Examples
>>> G=nx.Graph()
>>> G.add_cycle([0,1,2,3])
>>> G.add_cycle([0,3,4,5])
>>> print(nx.cycle_basis(G,0))
[[3, 4, 5, 0], [1, 2, 3, 0]]
4.15.2 simple_cycles
simple_cycles(G)
Find simple cycles (elementary circuits) of a directed graph.
An simple cycle, or elementary circuit, is a closed path where no node appears twice, except that the first and
last node are the same. Two elementary circuits are distinct if they are not cyclic permutations of each other.
This is a nonrecursive, iterator/generator version of Johnson’s algorithm [R261]. There may be better algorithms
for some cases [R262] [R263].
Parameters G : NetworkX DiGraph
A directed graph
Returns cycle_generator: generator
A generator that produces elementary cycles of the graph. Each cycle is a list of nodes
with the first and last nodes being the same.
See also:
cycle_basis
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Notes
The implementation follows pp. 79-80 in [R261].
The time complexity is рќ‘‚((рќ‘› + рќ‘’)(рќ‘ђ + 1)) for рќ‘› nodes, рќ‘’ edges and рќ‘ђ elementary circuits.
To filter the cycles so that they don’t include certain nodes or edges, copy your graph and eliminate those nodes
or edges before calling:
>>> copyG = G.copy()
>>> copyG.remove_nodes_from([1])
>>> copyG.remove_edges_from([(0,1)])
>>> list(nx.simple_cycles(copyG))
[[2], [2, 0], [0]]
References
[R261], [R262], [R263]
Examples
>>> G = nx.DiGraph([(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)])
>>> list(nx.simple_cycles(G))
[[2], [2, 1], [2, 0], [2, 0, 1], [0]]
4.15.3 find_cycle
find_cycle(G, source=None, orientation=’original’)
Returns the edges of a cycle found via a directed, depth-first traversal.
Parameters G : graph
A directed/undirected graph/multigraph.
source : node, list of nodes
The node from which the traversal begins. If None, then a source is chosen arbitrarily
and repeatedly until all edges from each node in the graph are searched.
orientation : �original’ | �reverse’ | �ignore’
For directed graphs and directed multigraphs, edge traversals need not respect the original orientation of the edges. When set to �reverse’, then every edge will be traversed
in the reverse direction. When set to �ignore’, then each directed edge is treated as a
single undirected edge that can be traversed in either direction. For undirected graphs
and undirected multigraphs, this parameter is meaningless and is not consulted by the
algorithm.
Returns edges : directed edges
A list of directed edges indicating the path taken for the loop. If no cycle is found, then
edges will be an empty list. For graphs, an edge is of the form (u, v) where u and v are
the tail and head of the edge as determined by the traversal. For multigraphs, an edge is
of the form (u, v, key), where key is the key of the edge. When the graph is directed,
then u and v are always in the order of the actual directed edge. If orientation is �ignore’,
then an edge takes the form (u, v, key, direction) where direction indicates if the edge
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was followed in the forward (tail to head) or reverse (head to tail) direction. When
the direction is forward, the value of direction is �forward’. When the direction is
reverse, the value of direction is �reverse’.
Examples
In this example, we construct a DAG and find, in the first call, that there are no directed cycles, and so an
exception is raised. In the second call, we ignore edge orientations and find that there is an undirected cycle.
Note that the second call finds a directed cycle while effectively traversing an undirected graph, and so, we
found an “undirected cycle”. This means that this DAG structure does not form a directed tree (which is also
known as a polytree).
>>> import networkx as nx
>>> G = nx.DiGraph([(0,1), (0,2), (1,2)])
>>> try:
...
find_cycle(G, orientation='original')
... except:
...
pass
...
>>> list(find_cycle(G, orientation='ignore'))
[(0, 1, 'forward'), (1, 2, 'forward'), (0, 2, 'reverse')]
4.16 Directed Acyclic Graphs
ancestors(G, source)
descendants(G, source)
topological_sort(G[, nbunch, reverse])
topological_sort_recursive(G[, nbunch, reverse])
is_directed_acyclic_graph(G)
is_aperiodic(G)
Return all nodes having a path to рќ‘ рќ‘њрќ‘ўрќ‘џрќ‘ђрќ‘’ in G.
Return all nodes reachable from рќ‘ рќ‘њрќ‘ўрќ‘џрќ‘ђрќ‘’ in G.
Return a list of nodes in topological sort order.
Return a list of nodes in topological sort order.
Return True if the graph G is a directed acyclic graph (DAG) or False
Return True if G is aperiodic.
4.16.1 ancestors
ancestors(G, source)
Return all nodes having a path to рќ‘ рќ‘њрќ‘ўрќ‘џрќ‘ђрќ‘’ in G.
Parameters G : NetworkX DiGraph
source : node in G
Returns ancestors : set()
The ancestors of source in G
4.16.2 descendants
descendants(G, source)
Return all nodes reachable from рќ‘ рќ‘њрќ‘ўрќ‘џрќ‘ђрќ‘’ in G.
Parameters G : NetworkX DiGraph
source : node in G
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Returns des : set()
The descendants of source in G
4.16.3 topological_sort
topological_sort(G, nbunch=None, reverse=False)
Return a list of nodes in topological sort order.
A topological sort is a nonunique permutation of the nodes such that an edge from u to v implies that u appears
before v in the topological sort order.
Parameters G : NetworkX digraph
A directed graph
nbunch : container of nodes (optional)
Explore graph in specified order given in nbunch
reverse : bool, optional
Return postorder instead of preorder if True. Reverse mode is a bit more efficient.
Raises NetworkXError
Topological sort is defined for directed graphs only. If the graph G is undirected, a
NetworkXError is raised.
NetworkXUnfeasible
If G is not a directed acyclic graph (DAG) no topological sort exists and a NetworkXUnfeasible exception is raised.
See also:
is_directed_acyclic_graph
Notes
This algorithm is based on a description and proof in The Algorithm Design Manual [R265] .
References
[R265]
4.16.4 topological_sort_recursive
topological_sort_recursive(G, nbunch=None, reverse=False)
Return a list of nodes in topological sort order.
A topological sort is a nonunique permutation of the nodes such that an edge from u to v implies that u appears
before v in the topological sort order.
Parameters G : NetworkX digraph
nbunch : container of nodes (optional)
Explore graph in specified order given in nbunch
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reverse : bool, optional
Return postorder instead of preorder if True. Reverse mode is a bit more efficient.
Raises NetworkXError
Topological sort is defined for directed graphs only. If the graph G is undirected, a
NetworkXError is raised.
NetworkXUnfeasible
If G is not a directed acyclic graph (DAG) no topological sort exists and a NetworkXUnfeasible exception is raised.
See also:
topological_sort, is_directed_acyclic_graph
Notes
This is a recursive version of topological sort.
4.16.5 is_directed_acyclic_graph
is_directed_acyclic_graph(G)
Return True if the graph G is a directed acyclic graph (DAG) or False if not.
Parameters G : NetworkX graph
A graph
Returns is_dag : bool
True if G is a DAG, false otherwise
4.16.6 is_aperiodic
is_aperiodic(G)
Return True if G is aperiodic.
A directed graph is aperiodic if there is no integer k > 1 that divides the length of every cycle in the graph.
Parameters G : NetworkX DiGraph
Graph
Returns aperiodic : boolean
True if the graph is aperiodic False otherwise
Raises NetworkXError
If G is not directed
Notes
This uses the method outlined in [R264], which runs in O(m) time given m edges in G. Note that a graph is not
aperiodic if it is acyclic as every integer trivial divides length 0 cycles.
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References
[R264]
4.17 Distance Measures
Graph diameter, radius, eccentricity and other properties.
center(G[, e])
diameter(G[, e])
eccentricity(G[, v, sp])
periphery(G[, e])
radius(G[, e])
Return the center of the graph G.
Return the diameter of the graph G.
Return the eccentricity of nodes in G.
Return the periphery of the graph G.
Return the radius of the graph G.
4.17.1 center
center(G, e=None)
Return the center of the graph G.
The center is the set of nodes with eccentricity equal to radius.
Parameters G : NetworkX graph
A graph
e : eccentricity dictionary, optional
A precomputed dictionary of eccentricities.
Returns c : list
List of nodes in center
4.17.2 diameter
diameter(G, e=None)
Return the diameter of the graph G.
The diameter is the maximum eccentricity.
Parameters G : NetworkX graph
A graph
e : eccentricity dictionary, optional
A precomputed dictionary of eccentricities.
Returns d : integer
Diameter of graph
See also:
eccentricity
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4.17.3 eccentricity
eccentricity(G, v=None, sp=None)
Return the eccentricity of nodes in G.
The eccentricity of a node v is the maximum distance from v to all other nodes in G.
Parameters G : NetworkX graph
A graph
v : node, optional
Return value of specified node
sp : dict of dicts, optional
All pairs shortest path lengths as a dictionary of dictionaries
Returns ecc : dictionary
A dictionary of eccentricity values keyed by node.
4.17.4 periphery
periphery(G, e=None)
Return the periphery of the graph G.
The periphery is the set of nodes with eccentricity equal to the diameter.
Parameters G : NetworkX graph
A graph
e : eccentricity dictionary, optional
A precomputed dictionary of eccentricities.
Returns p : list
List of nodes in periphery
4.17.5 radius
radius(G, e=None)
Return the radius of the graph G.
The radius is the minimum eccentricity.
Parameters G : NetworkX graph
A graph
e : eccentricity dictionary, optional
A precomputed dictionary of eccentricities.
Returns r : integer
Radius of graph
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4.18 Distance-Regular Graphs
is_distance_regular(G)
intersection_array(G)
global_parameters(b, c)
Returns True if the graph is distance regular, False otherwise.
Returns the intersection array of a distance-regular graph.
Return global parameters for a given intersection array.
4.18.1 is_distance_regular
is_distance_regular(G)
Returns True if the graph is distance regular, False otherwise.
A connected graph G is distance-regular if for any nodes x,y and any integers i,j=0,1,...,d (where d is the graph
diameter), the number of vertices at distance i from x and distance j from y depends only on i,j and the graph
distance between x and y, independently of the choice of x and y.
Parameters G: Networkx graph (undirected)
Returns bool
True if the graph is Distance Regular, False otherwise
See also:
intersection_array, global_parameters
Notes
For undirected and simple graphs only
References
[R268], [R269]
Examples
>>> G=nx.hypercube_graph(6)
>>> nx.is_distance_regular(G)
True
4.18.2 intersection_array
intersection_array(G)
Returns the intersection array of a distance-regular graph.
Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d such that for any 2 vertices x,y in G at a
distance i=d(x,y), there are exactly c_i neighbors of y at a distance of i-1 from x and b_i neighbors of y at a
distance of i+1 from x.
A distance regular graph’sintersection array is given by, [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d]
Parameters G: Networkx graph (undirected)
Returns b,c: tuple of lists
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See also:
global_parameters
References
[R267]
Examples
>>> G=nx.icosahedral_graph()
>>> nx.intersection_array(G)
([5, 2, 1], [1, 2, 5])
4.18.3 global_parameters
global_parameters(b, c)
Return global parameters for a given intersection array.
Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d such that for any 2 vertices x,y in G at a
distance i=d(x,y), there are exactly c_i neighbors of y at a distance of i-1 from x and b_i neighbors of y at a
distance of i+1 from x.
Thus, a distance regular graph has the global parameters, [[c_0,a_0,b_0],[c_1,a_1,b_1],......,[c_d,a_d,b_d]] for
the intersection array [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d] where a_i+b_i+c_i=k , k= degree of every vertex.
Parameters b,c: tuple of lists
Returns p : list of three-tuples
See also:
intersection_array
References
[R266]
Examples
>>> G=nx.dodecahedral_graph()
>>> b,c=nx.intersection_array(G)
>>> list(nx.global_parameters(b,c))
[(0, 0, 3), (1, 0, 2), (1, 1, 1), (1, 1, 1), (2, 0, 1), (3, 0, 0)]
4.19 Dominance
Dominance algorithms.
immediate_dominators(G, start)
240
Returns the immediate dominators of all nodes of a directed graph.
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Table 4.50 – continued from previous page
dominance_frontiers(G, start)
Returns the dominance frontiers of all nodes of a directed graph.
4.19.1 immediate_dominators
immediate_dominators(G, start)
Returns the immediate dominators of all nodes of a directed graph.
Parameters G : a DiGraph or MultiDiGraph
The graph where dominance is to be computed.
start : node
The start node of dominance computation.
Returns idom : dict keyed by nodes
A dict containing the immediate dominators of each node reachable from start.
Raises NetworkXNotImplemented
If G is undirected.
NetworkXError
If start is not in G.
Notes
Except for start, the immediate dominators are the parents of their corresponding nodes in the dominator tree.
References
[R271]
Examples
>>> G = nx.DiGraph([(1, 2), (1, 3), (2, 5), (3, 4), (4, 5)])
>>> sorted(nx.immediate_dominators(G, 1).items())
[(1, 1), (2, 1), (3, 1), (4, 3), (5, 1)]
4.19.2 dominance_frontiers
dominance_frontiers(G, start)
Returns the dominance frontiers of all nodes of a directed graph.
Parameters G : a DiGraph or MultiDiGraph
The graph where dominance is to be computed.
start : node
The start node of dominance computation.
Returns df : dict keyed by nodes
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A dict containing the dominance frontiers of each node reachable from start as lists.
Raises NetworkXNotImplemented
If G is undirected.
NetworkXError
If start is not in G.
References
[R270]
Examples
>>> G = nx.DiGraph([(1, 2), (1, 3), (2, 5), (3, 4), (4, 5)])
>>> sorted((u, sorted(df)) for u, df in nx.dominance_frontiers(G, 1).items())
[(1, []), (2, [5]), (3, [5]), (4, [5]), (5, [])]
4.20 Dominating Sets
dominating_set(G[, start_with])
is_dominating_set(G, nbunch)
Finds a dominating set for the graph G.
Checks if nodes in nbunch are a dominating set for G.
4.20.1 dominating_set
dominating_set(G, start_with=None)
Finds a dominating set for the graph G.
A dominating set for a graph 𝐺 = (𝑉, 𝐸) is a node subset 𝐷 of 𝑉 such that every node not in 𝐷 is adjacent to
at least one member of рќђ· [R272].
Parameters G : NetworkX graph
start_with : Node (default=None)
Node to use as a starting point for the algorithm.
Returns D : set
A dominating set for G.
See also:
is_dominating_set
Notes
This function is an implementation of algorithm 7 in [R273] which finds some dominating set, not necessarily
the smallest one.
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References
[R272], [R273]
4.20.2 is_dominating_set
is_dominating_set(G, nbunch)
Checks if nodes in nbunch are a dominating set for G.
A dominating set for a graph 𝐺 = (𝑉, 𝐸) is a node subset 𝐷 of 𝑉 such that every node not in 𝐷 is adjacent to
at least one member of рќђ· [R274].
Parameters G : NetworkX graph
nbunch : Node container
See also:
dominating_set
References
[R274]
4.21 Eulerian
Eulerian circuits and graphs.
is_eulerian(G)
eulerian_circuit(G[, source])
Return True if G is an Eulerian graph, False otherwise.
Return the edges of an Eulerian circuit in G.
4.21.1 is_eulerian
is_eulerian(G)
Return True if G is an Eulerian graph, False otherwise.
An Eulerian graph is a graph with an Eulerian circuit.
Parameters G : graph
A NetworkX Graph
Notes
This implementation requires the graph to be connected (or strongly connected for directed graphs).
Examples
>>> nx.is_eulerian(nx.DiGraph({0:[3], 1:[2], 2:[3], 3:[0, 1]}))
True
>>> nx.is_eulerian(nx.complete_graph(5))
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True
>>> nx.is_eulerian(nx.petersen_graph())
False
4.21.2 eulerian_circuit
eulerian_circuit(G, source=None)
Return the edges of an Eulerian circuit in G.
An Eulerian circuit is a path that crosses every edge in G exactly once and finishes at the starting node.
Parameters G : NetworkX Graph or DiGraph
A directed or undirected graph
source : node, optional
Starting node for circuit.
Returns edges : generator
A generator that produces edges in the Eulerian circuit.
Raises NetworkXError
If the graph is not Eulerian.
See also:
is_eulerian
Notes
Linear time algorithm, adapted from [R275]. General information about Euler tours [R276].
References
[R275], [R276]
Examples
>>> G=nx.complete_graph(3)
>>> list(nx.eulerian_circuit(G))
[(0, 2), (2, 1), (1, 0)]
>>> list(nx.eulerian_circuit(G,source=1))
[(1, 2), (2, 0), (0, 1)]
>>> [u for u,v in nx.eulerian_circuit(G)]
[0, 2, 1]
# nodes in circuit
4.22 Flows
4.22.1 Maximum Flow
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maximum_flow(G, s, t[, capacity, flow_func])
maximum_flow_value(G, s, t[, capacity, ...])
minimum_cut(G, s, t[, capacity, flow_func])
minimum_cut_value(G, s, t[, capacity, flow_func])
Find a maximum single-commodity flow.
Find the value of maximum single-commodity flow.
Compute the value and the node partition of a minimum (s, t)-cut.
Compute the value of a minimum (s, t)-cut.
maximum_flow
maximum_flow(G, s, t, capacity=’capacity’, flow_func=None, **kwargs)
Find a maximum single-commodity flow.
Parameters G : NetworkX graph
Edges of the graph are expected to have an attribute called �capacity’. If this attribute is
not present, the edge is considered to have infinite capacity.
s : node
Source node for the flow.
t : node
Sink node for the flow.
capacity : string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
flow_func : function
A function for computing the maximum flow among a pair of nodes in a capacitated
graph. The function has to accept at least three parameters: a Graph or Digraph, a
source node, and a target node. And return a residual network that follows NetworkX
conventions (see Notes). If flow_func is None, the default maximum flow function
(preflow_push()) is used. See below for alternative algorithms. The choice of
the default function may change from version to version and should not be relied on.
Default value: None.
kwargs : Any other keyword parameter is passed to the function that
computes the maximum flow.
Returns flow_value : integer, float
Value of the maximum flow, i.e., net outflow from the source.
flow_dict : dict
A dictionary containing the value of the flow that went through each edge.
Raises NetworkXError
The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an
instance of one of these two classes, a NetworkXError is raised.
NetworkXUnbounded
If the graph has a path of infinite capacity, the value of a feasible flow on the graph is
unbounded above and the function raises a NetworkXUnbounded.
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See also:
maximum_flow_value(), minimum_cut(), minimum_cut_value(),
preflow_push(), shortest_augmenting_path()
edmonds_karp(),
Notes
The function used in the flow_func paramter has to return a residual network that follows NetworkX conventions:
The residual network R from an input graph G has the same nodes as G. R is a DiGraph that contains a pair of
edges (u, v) and (v, u) iff (u, v) is not a self-loop, and at least one of (u, v) and (v, u) exists in
G.
For each edge (u, v) in R, R[u][v][’capacity’] is equal to the capacity of (u, v) in G if it exists in
G or zero otherwise. If the capacity is infinite, R[u][v][’capacity’] will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in R.graph[’inf’]. For each edge (u,
v) in R, R[u][v][’flow’] represents the flow function of (u, v) and satisfies R[u][v][’flow’]
== -R[v][u][’flow’].
The flow value, defined as the total flow into t, the sink, is stored in R.graph[’flow_value’]. Reachability to t using only edges (u, v) such that R[u][v][’flow’] < R[u][v][’capacity’] induces
a minimum s-t cut.
Specific algorithms may store extra data in R.
The function should supports an optional boolean parameter value_only. When True, it can optionally terminate
the algorithm as soon as the maximum flow value and the minimum cut can be determined.
Examples
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
import networkx as nx
G = nx.DiGraph()
G.add_edge('x','a', capacity=3.0)
G.add_edge('x','b', capacity=1.0)
G.add_edge('a','c', capacity=3.0)
G.add_edge('b','c', capacity=5.0)
G.add_edge('b','d', capacity=4.0)
G.add_edge('d','e', capacity=2.0)
G.add_edge('c','y', capacity=2.0)
G.add_edge('e','y', capacity=3.0)
maximum_flow returns both the value of the maximum flow and a dictionary with all flows.
>>> flow_value, flow_dict = nx.maximum_flow(G, 'x', 'y')
>>> flow_value
3.0
>>> print(flow_dict['x']['b'])
1.0
You can also use alternative algorithms for computing the maximum flow by using the flow_func parameter.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> flow_value == nx.maximum_flow(G, 'x', 'y',
...
flow_func=shortest_augmenting_path)[0]
True
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maximum_flow_value
maximum_flow_value(G, s, t, capacity=’capacity’, flow_func=None, **kwargs)
Find the value of maximum single-commodity flow.
Parameters G : NetworkX graph
Edges of the graph are expected to have an attribute called �capacity’. If this attribute is
not present, the edge is considered to have infinite capacity.
s : node
Source node for the flow.
t : node
Sink node for the flow.
capacity : string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
flow_func : function
A function for computing the maximum flow among a pair of nodes in a capacitated
graph. The function has to accept at least three parameters: a Graph or Digraph, a
source node, and a target node. And return a residual network that follows NetworkX
conventions (see Notes). If flow_func is None, the default maximum flow function
(preflow_push()) is used. See below for alternative algorithms. The choice of
the default function may change from version to version and should not be relied on.
Default value: None.
kwargs : Any other keyword parameter is passed to the function that
computes the maximum flow.
Returns flow_value : integer, float
Value of the maximum flow, i.e., net outflow from the source.
Raises NetworkXError
The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an
instance of one of these two classes, a NetworkXError is raised.
NetworkXUnbounded
If the graph has a path of infinite capacity, the value of a feasible flow on the graph is
unbounded above and the function raises a NetworkXUnbounded.
See also:
maximum_flow(),
minimum_cut(),
minimum_cut_value(),
preflow_push(), shortest_augmenting_path()
edmonds_karp(),
Notes
The function used in the flow_func paramter has to return a residual network that follows NetworkX conventions:
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The residual network R from an input graph G has the same nodes as G. R is a DiGraph that contains a pair of
edges (u, v) and (v, u) iff (u, v) is not a self-loop, and at least one of (u, v) and (v, u) exists in
G.
For each edge (u, v) in R, R[u][v][’capacity’] is equal to the capacity of (u, v) in G if it exists in
G or zero otherwise. If the capacity is infinite, R[u][v][’capacity’] will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in R.graph[’inf’]. For each edge (u,
v) in R, R[u][v][’flow’] represents the flow function of (u, v) and satisfies R[u][v][’flow’]
== -R[v][u][’flow’].
The flow value, defined as the total flow into t, the sink, is stored in R.graph[’flow_value’]. Reachability to t using only edges (u, v) such that R[u][v][’flow’] < R[u][v][’capacity’] induces
a minimum s-t cut.
Specific algorithms may store extra data in R.
The function should supports an optional boolean parameter value_only. When True, it can optionally terminate
the algorithm as soon as the maximum flow value and the minimum cut can be determined.
Examples
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
import networkx as nx
G = nx.DiGraph()
G.add_edge('x','a', capacity=3.0)
G.add_edge('x','b', capacity=1.0)
G.add_edge('a','c', capacity=3.0)
G.add_edge('b','c', capacity=5.0)
G.add_edge('b','d', capacity=4.0)
G.add_edge('d','e', capacity=2.0)
G.add_edge('c','y', capacity=2.0)
G.add_edge('e','y', capacity=3.0)
maximum_flow_value computes only the value of the maximum flow:
>>> flow_value = nx.maximum_flow_value(G, 'x', 'y')
>>> flow_value
3.0
You can also use alternative algorithms for computing the maximum flow by using the flow_func parameter.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> flow_value == nx.maximum_flow_value(G, 'x', 'y',
...
flow_func=shortest_augmenting_path)
True
minimum_cut
minimum_cut(G, s, t, capacity=’capacity’, flow_func=None, **kwargs)
Compute the value and the node partition of a minimum (s, t)-cut.
Use the max-flow min-cut theorem, i.e., the capacity of a minimum capacity cut is equal to the flow value of a
maximum flow.
Parameters G : NetworkX graph
Edges of the graph are expected to have an attribute called �capacity’. If this attribute is
not present, the edge is considered to have infinite capacity.
s : node
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Source node for the flow.
t : node
Sink node for the flow.
capacity : string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
flow_func : function
A function for computing the maximum flow among a pair of nodes in a capacitated
graph. The function has to accept at least three parameters: a Graph or Digraph, a
source node, and a target node. And return a residual network that follows NetworkX
conventions (see Notes). If flow_func is None, the default maximum flow function
(preflow_push()) is used. See below for alternative algorithms. The choice of
the default function may change from version to version and should not be relied on.
Default value: None.
kwargs : Any other keyword parameter is passed to the function that
computes the maximum flow.
Returns cut_value : integer, float
Value of the minimum cut.
partition : pair of node sets
A partitioning of the nodes that defines a minimum cut.
Raises NetworkXUnbounded
If the graph has a path of infinite capacity, all cuts have infinite capacity and the function
raises a NetworkXError.
See also:
maximum_flow(), maximum_flow_value(), minimum_cut_value(),
preflow_push(), shortest_augmenting_path()
edmonds_karp(),
Notes
The function used in the flow_func paramter has to return a residual network that follows NetworkX conventions:
The residual network R from an input graph G has the same nodes as G. R is a DiGraph that contains a pair of
edges (u, v) and (v, u) iff (u, v) is not a self-loop, and at least one of (u, v) and (v, u) exists in
G.
For each edge (u, v) in R, R[u][v][’capacity’] is equal to the capacity of (u, v) in G if it exists in
G or zero otherwise. If the capacity is infinite, R[u][v][’capacity’] will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in R.graph[’inf’]. For each edge (u,
v) in R, R[u][v][’flow’] represents the flow function of (u, v) and satisfies R[u][v][’flow’]
== -R[v][u][’flow’].
The flow value, defined as the total flow into t, the sink, is stored in R.graph[’flow_value’]. Reachability to t using only edges (u, v) such that R[u][v][’flow’] < R[u][v][’capacity’] induces
a minimum s-t cut.
Specific algorithms may store extra data in R.
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The function should supports an optional boolean parameter value_only. When True, it can optionally terminate
the algorithm as soon as the maximum flow value and the minimum cut can be determined.
Examples
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
import networkx as nx
G = nx.DiGraph()
G.add_edge('x','a', capacity
G.add_edge('x','b', capacity
G.add_edge('a','c', capacity
G.add_edge('b','c', capacity
G.add_edge('b','d', capacity
G.add_edge('d','e', capacity
G.add_edge('c','y', capacity
G.add_edge('e','y', capacity
=
=
=
=
=
=
=
=
3.0)
1.0)
3.0)
5.0)
4.0)
2.0)
2.0)
3.0)
minimum_cut computes both the value of the minimum cut and the node partition:
>>> cut_value, partition = nx.minimum_cut(G, 'x', 'y')
>>> reachable, non_reachable = partition
�partition’ here is a tuple with the two sets of nodes that define the minimum cut. You can compute the cut set
of edges that induce the minimum cut as follows:
>>> cutset = set()
>>> for u, nbrs in ((n, G[n]) for n in reachable):
...
cutset.update((u, v) for v in nbrs if v in non_reachable)
>>> print(sorted(cutset))
[('c', 'y'), ('x', 'b')]
>>> cut_value == sum(G.edge[u][v]['capacity'] for (u, v) in cutset)
True
You can also use alternative algorithms for computing the minimum cut by using the flow_func parameter.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> cut_value == nx.minimum_cut(G, 'x', 'y',
...
flow_func=shortest_augmenting_path)[0]
True
minimum_cut_value
minimum_cut_value(G, s, t, capacity=’capacity’, flow_func=None, **kwargs)
Compute the value of a minimum (s, t)-cut.
Use the max-flow min-cut theorem, i.e., the capacity of a minimum capacity cut is equal to the flow value of a
maximum flow.
Parameters G : NetworkX graph
Edges of the graph are expected to have an attribute called �capacity’. If this attribute is
not present, the edge is considered to have infinite capacity.
s : node
Source node for the flow.
t : node
Sink node for the flow.
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capacity : string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
flow_func : function
A function for computing the maximum flow among a pair of nodes in a capacitated
graph. The function has to accept at least three parameters: a Graph or Digraph, a
source node, and a target node. And return a residual network that follows NetworkX
conventions (see Notes). If flow_func is None, the default maximum flow function
(preflow_push()) is used. See below for alternative algorithms. The choice of
the default function may change from version to version and should not be relied on.
Default value: None.
kwargs : Any other keyword parameter is passed to the function that
computes the maximum flow.
Returns cut_value : integer, float
Value of the minimum cut.
Raises NetworkXUnbounded
If the graph has a path of infinite capacity, all cuts have infinite capacity and the function
raises a NetworkXError.
See also:
maximum_flow(),
maximum_flow_value(),
minimum_cut(),
preflow_push(), shortest_augmenting_path()
edmonds_karp(),
Notes
The function used in the flow_func paramter has to return a residual network that follows NetworkX conventions:
The residual network R from an input graph G has the same nodes as G. R is a DiGraph that contains a pair of
edges (u, v) and (v, u) iff (u, v) is not a self-loop, and at least one of (u, v) and (v, u) exists in
G.
For each edge (u, v) in R, R[u][v][’capacity’] is equal to the capacity of (u, v) in G if it exists in
G or zero otherwise. If the capacity is infinite, R[u][v][’capacity’] will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in R.graph[’inf’]. For each edge (u,
v) in R, R[u][v][’flow’] represents the flow function of (u, v) and satisfies R[u][v][’flow’]
== -R[v][u][’flow’].
The flow value, defined as the total flow into t, the sink, is stored in R.graph[’flow_value’]. Reachability to t using only edges (u, v) such that R[u][v][’flow’] < R[u][v][’capacity’] induces
a minimum s-t cut.
Specific algorithms may store extra data in R.
The function should supports an optional boolean parameter value_only. When True, it can optionally terminate
the algorithm as soon as the maximum flow value and the minimum cut can be determined.
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Examples
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
import networkx as nx
G = nx.DiGraph()
G.add_edge('x','a', capacity
G.add_edge('x','b', capacity
G.add_edge('a','c', capacity
G.add_edge('b','c', capacity
G.add_edge('b','d', capacity
G.add_edge('d','e', capacity
G.add_edge('c','y', capacity
G.add_edge('e','y', capacity
=
=
=
=
=
=
=
=
3.0)
1.0)
3.0)
5.0)
4.0)
2.0)
2.0)
3.0)
minimum_cut_value computes only the value of the minimum cut:
>>> cut_value = nx.minimum_cut_value(G, 'x', 'y')
>>> cut_value
3.0
You can also use alternative algorithms for computing the minimum cut by using the flow_func parameter.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> cut_value == nx.minimum_cut_value(G, 'x', 'y',
...
flow_func=shortest_augmenting_path)
True
4.22.2 Edmonds-Karp
edmonds_karp(G, s, t[, capacity, residual, ...])
Find a maximum single-commodity flow using the Edmonds-Karp algorithm.
edmonds_karp
edmonds_karp(G, s, t, capacity=’capacity’, residual=None, value_only=False, cutoff=None)
Find a maximum single-commodity flow using the Edmonds-Karp algorithm.
This function returns the residual network resulting after computing the maximum flow. See below for details
about the conventions NetworkX uses for defining residual networks.
This algorithm has a running time of рќ‘‚(рќ‘›рќ‘љ2 ) for рќ‘› nodes and рќ‘љ edges.
Parameters G : NetworkX graph
Edges of the graph are expected to have an attribute called �capacity’. If this attribute is
not present, the edge is considered to have infinite capacity.
s : node
Source node for the flow.
t : node
Sink node for the flow.
capacity : string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
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residual : NetworkX graph
Residual network on which the algorithm is to be executed. If None, a new residual
network is created. Default value: None.
value_only : bool
If True compute only the value of the maximum flow. This parameter will be ignored
by this algorithm because it is not applicable.
cutoff : integer, float
If specified, the algorithm will terminate when the flow value reaches or exceeds the
cutoff. In this case, it may be unable to immediately determine a minimum cut. Default
value: None.
Returns R : NetworkX DiGraph
Residual network after computing the maximum flow.
Raises NetworkXError
The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an
instance of one of these two classes, a NetworkXError is raised.
NetworkXUnbounded
If the graph has a path of infinite capacity, the value of a feasible flow on the graph is
unbounded above and the function raises a NetworkXUnbounded.
See also:
maximum_flow(), minimum_cut(), preflow_push(), shortest_augmenting_path()
Notes
The residual network R from an input graph G has the same nodes as G. R is a DiGraph that contains a pair of
edges (u, v) and (v, u) iff (u, v) is not a self-loop, and at least one of (u, v) and (v, u) exists in
G.
For each edge (u, v) in R, R[u][v][’capacity’] is equal to the capacity of (u, v) in G if it exists in
G or zero otherwise. If the capacity is infinite, R[u][v][’capacity’] will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in R.graph[’inf’]. For each edge (u,
v) in R, R[u][v][’flow’] represents the flow function of (u, v) and satisfies R[u][v][’flow’]
== -R[v][u][’flow’].
The flow value, defined as the total flow into t, the sink, is stored in R.graph[’flow_value’]. If
cutoff is not specified, reachability to t using only edges (u, v) such that R[u][v][’flow’] <
R[u][v][’capacity’] induces a minimum s-t cut.
Examples
>>> import networkx as nx
>>> from networkx.algorithms.flow import edmonds_karp
The functions that implement flow algorithms and output a residual network, such as this one, are not imported
to the base NetworkX namespace, so you have to explicitly import them from the flow package.
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>>> G = nx.DiGraph()
>>> G.add_edge('x','a', capacity=3.0)
>>> G.add_edge('x','b', capacity=1.0)
>>> G.add_edge('a','c', capacity=3.0)
>>> G.add_edge('b','c', capacity=5.0)
>>> G.add_edge('b','d', capacity=4.0)
>>> G.add_edge('d','e', capacity=2.0)
>>> G.add_edge('c','y', capacity=2.0)
>>> G.add_edge('e','y', capacity=3.0)
>>> R = edmonds_karp(G, 'x', 'y')
>>> flow_value = nx.maximum_flow_value(G, 'x', 'y')
>>> flow_value
3.0
>>> flow_value == R.graph['flow_value']
True
4.22.3 Shortest Augmenting Path
shortest_augmenting_path(G, s, t[, ...])
Find a maximum single-commodity flow using the shortest augmenting path algori
shortest_augmenting_path
shortest_augmenting_path(G, s, t, capacity=’capacity’, residual=None, value_only=False,
two_phase=False, cutoff=None)
Find a maximum single-commodity flow using the shortest augmenting path algorithm.
This function returns the residual network resulting after computing the maximum flow. See below for details
about the conventions NetworkX uses for defining residual networks.
This algorithm has a running time of рќ‘‚(рќ‘›2 рќ‘љ) for рќ‘› nodes and рќ‘љ edges.
Parameters G : NetworkX graph
Edges of the graph are expected to have an attribute called �capacity’. If this attribute is
not present, the edge is considered to have infinite capacity.
s : node
Source node for the flow.
t : node
Sink node for the flow.
capacity : string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
residual : NetworkX graph
Residual network on which the algorithm is to be executed. If None, a new residual
network is created. Default value: None.
value_only : bool
If True compute only the value of the maximum flow. This parameter will be ignored
by this algorithm because it is not applicable.
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two_phase : bool
If True, a two-phase variant is used. The two-phase variant improves the running time on
unit-capacity networks from рќ‘‚(рќ‘›рќ‘љ) to рќ‘‚(min(рќ‘›2/3 , рќ‘љ1/2 )рќ‘љ). Default value: False.
cutoff : integer, float
If specified, the algorithm will terminate when the flow value reaches or exceeds the
cutoff. In this case, it may be unable to immediately determine a minimum cut. Default
value: None.
Returns R : NetworkX DiGraph
Residual network after computing the maximum flow.
Raises NetworkXError
The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an
instance of one of these two classes, a NetworkXError is raised.
NetworkXUnbounded
If the graph has a path of infinite capacity, the value of a feasible flow on the graph is
unbounded above and the function raises a NetworkXUnbounded.
See also:
maximum_flow(), minimum_cut(), edmonds_karp(), preflow_push()
Notes
The residual network R from an input graph G has the same nodes as G. R is a DiGraph that contains a pair of
edges (u, v) and (v, u) iff (u, v) is not a self-loop, and at least one of (u, v) and (v, u) exists in
G.
For each edge (u, v) in R, R[u][v][’capacity’] is equal to the capacity of (u, v) in G if it exists in
G or zero otherwise. If the capacity is infinite, R[u][v][’capacity’] will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in R.graph[’inf’]. For each edge (u,
v) in R, R[u][v][’flow’] represents the flow function of (u, v) and satisfies R[u][v][’flow’]
== -R[v][u][’flow’].
The flow value, defined as the total flow into t, the sink, is stored in R.graph[’flow_value’]. If
cutoff is not specified, reachability to t using only edges (u, v) such that R[u][v][’flow’] <
R[u][v][’capacity’] induces a minimum s-t cut.
Examples
>>> import networkx as nx
>>> from networkx.algorithms.flow import shortest_augmenting_path
The functions that implement flow algorithms and output a residual network, such as this one, are not imported
to the base NetworkX namespace, so you have to explicitly import them from the flow package.
>>>
>>>
>>>
>>>
>>>
>>>
G = nx.DiGraph()
G.add_edge('x','a',
G.add_edge('x','b',
G.add_edge('a','c',
G.add_edge('b','c',
G.add_edge('b','d',
4.22. Flows
capacity=3.0)
capacity=1.0)
capacity=3.0)
capacity=5.0)
capacity=4.0)
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>>> G.add_edge('d','e', capacity=2.0)
>>> G.add_edge('c','y', capacity=2.0)
>>> G.add_edge('e','y', capacity=3.0)
>>> R = shortest_augmenting_path(G, 'x', 'y')
>>> flow_value = nx.maximum_flow_value(G, 'x', 'y')
>>> flow_value
3.0
>>> flow_value == R.graph['flow_value']
True
4.22.4 Preflow-Push
preflow_push(G, s, t[, capacity, residual, ...])
Find a maximum single-commodity flow using the highest-label preflow-push algo
preflow_push
preflow_push(G, s, t, capacity=’capacity’, residual=None, global_relabel_freq=1, value_only=False)
Find a maximum single-commodity flow using the highest-label preflow-push algorithm.
This function returns the residual network resulting after computing the maximum flow. See below for details
about the conventions NetworkX uses for defining residual networks.
в€љ
This algorithm has a running time of рќ‘‚(рќ‘›2 рќ‘љ) for рќ‘› nodes and рќ‘љ edges.
Parameters G : NetworkX graph
Edges of the graph are expected to have an attribute called �capacity’. If this attribute is
not present, the edge is considered to have infinite capacity.
s : node
Source node for the flow.
t : node
Sink node for the flow.
capacity : string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
residual : NetworkX graph
Residual network on which the algorithm is to be executed. If None, a new residual
network is created. Default value: None.
global_relabel_freq : integer, float
Relative frequency of applying the global relabeling heuristic to speed up the algorithm.
If it is None, the heuristic is disabled. Default value: 1.
value_only : bool
If False, compute a maximum flow; otherwise, compute a maximum preflow which is
enough for computing the maximum flow value. Default value: False.
Returns R : NetworkX DiGraph
Residual network after computing the maximum flow.
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Raises NetworkXError
The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an
instance of one of these two classes, a NetworkXError is raised.
NetworkXUnbounded
If the graph has a path of infinite capacity, the value of a feasible flow on the graph is
unbounded above and the function raises a NetworkXUnbounded.
See also:
maximum_flow(), minimum_cut(), edmonds_karp(), shortest_augmenting_path()
Notes
The residual network R from an input graph G has the same nodes as G. R is a DiGraph that contains a pair of
edges (u, v) and (v, u) iff (u, v) is not a self-loop, and at least one of (u, v) and (v, u) exists in
G. For each node u in R, R.node[u][’excess’] represents the difference between flow into u and flow out
of u.
For each edge (u, v) in R, R[u][v][’capacity’] is equal to the capacity of (u, v) in G if it exists in
G or zero otherwise. If the capacity is infinite, R[u][v][’capacity’] will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in R.graph[’inf’]. For each edge (u,
v) in R, R[u][v][’flow’] represents the flow function of (u, v) and satisfies R[u][v][’flow’]
== -R[v][u][’flow’].
The flow value, defined as the total flow into t, the sink, is stored in R.graph[’flow_value’]. Reachability to t using only edges (u, v) such that R[u][v][’flow’] < R[u][v][’capacity’] induces
a minimum s-t cut.
Examples
>>> import networkx as nx
>>> from networkx.algorithms.flow import preflow_push
The functions that implement flow algorithms and output a residual network, such as this one, are not imported
to the base NetworkX namespace, so you have to explicitly import them from the flow package.
>>> G = nx.DiGraph()
>>> G.add_edge('x','a', capacity=3.0)
>>> G.add_edge('x','b', capacity=1.0)
>>> G.add_edge('a','c', capacity=3.0)
>>> G.add_edge('b','c', capacity=5.0)
>>> G.add_edge('b','d', capacity=4.0)
>>> G.add_edge('d','e', capacity=2.0)
>>> G.add_edge('c','y', capacity=2.0)
>>> G.add_edge('e','y', capacity=3.0)
>>> R = preflow_push(G, 'x', 'y')
>>> flow_value = nx.maximum_flow_value(G, 'x', 'y')
>>> flow_value == R.graph['flow_value']
True
>>> # preflow_push also stores the maximum flow value
>>> # in the excess attribute of the sink node t
>>> flow_value == R.node['y']['excess']
True
>>> # For some problems, you might only want to compute a
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>>> # maximum preflow.
>>> R = preflow_push(G, 'x', 'y', value_only=True)
>>> flow_value == R.graph['flow_value']
True
>>> flow_value == R.node['y']['excess']
True
4.22.5 Utils
build_residual_network(G, capacity)
Build a residual network and initialize a zero flow.
build_residual_network
build_residual_network(G, capacity)
Build a residual network and initialize a zero flow.
The residual network R from an input graph G has the same nodes as G. R is a DiGraph that contains a pair of
edges (u, v) and (v, u) iff (u, v) is not a self-loop, and at least one of (u, v) and (v, u) exists in
G.
For each edge (u, v) in R, R[u][v][’capacity’] is equal to the capacity of (u, v) in G if it exists in
G or zero otherwise. If the capacity is infinite, R[u][v][’capacity’] will have a high arbitrary finite value
that does not affect the solution of the problem. This value is stored in R.graph[’inf’]. For each edge (u,
v) in R, R[u][v][’flow’] represents the flow function of (u, v) and satisfies R[u][v][’flow’]
== -R[v][u][’flow’].
The flow value, defined as the total flow into t, the sink, is stored in R.graph[’flow_value’]. If
cutoff is not specified, reachability to t using only edges (u, v) such that R[u][v][’flow’] <
R[u][v][’capacity’] induces a minimum s-t cut.
4.22.6 Network Simplex
network_simplex(G[, demand, capacity, weight])
min_cost_flow_cost(G[, demand, capacity, weight])
min_cost_flow(G[, demand, capacity, weight])
cost_of_flow(G, flowDict[, weight])
max_flow_min_cost(G, s, t[, capacity, weight])
Find a minimum cost flow satisfying all demands in digraph G.
Find the cost of a minimum cost flow satisfying all demands in digraph G
Return a minimum cost flow satisfying all demands in digraph G.
Compute the cost of the flow given by flowDict on graph G.
Return a maximum (s, t)-flow of minimum cost.
network_simplex
network_simplex(G, demand=’demand’, capacity=’capacity’, weight=’weight’)
Find a minimum cost flow satisfying all demands in digraph G.
This is a primal network simplex algorithm that uses the leaving arc rule to prevent cycling.
G is a digraph with edge costs and capacities and in which nodes have demand, i.e., they want to send or receive
some amount of flow. A negative demand means that the node wants to send flow, a positive demand means
that the node want to receive flow. A flow on the digraph G satisfies all demand if the net flow into each node is
equal to the demand of that node.
Parameters G : NetworkX graph
DiGraph on which a minimum cost flow satisfying all demands is to be found.
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demand: string
Nodes of the graph G are expected to have an attribute demand that indicates how much
flow a node wants to send (negative demand) or receive (positive demand). Note that the
sum of the demands should be 0 otherwise the problem in not feasible. If this attribute
is not present, a node is considered to have 0 demand. Default value: �demand’.
capacity: string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
weight: string
Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered
to be 0. Default value: �weight’.
Returns flowCost: integer, float
Cost of a minimum cost flow satisfying all demands.
flowDict: dictionary
Dictionary of dictionaries keyed by nodes such that flowDict[u][v] is the flow edge (u,
v).
Raises NetworkXError
This exception is raised if the input graph is not directed, not connected or is a multigraph.
NetworkXUnfeasible
This exception is raised in the following situations:
• The sum of the demands is not zero. Then, there is no flow satisfying all demands.
• There is no flow satisfying all demand.
NetworkXUnbounded
This exception is raised if the digraph G has a cycle of negative cost and infinite capacity.
Then, the cost of a flow satisfying all demands is unbounded below.
See also:
cost_of_flow, max_flow_min_cost, min_cost_flow, min_cost_flow_cost
Notes
This algorithm is not guaranteed to work if edge weights are floating point numbers (overflows and roundoff
errors can cause problems).
References
[R277], [R278]
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Examples
A simple example of a min cost flow problem.
>>> import networkx as nx
>>> G = nx.DiGraph()
>>> G.add_node('a', demand=-5)
>>> G.add_node('d', demand=5)
>>> G.add_edge('a', 'b', weight=3, capacity=4)
>>> G.add_edge('a', 'c', weight=6, capacity=10)
>>> G.add_edge('b', 'd', weight=1, capacity=9)
>>> G.add_edge('c', 'd', weight=2, capacity=5)
>>> flowCost, flowDict = nx.network_simplex(G)
>>> flowCost
24
>>> flowDict
{'a': {'c': 1, 'b': 4}, 'c': {'d': 1}, 'b': {'d': 4}, 'd': {}}
The mincost flow algorithm can also be used to solve shortest path problems. To find the shortest path between
two nodes u and v, give all edges an infinite capacity, give node u a demand of -1 and node v a demand a 1. Then
run the network simplex. The value of a min cost flow will be the distance between u and v and edges carrying
positive flow will indicate the path.
>>> G=nx.DiGraph()
>>> G.add_weighted_edges_from([('s', 'u' ,10), ('s' ,'x' ,5),
...
('u', 'v' ,1), ('u' ,'x' ,2),
...
('v', 'y' ,1), ('x' ,'u' ,3),
...
('x', 'v' ,5), ('x' ,'y' ,2),
...
('y', 's' ,7), ('y' ,'v' ,6)])
>>> G.add_node('s', demand = -1)
>>> G.add_node('v', demand = 1)
>>> flowCost, flowDict = nx.network_simplex(G)
>>> flowCost == nx.shortest_path_length(G, 's', 'v', weight='weight')
True
>>> sorted([(u, v) for u in flowDict for v in flowDict[u] if flowDict[u][v] > 0])
[('s', 'x'), ('u', 'v'), ('x', 'u')]
>>> nx.shortest_path(G, 's', 'v', weight = 'weight')
['s', 'x', 'u', 'v']
It is possible to change the name of the attributes used for the algorithm.
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
...
...
>>>
260
G = nx.DiGraph()
G.add_node('p', spam=-4)
G.add_node('q', spam=2)
G.add_node('a', spam=-2)
G.add_node('d', spam=-1)
G.add_node('t', spam=2)
G.add_node('w', spam=3)
G.add_edge('p', 'q', cost=7, vacancies=5)
G.add_edge('p', 'a', cost=1, vacancies=4)
G.add_edge('q', 'd', cost=2, vacancies=3)
G.add_edge('t', 'q', cost=1, vacancies=2)
G.add_edge('a', 't', cost=2, vacancies=4)
G.add_edge('d', 'w', cost=3, vacancies=4)
G.add_edge('t', 'w', cost=4, vacancies=1)
flowCost, flowDict = nx.network_simplex(G, demand='spam',
capacity='vacancies',
weight='cost')
flowCost
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>>> flowDict
{'a': {'t': 4}, 'd': {'w': 2}, 'q': {'d': 1}, 'p': {'q': 2, 'a': 2}, 't': {'q': 1, 'w': 1}, 'w':
min_cost_flow_cost
min_cost_flow_cost(G, demand=’demand’, capacity=’capacity’, weight=’weight’)
Find the cost of a minimum cost flow satisfying all demands in digraph G.
G is a digraph with edge costs and capacities and in which nodes have demand, i.e., they want to send or receive
some amount of flow. A negative demand means that the node wants to send flow, a positive demand means
that the node want to receive flow. A flow on the digraph G satisfies all demand if the net flow into each node is
equal to the demand of that node.
Parameters G : NetworkX graph
DiGraph on which a minimum cost flow satisfying all demands is to be found.
demand: string
Nodes of the graph G are expected to have an attribute demand that indicates how much
flow a node wants to send (negative demand) or receive (positive demand). Note that the
sum of the demands should be 0 otherwise the problem in not feasible. If this attribute
is not present, a node is considered to have 0 demand. Default value: �demand’.
capacity: string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
weight: string
Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered
to be 0. Default value: �weight’.
Returns flowCost: integer, float
Cost of a minimum cost flow satisfying all demands.
Raises NetworkXError
This exception is raised if the input graph is not directed or not connected.
NetworkXUnfeasible
This exception is raised in the following situations:
• The sum of the demands is not zero. Then, there is no flow satisfying all demands.
• There is no flow satisfying all demand.
NetworkXUnbounded
This exception is raised if the digraph G has a cycle of negative cost and infinite capacity.
Then, the cost of a flow satisfying all demands is unbounded below.
See also:
cost_of_flow, max_flow_min_cost, min_cost_flow, network_simplex
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Examples
A simple example of a min cost flow problem.
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
24
import networkx as nx
G = nx.DiGraph()
G.add_node('a', demand = -5)
G.add_node('d', demand = 5)
G.add_edge('a', 'b', weight = 3, capacity
G.add_edge('a', 'c', weight = 6, capacity
G.add_edge('b', 'd', weight = 1, capacity
G.add_edge('c', 'd', weight = 2, capacity
flowCost = nx.min_cost_flow_cost(G)
flowCost
=
=
=
=
4)
10)
9)
5)
min_cost_flow
min_cost_flow(G, demand=’demand’, capacity=’capacity’, weight=’weight’)
Return a minimum cost flow satisfying all demands in digraph G.
G is a digraph with edge costs and capacities and in which nodes have demand, i.e., they want to send or receive
some amount of flow. A negative demand means that the node wants to send flow, a positive demand means
that the node want to receive flow. A flow on the digraph G satisfies all demand if the net flow into each node is
equal to the demand of that node.
Parameters G : NetworkX graph
DiGraph on which a minimum cost flow satisfying all demands is to be found.
demand: string
Nodes of the graph G are expected to have an attribute demand that indicates how much
flow a node wants to send (negative demand) or receive (positive demand). Note that the
sum of the demands should be 0 otherwise the problem in not feasible. If this attribute
is not present, a node is considered to have 0 demand. Default value: �demand’.
capacity: string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
weight: string
Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered
to be 0. Default value: �weight’.
Returns flowDict: dictionary
Dictionary of dictionaries keyed by nodes such that flowDict[u][v] is the flow edge (u,
v).
Raises NetworkXError
This exception is raised if the input graph is not directed or not connected.
NetworkXUnfeasible
This exception is raised in the following situations:
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• The sum of the demands is not zero. Then, there is no flow satisfying all demands.
• There is no flow satisfying all demand.
NetworkXUnbounded
This exception is raised if the digraph G has a cycle of negative cost and infinite capacity.
Then, the cost of a flow satisfying all demands is unbounded below.
See also:
cost_of_flow, max_flow_min_cost, min_cost_flow_cost, network_simplex
Examples
A simple example of a min cost flow problem.
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
import networkx as nx
G = nx.DiGraph()
G.add_node('a', demand = -5)
G.add_node('d', demand = 5)
G.add_edge('a', 'b', weight = 3,
G.add_edge('a', 'c', weight = 6,
G.add_edge('b', 'd', weight = 1,
G.add_edge('c', 'd', weight = 2,
flowDict = nx.min_cost_flow(G)
capacity
capacity
capacity
capacity
=
=
=
=
4)
10)
9)
5)
cost_of_flow
cost_of_flow(G, flowDict, weight=’weight’)
Compute the cost of the flow given by flowDict on graph G.
Note that this function does not check for the validity of the flow flowDict. This function will fail if the graph G
and the flow don’t have the same edge set.
Parameters G : NetworkX graph
DiGraph on which a minimum cost flow satisfying all demands is to be found.
weight: string
Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered
to be 0. Default value: �weight’.
flowDict: dictionary
Dictionary of dictionaries keyed by nodes such that flowDict[u][v] is the flow edge (u,
v).
Returns cost: Integer, float
The total cost of the flow. This is given by the sum over all edges of the product of the
edge’s flow and the edge’s weight.
See also:
max_flow_min_cost, min_cost_flow, min_cost_flow_cost, network_simplex
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max_flow_min_cost
max_flow_min_cost(G, s, t, capacity=’capacity’, weight=’weight’)
Return a maximum (s, t)-flow of minimum cost.
G is a digraph with edge costs and capacities. There is a source node s and a sink node t. This function finds a
maximum flow from s to t whose total cost is minimized.
Parameters G : NetworkX graph
DiGraph on which a minimum cost flow satisfying all demands is to be found.
s: node label
Source of the flow.
t: node label
Destination of the flow.
capacity: string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
weight: string
Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered
to be 0. Default value: �weight’.
Returns flowDict: dictionary
Dictionary of dictionaries keyed by nodes such that flowDict[u][v] is the flow edge (u,
v).
Raises NetworkXError
This exception is raised if the input graph is not directed or not connected.
NetworkXUnbounded
This exception is raised if there is an infinite capacity path from s to t in G. In this case
there is no maximum flow. This exception is also raised if the digraph G has a cycle of
negative cost and infinite capacity. Then, the cost of a flow is unbounded below.
See also:
cost_of_flow, min_cost_flow, min_cost_flow_cost, network_simplex
Examples
>>> G = nx.DiGraph()
>>> G.add_edges_from([(1,
...
(1,
...
(2,
...
(2,
...
(3,
...
(3,
...
(4,
...
(4,
...
(5,
264
2,
3,
3,
6,
4,
5,
2,
5,
7,
{'capacity': 12, 'weight': 4}),
{'capacity': 20, 'weight': 6}),
{'capacity': 6, 'weight': -3}),
{'capacity': 14, 'weight': 1}),
{'weight': 9}),
{'capacity': 10, 'weight': 5}),
{'capacity': 19, 'weight': 13}),
{'capacity': 4, 'weight': 0}),
{'capacity': 28, 'weight': 2}),
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...
(6, 5, {'capacity': 11, 'weight': 1}),
...
(6, 7, {'weight': 8}),
...
(7, 4, {'capacity': 6, 'weight': 6})])
>>> mincostFlow = nx.max_flow_min_cost(G, 1, 7)
>>> mincost = nx.cost_of_flow(G, mincostFlow)
>>> mincost
373
>>> from networkx.algorithms.flow import maximum_flow
>>> maxFlow = maximum_flow(G, 1, 7)[1]
>>> nx.cost_of_flow(G, maxFlow) >= mincost
True
>>> mincostFlowValue = (sum((mincostFlow[u][7] for u in G.predecessors(7)))
...
- sum((mincostFlow[7][v] for v in G.successors(7))))
>>> mincostFlowValue == nx.maximum_flow_value(G, 1, 7)
True
4.22.7 Capacity Scaling Minimum Cost Flow
capacity_scaling(G[, demand, capacity, ...])
Find a minimum cost flow satisfying all demands in digraph G.
capacity_scaling
capacity_scaling(G, demand=’demand’, capacity=’capacity’, weight=’weight’, heap=<class �networkx.utils.heaps.BinaryHeap’>)
Find a minimum cost flow satisfying all demands in digraph G.
This is a capacity scaling successive shortest augmenting path algorithm.
G is a digraph with edge costs and capacities and in which nodes have demand, i.e., they want to send or receive
some amount of flow. A negative demand means that the node wants to send flow, a positive demand means
that the node want to receive flow. A flow on the digraph G satisfies all demand if the net flow into each node is
equal to the demand of that node.
Parameters G : NetworkX graph
DiGraph or MultiDiGraph on which a minimum cost flow satisfying all demands is to
be found.
demand : string
Nodes of the graph G are expected to have an attribute demand that indicates how much
flow a node wants to send (negative demand) or receive (positive demand). Note that the
sum of the demands should be 0 otherwise the problem in not feasible. If this attribute
is not present, a node is considered to have 0 demand. Default value: �demand’.
capacity : string
Edges of the graph G are expected to have an attribute capacity that indicates how much
flow the edge can support. If this attribute is not present, the edge is considered to have
infinite capacity. Default value: �capacity’.
weight : string
Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered
to be 0. Default value: �weight’.
heap : class
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Type of heap to be used in the algorithm. It should be a subclass of MinHeap or
implement a compatible interface.
If a stock heap implementation is to be used, BinaryHeap is recommeded over
PairingHeap for Python implementations without optimized attribute accesses (e.g.,
CPython) despite a slower asymptotic running time. For Python implementations with
optimized attribute accesses (e.g., PyPy), PairingHeap provides better performance.
Default value: BinaryHeap.
Returns flowCost: integer
Cost of a minimum cost flow satisfying all demands.
flowDict: dictionary
Dictionary of dictionaries keyed by nodes such that flowDict[u][v] is the flow edge (u,
v) if G is a digraph.
Dictionary of dictionaries of dictionaries keyed by nodes such that flowDict[u][v][key]
is the flow edge (u, v, key) if G is a multidigraph.
Raises NetworkXError
This exception is raised if the input graph is not directed, not connected.
NetworkXUnfeasible
This exception is raised in the following situations:
• The sum of the demands is not zero. Then, there is no flow satisfying all demands.
• There is no flow satisfying all demand.
NetworkXUnbounded
This exception is raised if the digraph G has a cycle of negative cost and infinite capacity.
Then, the cost of a flow satisfying all demands is unbounded below.
See also:
network_simplex()
Notes
This algorithm does not work if edge weights are floating-point numbers.
Examples
A simple example of a min cost flow problem.
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
24
266
import networkx as nx
G = nx.DiGraph()
G.add_node('a', demand = -5)
G.add_node('d', demand = 5)
G.add_edge('a', 'b', weight = 3, capacity =
G.add_edge('a', 'c', weight = 6, capacity =
G.add_edge('b', 'd', weight = 1, capacity =
G.add_edge('c', 'd', weight = 2, capacity =
flowCost, flowDict = nx.capacity_scaling(G)
flowCost
4)
10)
9)
5)
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>>> flowDict
{'a': {'c': 1, 'b': 4}, 'c': {'d': 1}, 'b': {'d': 4}, 'd': {}}
It is possible to change the name of the attributes used for the algorithm.
>>> G = nx.DiGraph()
>>> G.add_node('p', spam = -4)
>>> G.add_node('q', spam = 2)
>>> G.add_node('a', spam = -2)
>>> G.add_node('d', spam = -1)
>>> G.add_node('t', spam = 2)
>>> G.add_node('w', spam = 3)
>>> G.add_edge('p', 'q', cost = 7, vacancies = 5)
>>> G.add_edge('p', 'a', cost = 1, vacancies = 4)
>>> G.add_edge('q', 'd', cost = 2, vacancies = 3)
>>> G.add_edge('t', 'q', cost = 1, vacancies = 2)
>>> G.add_edge('a', 't', cost = 2, vacancies = 4)
>>> G.add_edge('d', 'w', cost = 3, vacancies = 4)
>>> G.add_edge('t', 'w', cost = 4, vacancies = 1)
>>> flowCost, flowDict = nx.capacity_scaling(G, demand = 'spam',
...
capacity = 'vacancies',
...
weight = 'cost')
>>> flowCost
37
>>> flowDict
{'a': {'t': 4}, 'd': {'w': 2}, 'q': {'d': 1}, 'p': {'q': 2, 'a': 2}, 't': {'q': 1, 'w': 1}, 'w':
4.23 Graphical degree sequence
Test sequences for graphiness.
is_graphical(sequence[, method])
is_digraphical(in_sequence, out_sequence)
is_multigraphical(sequence)
is_pseudographical(sequence)
is_valid_degree_sequence_havel_hakimi(...)
is_valid_degree_sequence_erdos_gallai(...)
Returns True if sequence is a valid degree sequence.
Returns True if some directed graph can realize the in- and out-degree se
Returns True if some multigraph can realize the sequence.
Returns True if some pseudograph can realize the sequence.
Returns True if deg_sequence can be realized by a simple graph.
Returns True if deg_sequence can be realized by a simple graph.
4.23.1 is_graphical
is_graphical(sequence, method=’eg’)
Returns True if sequence is a valid degree sequence.
A degree sequence is valid if some graph can realize it.
Parameters sequence : list or iterable container
A sequence of integer node degrees
method : “eg” | “hh”
The method used to validate the degree sequence. “eg” corresponds to the Erd˝os-Gallai
algorithm, and “hh” to the Havel-Hakimi algorithm.
Returns valid : bool
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True if the sequence is a valid degree sequence and False if not.
References
ErdЛќos-Gallai [EG1960], [choudum1986]
Havel-Hakimi [havel1955], [hakimi1962], [CL1996]
Examples
>>> G = nx.path_graph(4)
>>> sequence = G.degree().values()
>>> nx.is_valid_degree_sequence(sequence)
True
4.23.2 is_digraphical
is_digraphical(in_sequence, out_sequence)
Returns True if some directed graph can realize the in- and out-degree sequences.
Parameters in_sequence : list or iterable container
A sequence of integer node in-degrees
out_sequence : list or iterable container
A sequence of integer node out-degrees
Returns valid : bool
True if in and out-sequences are digraphic False if not.
Notes
This algorithm is from Kleitman and Wang [R279]. The worst case runtime is O(s * log n) where s and n are
the sum and length of the sequences respectively.
References
[R279]
4.23.3 is_multigraphical
is_multigraphical(sequence)
Returns True if some multigraph can realize the sequence.
Parameters deg_sequence : list
A list of integers
Returns valid : bool
True if deg_sequence is a multigraphic degree sequence and False if not.
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Notes
The worst-case run time is O(n) where n is the length of the sequence.
References
[R280]
4.23.4 is_pseudographical
is_pseudographical(sequence)
Returns True if some pseudograph can realize the sequence.
Every nonnegative integer sequence with an even sum is pseudographical (see [R281]).
Parameters sequence : list or iterable container
A sequence of integer node degrees
Returns valid : bool
True if the sequence is a pseudographic degree sequence and False if not.
Notes
The worst-case run time is O(n) where n is the length of the sequence.
References
[R281]
4.23.5 is_valid_degree_sequence_havel_hakimi
is_valid_degree_sequence_havel_hakimi(deg_sequence)
Returns True if deg_sequence can be realized by a simple graph.
The validation proceeds using the Havel-Hakimi theorem. Worst-case run time is: O(s) where s is the sum of
the sequence.
Parameters deg_sequence : list
A list of integers where each element specifies the degree of a node in a graph.
Returns valid : bool
True if deg_sequence is graphical and False if not.
Notes
The ZZ condition says that for the sequence d if
|рќ‘‘| >=
(max(рќ‘‘) + min(рќ‘‘) + 1)2
4 * min(рќ‘‘)
then d is graphical. This was shown in Theorem 6 in [R284].
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References
[havel1955], [hakimi1962], [CL1996]
[R284]
4.23.6 is_valid_degree_sequence_erdos_gallai
is_valid_degree_sequence_erdos_gallai(deg_sequence)
Returns True if deg_sequence can be realized by a simple graph.
The validation is done using the ErdЛќos-Gallai theorem [EG1960].
Parameters deg_sequence : list
A list of integers
Returns valid : bool
True if deg_sequence is graphical and False if not.
Notes
This implementation uses an equivalent form of the ErdЛќos-Gallai criterion. Worst-case run time is: O(n) where
n is the length of the sequence.
Specifically, a sequence d is graphical if and only if the sum of the sequence is even and for all strong indices k
in the sequence,
�
∑︁
𝑑𝑖 ≤ �(� − 1) +
рќ‘–=1
рќ‘›
∑︁
min(𝑑𝑖 , �) = �(𝑛 − 1) − (�
�−1
∑︁
рќ‘—=0
𝑗=�+1
рќ‘›рќ‘— в€’
�−1
∑︁
рќ‘—рќ‘›рќ‘— )
рќ‘—=0
A strong index k is any index where 𝑑� ≥ � and the value 𝑛𝑗 is the number of occurrences of j in d. The
maximal strong index is called the Durfee index.
This particular rearrangement comes from the proof of Theorem 3 in [R283].
The ZZ condition says that for the sequence d if
|рќ‘‘| >=
(max(рќ‘‘) + min(рќ‘‘) + 1)2
4 * min(рќ‘‘)
then d is graphical. This was shown in Theorem 6 in [R283].
References
[EG1960], [choudum1986]
[R282], [R283]
4.24 Hierarchy
Flow Hierarchy.
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flow_hierarchy(G[, weight])
Returns the flow hierarchy of a directed network.
4.24.1 flow_hierarchy
flow_hierarchy(G, weight=None)
Returns the flow hierarchy of a directed network.
Flow hierarchy is defined as the fraction of edges not participating in cycles in a directed graph [R285].
Parameters G : DiGraph or MultiDiGraph
A directed graph
weight : key,optional (default=None)
Attribute to use for node weights. If None the weight defaults to 1.
Returns h : float
Flow heirarchy value
Notes
The algorithm described in [R285] computes the flow hierarchy through exponentiation of the adjacency matrix.
This function implements an alternative approach that finds strongly connected components. An edge is in a
cycle if and only if it is in a strongly connected component, which can be found in 𝑂(𝑚) time using Tarjan’s
algorithm.
References
[R285]
4.25 Isolates
Functions for identifying isolate (degree zero) nodes.
is_isolate(G, n)
isolates(G)
Determine of node n is an isolate (degree zero).
Return list of isolates in the graph.
4.25.1 is_isolate
is_isolate(G, n)
Determine of node n is an isolate (degree zero).
Parameters G : graph
A networkx graph
n : node
A node in G
Returns isolate : bool
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True if n has no neighbors, False otherwise.
Examples
>>> G=nx.Graph()
>>> G.add_edge(1,2)
>>> G.add_node(3)
>>> nx.is_isolate(G,2)
False
>>> nx.is_isolate(G,3)
True
4.25.2 isolates
isolates(G)
Return list of isolates in the graph.
Isolates are nodes with no neighbors (degree zero).
Parameters G : graph
A networkx graph
Returns isolates : list
List of isolate nodes.
Examples
>>>
>>>
>>>
>>>
[3]
G = nx.Graph()
G.add_edge(1,2)
G.add_node(3)
nx.isolates(G)
To remove all isolates in the graph use >>> G.remove_nodes_from(nx.isolates(G)) >>> G.nodes() [1, 2]
For digraphs isolates have zero in-degree and zero out_degre >>> G = nx.DiGraph([(0,1),(1,2)]) >>>
G.add_node(3) >>> nx.isolates(G) [3]
4.26 Isomorphism
is_isomorphic(G1, G2[, node_match, edge_match])
could_be_isomorphic(G1, G2)
fast_could_be_isomorphic(G1, G2)
faster_could_be_isomorphic(G1, G2)
Returns True if the graphs G1 and G2 are isomorphic and False otherwise.
Returns False if graphs are definitely not isomorphic.
Returns False if graphs are definitely not isomorphic.
Returns False if graphs are definitely not isomorphic.
4.26.1 is_isomorphic
is_isomorphic(G1, G2, node_match=None, edge_match=None)
Returns True if the graphs G1 and G2 are isomorphic and False otherwise.
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Parameters G1, G2: graphs
The two graphs G1 and G2 must be the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2 should be considered equal
during the isomorphism test. If node_match is not specified then node attributes are not
considered.
The function will be called like
node_match(G1.node[n1], G2.node[n2]).
That is, the function will receive the node attribute dictionaries for n1 and n2 as inputs.
edge_match : callable
A function that returns True if the edge attribute dictionary for the pair of nodes (u1,
v1) in G1 and (u2, v2) in G2 should be considered equal during the isomorphism test.
If edge_match is not specified then edge attributes are not considered.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute dictionaries of the edges under consideration.
See also:
numerical_node_match,
numerical_edge_match,
numerical_multiedge_match,
categorical_node_match, categorical_edge_match, categorical_multiedge_match
Notes
Uses the vf2 algorithm [R286].
References
[R286]
Examples
>>> import networkx.algorithms.isomorphism as iso
For digraphs G1 and G2, using �weight’ edge attribute (default: 1)
>>> G1 = nx.DiGraph()
>>> G2 = nx.DiGraph()
>>> G1.add_path([1,2,3,4],weight=1)
>>> G2.add_path([10,20,30,40],weight=2)
>>> em = iso.numerical_edge_match('weight', 1)
>>> nx.is_isomorphic(G1, G2) # no weights considered
True
>>> nx.is_isomorphic(G1, G2, edge_match=em) # match weights
False
For multidigraphs G1 and G2, using �fill’ node attribute (default: �’)
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>>> G1 = nx.MultiDiGraph()
>>> G2 = nx.MultiDiGraph()
>>> G1.add_nodes_from([1,2,3],fill='red')
>>> G2.add_nodes_from([10,20,30,40],fill='red')
>>> G1.add_path([1,2,3,4],weight=3, linewidth=2.5)
>>> G2.add_path([10,20,30,40],weight=3)
>>> nm = iso.categorical_node_match('fill', 'red')
>>> nx.is_isomorphic(G1, G2, node_match=nm)
True
For multidigraphs G1 and G2, using �weight’ edge attribute (default: 7)
>>> G1.add_edge(1,2, weight=7)
>>> G2.add_edge(10,20)
>>> em = iso.numerical_multiedge_match('weight', 7, rtol=1e-6)
>>> nx.is_isomorphic(G1, G2, edge_match=em)
True
For multigraphs G1 and G2, using �weight’ and �linewidth’ edge attributes with default values 7 and 2.5. Also
using �fill’ node attribute with default value �red’.
>>> em = iso.numerical_multiedge_match(['weight', 'linewidth'], [7, 2.5])
>>> nm = iso.categorical_node_match('fill', 'red')
>>> nx.is_isomorphic(G1, G2, edge_match=em, node_match=nm)
True
4.26.2 could_be_isomorphic
could_be_isomorphic(G1, G2)
Returns False if graphs are definitely not isomorphic. True does NOT guarantee isomorphism.
Parameters G1, G2 : graphs
The two graphs G1 and G2 must be the same type.
Notes
Checks for matching degree, triangle, and number of cliques sequences.
4.26.3 fast_could_be_isomorphic
fast_could_be_isomorphic(G1, G2)
Returns False if graphs are definitely not isomorphic.
True does NOT guarantee isomorphism.
Parameters G1, G2 : graphs
The two graphs G1 and G2 must be the same type.
Notes
Checks for matching degree and triangle sequences.
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4.26.4 faster_could_be_isomorphic
faster_could_be_isomorphic(G1, G2)
Returns False if graphs are definitely not isomorphic.
True does NOT guarantee isomorphism.
Parameters G1, G2 : graphs
The two graphs G1 and G2 must be the same type.
Notes
Checks for matching degree sequences.
4.26.5 Advanced Interface to VF2 Algorithm
VF2 Algorithm
An implementation of VF2 algorithm for graph ismorphism testing.
The simplest interface to use this module is to call networkx.is_isomorphic().
Introduction
The GraphMatcher and DiGraphMatcher are responsible for matching graphs or directed graphs in a predetermined
manner. This usually means a check for an isomorphism, though other checks are also possible. For example, a
subgraph of one graph can be checked for isomorphism to a second graph.
Matching is done via syntactic feasibility. It is also possible to check for semantic feasibility. Feasibility, then, is
defined as the logical AND of the two functions.
To include a semantic check, the (Di)GraphMatcher class should be subclassed, and the semantic_feasibility() function
should be redefined. By default, the semantic feasibility function always returns True. The effect of this is that
semantics are not considered in the matching of G1 and G2.
Examples
Suppose G1 and G2 are isomorphic graphs. Verification is as follows:
>>> from networkx.algorithms import isomorphism
>>> G1 = nx.path_graph(4)
>>> G2 = nx.path_graph(4)
>>> GM = isomorphism.GraphMatcher(G1,G2)
>>> GM.is_isomorphic()
True
GM.mapping stores the isomorphism mapping from G1 to G2.
>>> GM.mapping
{0: 0, 1: 1, 2: 2, 3: 3}
Suppose G1 and G2 are isomorphic directed graphs graphs. Verification is as follows:
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>>> G1 = nx.path_graph(4, create_using=nx.DiGraph())
>>> G2 = nx.path_graph(4, create_using=nx.DiGraph())
>>> DiGM = isomorphism.DiGraphMatcher(G1,G2)
>>> DiGM.is_isomorphic()
True
DiGM.mapping stores the isomorphism mapping from G1 to G2.
>>> DiGM.mapping
{0: 0, 1: 1, 2: 2, 3: 3}
Subgraph Isomorphism
Graph theory literature can be ambiguious about the meaning of the above statement, and we seek to clarify it now.
In the VF2 literature, a mapping M is said to be a graph-subgraph isomorphism iff M is an isomorphism between G2
and a subgraph of G1. Thus, to say that G1 and G2 are graph-subgraph isomorphic is to say that a subgraph of G1 is
isomorphic to G2.
Other literature uses the phrase �subgraph isomorphic’ as in �G1 does not have a subgraph isomorphic to G2’. Another
use is as an in adverb for isomorphic. Thus, to say that G1 and G2 are subgraph isomorphic is to say that a subgraph
of G1 is isomorphic to G2.
Finally, the term �subgraph’ can have multiple meanings. In this context, �subgraph’ always means a �node-induced
subgraph’. Edge-induced subgraph isomorphisms are not directly supported, but one should be able to perform the
check by making use of nx.line_graph(). For subgraphs which are not induced, the term �monomorphism’ is preferred
over �isomorphism’. Currently, it is not possible to check for monomorphisms.
Let G=(N,E) be a graph with a set of nodes N and set of edges E.
If G’=(N’,E’) is a subgraph, then: N’ is a subset of N E’ is a subset of E
If G’=(N’,E’) is a node-induced subgraph, then: N’ is a subset of N E’ is the subset of edges in E relating nodes in
N’
If G’=(N’,E’) is an edge-induced subgrpah, then: N’ is the subset of nodes in N related by edges in E’ E’ is a subset
of E
References
[1] Luigi P. Cordella, Pasquale Foggia, Carlo Sansone, Mario Vento, “A (Sub)Graph Isomorphism Algorithm for
Matching Large Graphs”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 10, pp.
1367-1372, Oct., 2004. http://ieeexplore.ieee.org/iel5/34/29305/01323804.pdf
[2] L. P. Cordella, P. Foggia, C. Sansone, M. Vento, “An Improved Algorithm for Matching Large Graphs”, 3rd
IAPR-TC15 Workshop on Graph-based Representations in Pattern Recognition, Cuen, pp. 149-159, 2001.
http://amalfi.dis.unina.it/graph/db/papers/vf-algorithm.pdf
See Also
syntactic_feasibliity(), semantic_feasibility()
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Modified to handle undirected graphs. Modified to handle multiple edges.
In general, this problem is NP-Complete.
Graph Matcher
GraphMatcher.__init__(G1, G2[, node_match, ...])
GraphMatcher.initialize()
GraphMatcher.is_isomorphic()
GraphMatcher.subgraph_is_isomorphic()
GraphMatcher.isomorphisms_iter()
GraphMatcher.subgraph_isomorphisms_iter()
GraphMatcher.candidate_pairs_iter()
GraphMatcher.match()
GraphMatcher.semantic_feasibility(G1_node, ...)
GraphMatcher.syntactic_feasibility(G1_node, ...)
Initialize graph matcher.
Reinitializes the state of the algorithm.
Returns True if G1 and G2 are isomorphic graphs.
Returns True if a subgraph of G1 is isomorphic to G2.
Generator over isomorphisms between G1 and G2.
Generator over isomorphisms between a subgraph of G1 and G2.
Iterator over candidate pairs of nodes in G1 and G2.
Extends the isomorphism mapping.
Returns True if mapping G1_node to G2_node is semantically feas
Returns True if adding (G1_node, G2_node) is syntactically feasib
__init__
GraphMatcher.__init__(G1, G2, node_match=None, edge_match=None)
Initialize graph matcher.
Parameters G1, G2: graph
The graphs to be tested.
node_match: callable
A function that returns True iff node n1 in G1 and n2 in G2 should be considered equal
during the isomorphism test. The function will be called like:
node_match(G1.node[n1], G2.node[n2])
That is, the function will receive the node attribute dictionaries of the nodes under consideration. If None, then no attributes are considered when testing for an isomorphism.
edge_match: callable
A function that returns True iff the edge attribute dictionary for the pair of nodes (u1,
v1) in G1 and (u2, v2) in G2 should be considered equal during the isomorphism test.
The function will be called like:
edge_match(G1[u1][v1], G2[u2][v2])
That is, the function will receive the edge attribute dictionaries of the edges under consideration. If None, then no attributes are considered when testing for an isomorphism.
initialize
GraphMatcher.initialize()
Reinitializes the state of the algorithm.
This method should be redefined if using something other than GMState. If only subclassing GraphMatcher, a
redefinition is not necessary.
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is_isomorphic
GraphMatcher.is_isomorphic()
Returns True if G1 and G2 are isomorphic graphs.
subgraph_is_isomorphic
GraphMatcher.subgraph_is_isomorphic()
Returns True if a subgraph of G1 is isomorphic to G2.
isomorphisms_iter
GraphMatcher.isomorphisms_iter()
Generator over isomorphisms between G1 and G2.
subgraph_isomorphisms_iter
GraphMatcher.subgraph_isomorphisms_iter()
Generator over isomorphisms between a subgraph of G1 and G2.
candidate_pairs_iter
GraphMatcher.candidate_pairs_iter()
Iterator over candidate pairs of nodes in G1 and G2.
match
GraphMatcher.match()
Extends the isomorphism mapping.
This function is called recursively to determine if a complete isomorphism can be found between G1 and G2. It
cleans up the class variables after each recursive call. If an isomorphism is found, we yield the mapping.
semantic_feasibility
GraphMatcher.semantic_feasibility(G1_node, G2_node)
Returns True if mapping G1_node to G2_node is semantically feasible.
syntactic_feasibility
GraphMatcher.syntactic_feasibility(G1_node, G2_node)
Returns True if adding (G1_node, G2_node) is syntactically feasible.
This function returns True if it is adding the candidate pair to the current partial isomorphism mapping is
allowable. The addition is allowable if the inclusion of the candidate pair does not make it impossible for an
isomorphism to be found.
DiGraph Matcher
DiGraphMatcher.__init__(G1, G2[, ...])
DiGraphMatcher.initialize()
DiGraphMatcher.is_isomorphic()
DiGraphMatcher.subgraph_is_isomorphic()
DiGraphMatcher.isomorphisms_iter()
DiGraphMatcher.subgraph_isomorphisms_iter()
278
Initialize graph matcher.
Reinitializes the state of the algorithm.
Returns True if G1 and G2 are isomorphic graphs.
Returns True if a subgraph of G1 is isomorphic to G2.
Generator over isomorphisms between G1 and G2.
Generator over isomorphisms between a subgraph of G1 and G2.
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DiGraphMatcher.candidate_pairs_iter()
Iterator over candidate pairs of nodes in G1 and G2.
DiGraphMatcher.match()
Extends the isomorphism mapping.
DiGraphMatcher.semantic_feasibility(G1_node, ...) Returns True if mapping G1_node to G2_node is semantically fe
DiGraphMatcher.syntactic_feasibility(...)
Returns True if adding (G1_node, G2_node) is syntactically feas
__init__
DiGraphMatcher.__init__(G1, G2, node_match=None, edge_match=None)
Initialize graph matcher.
Parameters G1, G2 : graph
The graphs to be tested.
node_match : callable
A function that returns True iff node n1 in G1 and n2 in G2 should be considered equal
during the isomorphism test. The function will be called like:
node_match(G1.node[n1], G2.node[n2])
That is, the function will receive the node attribute dictionaries of the nodes under consideration. If None, then no attributes are considered when testing for an isomorphism.
edge_match : callable
A function that returns True iff the edge attribute dictionary for the pair of nodes (u1,
v1) in G1 and (u2, v2) in G2 should be considered equal during the isomorphism test.
The function will be called like:
edge_match(G1[u1][v1], G2[u2][v2])
That is, the function will receive the edge attribute dictionaries of the edges under consideration. If None, then no attributes are considered when testing for an isomorphism.
initialize
DiGraphMatcher.initialize()
Reinitializes the state of the algorithm.
This method should be redefined if using something other than DiGMState. If only subclassing GraphMatcher,
a redefinition is not necessary.
is_isomorphic
DiGraphMatcher.is_isomorphic()
Returns True if G1 and G2 are isomorphic graphs.
subgraph_is_isomorphic
DiGraphMatcher.subgraph_is_isomorphic()
Returns True if a subgraph of G1 is isomorphic to G2.
isomorphisms_iter
DiGraphMatcher.isomorphisms_iter()
Generator over isomorphisms between G1 and G2.
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subgraph_isomorphisms_iter
DiGraphMatcher.subgraph_isomorphisms_iter()
Generator over isomorphisms between a subgraph of G1 and G2.
candidate_pairs_iter
DiGraphMatcher.candidate_pairs_iter()
Iterator over candidate pairs of nodes in G1 and G2.
match
DiGraphMatcher.match()
Extends the isomorphism mapping.
This function is called recursively to determine if a complete isomorphism can be found between G1 and G2. It
cleans up the class variables after each recursive call. If an isomorphism is found, we yield the mapping.
semantic_feasibility
DiGraphMatcher.semantic_feasibility(G1_node, G2_node)
Returns True if mapping G1_node to G2_node is semantically feasible.
syntactic_feasibility
DiGraphMatcher.syntactic_feasibility(G1_node, G2_node)
Returns True if adding (G1_node, G2_node) is syntactically feasible.
This function returns True if it is adding the candidate pair to the current partial isomorphism mapping is
allowable. The addition is allowable if the inclusion of the candidate pair does not make it impossible for an
isomorphism to be found.
Match helpers
categorical_node_match(attr, default)
categorical_edge_match(attr, default)
categorical_multiedge_match(attr, default)
numerical_node_match(attr, default[, rtol, atol])
numerical_edge_match(attr, default[, rtol, atol])
numerical_multiedge_match(attr, default[, ...])
generic_node_match(attr, default, op)
generic_edge_match(attr, default, op)
generic_multiedge_match(attr, default, op)
Returns a comparison function for a categorical node attribute.
Returns a comparison function for a categorical edge attribute.
Returns a comparison function for a categorical edge attribute.
Returns a comparison function for a numerical node attribute.
Returns a comparison function for a numerical edge attribute.
Returns a comparison function for a numerical edge attribute.
Returns a comparison function for a generic attribute.
Returns a comparison function for a generic attribute.
Returns a comparison function for a generic attribute.
categorical_node_match
categorical_node_match(attr, default)
Returns a comparison function for a categorical node attribute.
The value(s) of the attr(s) must be hashable and comparable via the == operator since they are placed into a
set([]) object. If the sets from G1 and G2 are the same, then the constructed function returns True.
Parameters attr : string | list
The categorical node attribute to compare, or a list of categorical node attributes to
compare.
default : value | list
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The default value for the categorical node attribute, or a list of default values for the
categorical node attributes.
Returns match : function
The customized, categorical рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘љ рќ‘Ћрќ‘Ўрќ‘ђв„Ћ function.
Examples
>>> import networkx.algorithms.isomorphism as iso
>>> nm = iso.categorical_node_match('size', 1)
>>> nm = iso.categorical_node_match(['color', 'size'], ['red', 2])
categorical_edge_match
categorical_edge_match(attr, default)
Returns a comparison function for a categorical edge attribute.
The value(s) of the attr(s) must be hashable and comparable via the == operator since they are placed into a
set([]) object. If the sets from G1 and G2 are the same, then the constructed function returns True.
Parameters attr : string | list
The categorical edge attribute to compare, or a list of categorical edge attributes to
compare.
default : value | list
The default value for the categorical edge attribute, or a list of default values for the
categorical edge attributes.
Returns match : function
The customized, categorical рќ‘’рќ‘‘рќ‘”рќ‘’рќ‘љ рќ‘Ћрќ‘Ўрќ‘ђв„Ћ function.
Examples
>>> import networkx.algorithms.isomorphism as iso
>>> nm = iso.categorical_edge_match('size', 1)
>>> nm = iso.categorical_edge_match(['color', 'size'], ['red', 2])
categorical_multiedge_match
categorical_multiedge_match(attr, default)
Returns a comparison function for a categorical edge attribute.
The value(s) of the attr(s) must be hashable and comparable via the == operator since they are placed into a
set([]) object. If the sets from G1 and G2 are the same, then the constructed function returns True.
Parameters attr : string | list
The categorical edge attribute to compare, or a list of categorical edge attributes to
compare.
default : value | list
The default value for the categorical edge attribute, or a list of default values for the
categorical edge attributes.
Returns match : function
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The customized, categorical рќ‘’рќ‘‘рќ‘”рќ‘’рќ‘љ рќ‘Ћрќ‘Ўрќ‘ђв„Ћ function.
Examples
>>> import networkx.algorithms.isomorphism as iso
>>> nm = iso.categorical_multiedge_match('size', 1)
>>> nm = iso.categorical_multiedge_match(['color', 'size'], ['red', 2])
numerical_node_match
numerical_node_match(attr, default, rtol=1e-05, atol=1e-08)
Returns a comparison function for a numerical node attribute.
The value(s) of the attr(s) must be numerical and sortable. If the sorted list of values from G1 and G2 are the
same within some tolerance, then the constructed function returns True.
Parameters attr : string | list
The numerical node attribute to compare, or a list of numerical node attributes to compare.
default : value | list
The default value for the numerical node attribute, or a list of default values for the
numerical node attributes.
rtol : float
The relative error tolerance.
atol : float
The absolute error tolerance.
Returns match : function
The customized, numerical рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘љ рќ‘Ћрќ‘Ўрќ‘ђв„Ћ function.
Examples
>>> import networkx.algorithms.isomorphism as iso
>>> nm = iso.numerical_node_match('weight', 1.0)
>>> nm = iso.numerical_node_match(['weight', 'linewidth'], [.25, .5])
numerical_edge_match
numerical_edge_match(attr, default, rtol=1e-05, atol=1e-08)
Returns a comparison function for a numerical edge attribute.
The value(s) of the attr(s) must be numerical and sortable. If the sorted list of values from G1 and G2 are the
same within some tolerance, then the constructed function returns True.
Parameters attr : string | list
The numerical edge attribute to compare, or a list of numerical edge attributes to compare.
default : value | list
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The default value for the numerical edge attribute, or a list of default values for the
numerical edge attributes.
rtol : float
The relative error tolerance.
atol : float
The absolute error tolerance.
Returns match : function
The customized, numerical рќ‘’рќ‘‘рќ‘”рќ‘’рќ‘љ рќ‘Ћрќ‘Ўрќ‘ђв„Ћ function.
Examples
>>> import networkx.algorithms.isomorphism as iso
>>> nm = iso.numerical_edge_match('weight', 1.0)
>>> nm = iso.numerical_edge_match(['weight', 'linewidth'], [.25, .5])
numerical_multiedge_match
numerical_multiedge_match(attr, default, rtol=1e-05, atol=1e-08)
Returns a comparison function for a numerical edge attribute.
The value(s) of the attr(s) must be numerical and sortable. If the sorted list of values from G1 and G2 are the
same within some tolerance, then the constructed function returns True.
Parameters attr : string | list
The numerical edge attribute to compare, or a list of numerical edge attributes to compare.
default : value | list
The default value for the numerical edge attribute, or a list of default values for the
numerical edge attributes.
rtol : float
The relative error tolerance.
atol : float
The absolute error tolerance.
Returns match : function
The customized, numerical рќ‘’рќ‘‘рќ‘”рќ‘’рќ‘љ рќ‘Ћрќ‘Ўрќ‘ђв„Ћ function.
Examples
>>> import networkx.algorithms.isomorphism as iso
>>> nm = iso.numerical_multiedge_match('weight', 1.0)
>>> nm = iso.numerical_multiedge_match(['weight', 'linewidth'], [.25, .5])
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generic_node_match
generic_node_match(attr, default, op)
Returns a comparison function for a generic attribute.
The value(s) of the attr(s) are compared using the specified operators. If all the attributes are equal, then the
constructed function returns True.
Parameters attr : string | list
The node attribute to compare, or a list of node attributes to compare.
default : value | list
The default value for the node attribute, or a list of default values for the node attributes.
op : callable | list
The operator to use when comparing attribute values, or a list of operators to use when
comparing values for each attribute.
Returns match : function
The customized, generic рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘љ рќ‘Ћрќ‘Ўрќ‘ђв„Ћ function.
Examples
>>>
>>>
>>>
>>>
>>>
>>>
from
from
from
nm =
nm =
nm =
operator import eq
networkx.algorithms.isomorphism.matchhelpers import close
networkx.algorithms.isomorphism import generic_node_match
generic_node_match('weight', 1.0, close)
generic_node_match('color', 'red', eq)
generic_node_match(['weight', 'color'], [1.0, 'red'], [close, eq])
generic_edge_match
generic_edge_match(attr, default, op)
Returns a comparison function for a generic attribute.
The value(s) of the attr(s) are compared using the specified operators. If all the attributes are equal, then the
constructed function returns True.
Parameters attr : string | list
The edge attribute to compare, or a list of edge attributes to compare.
default : value | list
The default value for the edge attribute, or a list of default values for the edge attributes.
op : callable | list
The operator to use when comparing attribute values, or a list of operators to use when
comparing values for each attribute.
Returns match : function
The customized, generic рќ‘’рќ‘‘рќ‘”рќ‘’рќ‘љ рќ‘Ћрќ‘Ўрќ‘ђв„Ћ function.
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Examples
>>>
>>>
>>>
>>>
>>>
>>>
from
from
from
nm =
nm =
nm =
operator import eq
networkx.algorithms.isomorphism.matchhelpers import close
networkx.algorithms.isomorphism import generic_edge_match
generic_edge_match('weight', 1.0, close)
generic_edge_match('color', 'red', eq)
generic_edge_match(['weight', 'color'], [1.0, 'red'], [close, eq])
generic_multiedge_match
generic_multiedge_match(attr, default, op)
Returns a comparison function for a generic attribute.
The value(s) of the attr(s) are compared using the specified operators. If all the attributes are equal, then the
constructed function returns True. Potentially, the constructed edge_match function can be slow since it must
verify that no isomorphism exists between the multiedges before it returns False.
Parameters attr : string | list
The edge attribute to compare, or a list of node attributes to compare.
default : value | list
The default value for the edge attribute, or a list of default values for the dgeattributes.
op : callable | list
The operator to use when comparing attribute values, or a list of operators to use when
comparing values for each attribute.
Returns match : function
The customized, generic рќ‘’рќ‘‘рќ‘”рќ‘’рќ‘љ рќ‘Ћрќ‘Ўрќ‘ђв„Ћ function.
Examples
>>>
>>>
>>>
>>>
>>>
>>>
...
...
...
from
from
from
nm =
nm =
nm =
operator import eq
networkx.algorithms.isomorphism.matchhelpers import close
networkx.algorithms.isomorphism import generic_node_match
generic_node_match('weight', 1.0, close)
generic_node_match('color', 'red', eq)
generic_node_match(['weight', 'color'],
[1.0, 'red'],
[close, eq])
4.27 Link Analysis
4.27.1 PageRank
PageRank analysis of graph structure.
pagerank(G[, alpha, personalization, ...])
pagerank_numpy(G[, alpha, personalization, ...])
4.27. Link Analysis
Return the PageRank of the nodes in the graph.
Return the PageRank of the nodes in the graph.
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pagerank_scipy(G[, alpha, personalization, ...]) Return the PageRank of the nodes in the graph.
google_matrix(G[, alpha, personalization, ...])
Return the Google matrix of the graph.
pagerank
pagerank(G, alpha=0.85, personalization=None, max_iter=100, tol=1e-06, nstart=None, weight=’weight’,
dangling=None)
Return the PageRank of the nodes in the graph.
PageRank computes a ranking of the nodes in the graph G based on the structure of the incoming links. It was
originally designed as an algorithm to rank web pages.
Parameters G : graph
A NetworkX graph. Undirected graphs will be converted to a directed graph with two
directed edges for each undirected edge.
alpha : float, optional
Damping parameter for PageRank, default=0.85.
personalization: dict, optional
The “personalization vector” consisting of a dictionary with a key for every graph node
and nonzero personalization value for each node. By default, a uniform distribution is
used.
max_iter : integer, optional
Maximum number of iterations in power method eigenvalue solver.
tol : float, optional
Error tolerance used to check convergence in power method solver.
nstart : dictionary, optional
Starting value of PageRank iteration for each node.
weight : key, optional
Edge data key to use as weight. If None weights are set to 1.
dangling: dict, optional
The outedges to be assigned to any “dangling” nodes, i.e., nodes without any outedges.
The dict key is the node the outedge points to and the dict value is the weight of that
outedge. By default, dangling nodes are given outedges according to the personalization
vector (uniform if not specified). This must be selected to result in an irreducible transition matrix (see notes under google_matrix). It may be common to have the dangling
dict to be the same as the personalization dict.
Returns pagerank : dictionary
Dictionary of nodes with PageRank as value
See also:
pagerank_numpy, pagerank_scipy, google_matrix
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Notes
The eigenvector calculation is done by the power iteration method and has no guarantee of convergence. The
iteration will stop after max_iter iterations or an error tolerance of number_of_nodes(G)*tol has been reached.
The PageRank algorithm was designed for directed graphs but this algorithm does not check if the input graph
is directed and will execute on undirected graphs by converting each edge in the directed graph to two edges.
References
[R293], [R294]
Examples
>>> G = nx.DiGraph(nx.path_graph(4))
>>> pr = nx.pagerank(G, alpha=0.9)
pagerank_numpy
pagerank_numpy(G, alpha=0.85, personalization=None, weight=’weight’, dangling=None)
Return the PageRank of the nodes in the graph.
PageRank computes a ranking of the nodes in the graph G based on the structure of the incoming links. It was
originally designed as an algorithm to rank web pages.
Parameters G : graph
A NetworkX graph. Undirected graphs will be converted to a directed graph with two
directed edges for each undirected edge.
alpha : float, optional
Damping parameter for PageRank, default=0.85.
personalization: dict, optional
The “personalization vector” consisting of a dictionary with a key for every graph node
and nonzero personalization value for each node. By default, a uniform distribution is
used.
weight : key, optional
Edge data key to use as weight. If None weights are set to 1.
dangling: dict, optional
The outedges to be assigned to any “dangling” nodes, i.e., nodes without any outedges.
The dict key is the node the outedge points to and the dict value is the weight of that
outedge. By default, dangling nodes are given outedges according to the personalization
vector (uniform if not specified) This must be selected to result in an irreducible transition matrix (see notes under google_matrix). It may be common to have the dangling
dict to be the same as the personalization dict.
Returns pagerank : dictionary
Dictionary of nodes with PageRank as value.
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See also:
pagerank, pagerank_scipy, google_matrix
Notes
The eigenvector calculation uses NumPy’s interface to the LAPACK eigenvalue solvers. This will be the fastest
and most accurate for small graphs.
This implementation works with Multi(Di)Graphs. For multigraphs the weight between two nodes is set to be
the sum of all edge weights between those nodes.
References
[R295], [R296]
Examples
>>> G = nx.DiGraph(nx.path_graph(4))
>>> pr = nx.pagerank_numpy(G, alpha=0.9)
pagerank_scipy
pagerank_scipy(G, alpha=0.85, personalization=None, max_iter=100, tol=1e-06, weight=’weight’, dangling=None)
Return the PageRank of the nodes in the graph.
PageRank computes a ranking of the nodes in the graph G based on the structure of the incoming links. It was
originally designed as an algorithm to rank web pages.
Parameters G : graph
A NetworkX graph. Undirected graphs will be converted to a directed graph with two
directed edges for each undirected edge.
alpha : float, optional
Damping parameter for PageRank, default=0.85.
personalization: dict, optional
The “personalization vector” consisting of a dictionary with a key for every graph node
and nonzero personalization value for each node. By default, a uniform distribution is
used.
max_iter : integer, optional
Maximum number of iterations in power method eigenvalue solver.
tol : float, optional
Error tolerance used to check convergence in power method solver.
weight : key, optional
Edge data key to use as weight. If None weights are set to 1.
dangling: dict, optional
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The outedges to be assigned to any “dangling” nodes, i.e., nodes without any outedges.
The dict key is the node the outedge points to and the dict value is the weight of that
outedge. By default, dangling nodes are given outedges according to the personalization
vector (uniform if not specified) This must be selected to result in an irreducible transition matrix (see notes under google_matrix). It may be common to have the dangling
dict to be the same as the personalization dict.
Returns pagerank : dictionary
Dictionary of nodes with PageRank as value
See also:
pagerank, pagerank_numpy, google_matrix
Notes
The eigenvector calculation uses power iteration with a SciPy sparse matrix representation.
This implementation works with Multi(Di)Graphs. For multigraphs the weight between two nodes is set to be
the sum of all edge weights between those nodes.
References
[R297], [R298]
Examples
>>> G = nx.DiGraph(nx.path_graph(4))
>>> pr = nx.pagerank_scipy(G, alpha=0.9)
google_matrix
google_matrix(G, alpha=0.85, personalization=None,
gling=None)
Return the Google matrix of the graph.
nodelist=None,
weight=’weight’,
dan-
Parameters G : graph
A NetworkX graph. Undirected graphs will be converted to a directed graph with two
directed edges for each undirected edge.
alpha : float
The damping factor.
personalization: dict, optional
The “personalization vector” consisting of a dictionary with a key for every graph node
and nonzero personalization value for each node. By default, a uniform distribution is
used.
nodelist : list, optional
The rows and columns are ordered according to the nodes in nodelist. If nodelist is
None, then the ordering is produced by G.nodes().
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weight : key, optional
Edge data key to use as weight. If None weights are set to 1.
dangling: dict, optional
The outedges to be assigned to any “dangling” nodes, i.e., nodes without any outedges.
The dict key is the node the outedge points to and the dict value is the weight of that
outedge. By default, dangling nodes are given outedges according to the personalization vector (uniform if not specified) This must be selected to result in an irreducible
transition matrix (see notes below). It may be common to have the dangling dict to be
the same as the personalization dict.
Returns A : NumPy matrix
Google matrix of the graph
See also:
pagerank, pagerank_numpy, pagerank_scipy
Notes
The matrix returned represents the transition matrix that describes the Markov chain used in PageRank. For
PageRank to converge to a unique solution (i.e., a unique stationary distribution in a Markov chain), the transition
matrix must be irreducible. In other words, it must be that there exists a path between every pair of nodes in the
graph, or else there is the potential of “rank sinks.”
This implementation works with Multi(Di)Graphs. For multigraphs the weight between two nodes is set to be
the sum of all edge weights between those nodes.
4.27.2 Hits
Hubs and authorities analysis of graph structure.
hits(G[, max_iter, tol, nstart, normalized])
hits_numpy(G[, normalized])
hits_scipy(G[, max_iter, tol, normalized])
hub_matrix(G[, nodelist])
authority_matrix(G[, nodelist])
Return HITS hubs and authorities values for nodes.
Return HITS hubs and authorities values for nodes.
Return HITS hubs and authorities values for nodes.
Return the HITS hub matrix.
Return the HITS authority matrix.
hits
hits(G, max_iter=100, tol=1e-08, nstart=None, normalized=True)
Return HITS hubs and authorities values for nodes.
The HITS algorithm computes two numbers for a node. Authorities estimates the node value based on the
incoming links. Hubs estimates the node value based on outgoing links.
Parameters G : graph
A NetworkX graph
max_iter : interger, optional
Maximum number of iterations in power method.
tol : float, optional
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Error tolerance used to check convergence in power method iteration.
nstart : dictionary, optional
Starting value of each node for power method iteration.
normalized : bool (default=True)
Normalize results by the sum of all of the values.
Returns (hubs,authorities) : two-tuple of dictionaries
Two dictionaries keyed by node containing the hub and authority values.
Notes
The eigenvector calculation is done by the power iteration method and has no guarantee of convergence. The
iteration will stop after max_iter iterations or an error tolerance of number_of_nodes(G)*tol has been reached.
The HITS algorithm was designed for directed graphs but this algorithm does not check if the input graph is
directed and will execute on undirected graphs.
References
[R287], [R288]
Examples
>>> G=nx.path_graph(4)
>>> h,a=nx.hits(G)
hits_numpy
hits_numpy(G, normalized=True)
Return HITS hubs and authorities values for nodes.
The HITS algorithm computes two numbers for a node. Authorities estimates the node value based on the
incoming links. Hubs estimates the node value based on outgoing links.
Parameters G : graph
A NetworkX graph
normalized : bool (default=True)
Normalize results by the sum of all of the values.
Returns (hubs,authorities) : two-tuple of dictionaries
Two dictionaries keyed by node containing the hub and authority values.
Notes
The eigenvector calculation uses NumPy’s interface to LAPACK.
The HITS algorithm was designed for directed graphs but this algorithm does not check if the input graph is
directed and will execute on undirected graphs.
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References
[R289], [R290]
Examples
>>> G=nx.path_graph(4)
>>> h,a=nx.hits(G)
hits_scipy
hits_scipy(G, max_iter=100, tol=1e-06, normalized=True)
Return HITS hubs and authorities values for nodes.
The HITS algorithm computes two numbers for a node. Authorities estimates the node value based on the
incoming links. Hubs estimates the node value based on outgoing links.
Parameters G : graph
A NetworkX graph
max_iter : interger, optional
Maximum number of iterations in power method.
tol : float, optional
Error tolerance used to check convergence in power method iteration.
nstart : dictionary, optional
Starting value of each node for power method iteration.
normalized : bool (default=True)
Normalize results by the sum of all of the values.
Returns (hubs,authorities) : two-tuple of dictionaries
Two dictionaries keyed by node containing the hub and authority values.
Notes
This implementation uses SciPy sparse matrices.
The eigenvector calculation is done by the power iteration method and has no guarantee of convergence. The
iteration will stop after max_iter iterations or an error tolerance of number_of_nodes(G)*tol has been reached.
The HITS algorithm was designed for directed graphs but this algorithm does not check if the input graph is
directed and will execute on undirected graphs.
References
[R291], [R292]
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Examples
>>> G=nx.path_graph(4)
>>> h,a=nx.hits(G)
hub_matrix
hub_matrix(G, nodelist=None)
Return the HITS hub matrix.
authority_matrix
authority_matrix(G, nodelist=None)
Return the HITS authority matrix.
4.28 Link Prediction
Link prediction algorithms.
resource_allocation_index(G[, ebunch])
jaccard_coefficient(G[, ebunch])
adamic_adar_index(G[, ebunch])
preferential_attachment(G[, ebunch])
cn_soundarajan_hopcroft(G[, ebunch, community])
ra_index_soundarajan_hopcroft(G[, ebunch, ...])
within_inter_cluster(G[, ebunch, delta, ...])
Compute the resource allocation index of all node pairs in ebunch.
Compute the Jaccard coefficient of all node pairs in ebunch.
Compute the Adamic-Adar index of all node pairs in ebunch.
Compute the preferential attachment score of all node pairs in ebunch.
Count the number of common neighbors of all node pairs in ebunch us
Compute the resource allocation index of all node pairs in ebunch usin
Compute the ratio of within- and inter-cluster common neighbors of al
4.28.1 resource_allocation_index
resource_allocation_index(G, ebunch=None)
Compute the resource allocation index of all node pairs in ebunch.
Resource allocation index of рќ‘ў and рќ‘Ј is defined as
∑︁
𝑤∈Γ(𝑢)∩Γ(𝑣)
1
|Γ(𝑤)|
where О“(рќ‘ў) denotes the set of neighbors of рќ‘ў.
Parameters G : graph
A NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
Resource allocation index will be computed for each pair of nodes given in the iterable.
The pairs must be given as 2-tuples (u, v) where u and v are nodes in the graph. If
ebunch is None then all non-existent edges in the graph will be used. Default value:
None.
Returns piter : iterator
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An iterator of 3-tuples in the form (u, v, p) where (u, v) is a pair of nodes and p is their
resource allocation index.
References
[R304]
Examples
>>> import networkx as nx
>>> G = nx.complete_graph(5)
>>> preds = nx.resource_allocation_index(G, [(0, 1), (2, 3)])
>>> for u, v, p in preds:
...
'(%d, %d) -> %.8f' % (u, v, p)
...
'(0, 1) -> 0.75000000'
'(2, 3) -> 0.75000000'
4.28.2 jaccard_coefficient
jaccard_coefficient(G, ebunch=None)
Compute the Jaccard coefficient of all node pairs in ebunch.
Jaccard coefficient of nodes рќ‘ў and рќ‘Ј is defined as
|Γ(𝑢) ∩ Γ(𝑣)|
|О“(рќ‘ў) в€Є О“(рќ‘Ј)|
where О“(рќ‘ў) denotes the set of neighbors of рќ‘ў.
Parameters G : graph
A NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
Jaccard coefficient will be computed for each pair of nodes given in the iterable. The
pairs must be given as 2-tuples (u, v) where u and v are nodes in the graph. If ebunch is
None then all non-existent edges in the graph will be used. Default value: None.
Returns piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a pair of nodes and p is their
Jaccard coefficient.
References
[R301]
Examples
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>>> import networkx as nx
>>> G = nx.complete_graph(5)
>>> preds = nx.jaccard_coefficient(G, [(0, 1), (2, 3)])
>>> for u, v, p in preds:
...
'(%d, %d) -> %.8f' % (u, v, p)
...
'(0, 1) -> 0.60000000'
'(2, 3) -> 0.60000000'
4.28.3 adamic_adar_index
adamic_adar_index(G, ebunch=None)
Compute the Adamic-Adar index of all node pairs in ebunch.
Adamic-Adar index of рќ‘ў and рќ‘Ј is defined as
∑︁
𝑤∈Γ(𝑢)∩Γ(𝑣)
1
log |Γ(𝑤)|
where О“(рќ‘ў) denotes the set of neighbors of рќ‘ў.
Parameters G : graph
NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
Adamic-Adar index will be computed for each pair of nodes given in the iterable. The
pairs must be given as 2-tuples (u, v) where u and v are nodes in the graph. If ebunch is
None then all non-existent edges in the graph will be used. Default value: None.
Returns piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a pair of nodes and p is their
Adamic-Adar index.
References
[R299]
Examples
>>> import networkx as nx
>>> G = nx.complete_graph(5)
>>> preds = nx.adamic_adar_index(G, [(0, 1), (2, 3)])
>>> for u, v, p in preds:
...
'(%d, %d) -> %.8f' % (u, v, p)
...
'(0, 1) -> 2.16404256'
'(2, 3) -> 2.16404256'
4.28.4 preferential_attachment
preferential_attachment(G, ebunch=None)
Compute the preferential attachment score of all node pairs in ebunch.
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Preferential attachment score of рќ‘ў and рќ‘Ј is defined as
|О“(рќ‘ў)||О“(рќ‘Ј)|
where О“(рќ‘ў) denotes the set of neighbors of рќ‘ў.
Parameters G : graph
NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
Preferential attachment score will be computed for each pair of nodes given in the iterable. The pairs must be given as 2-tuples (u, v) where u and v are nodes in the graph.
If ebunch is None then all non-existent edges in the graph will be used. Default value:
None.
Returns piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a pair of nodes and p is their
preferential attachment score.
References
[R302]
Examples
>>> import networkx as nx
>>> G = nx.complete_graph(5)
>>> preds = nx.preferential_attachment(G, [(0, 1), (2, 3)])
>>> for u, v, p in preds:
...
'(%d, %d) -> %d' % (u, v, p)
...
'(0, 1) -> 16'
'(2, 3) -> 16'
4.28.5 cn_soundarajan_hopcroft
cn_soundarajan_hopcroft(G, ebunch=None, community=’community’)
Count the number of common neighbors of all node pairs in ebunch using community information.
For two nodes рќ‘ў and рќ‘Ј, this function computes the number of common neighbors and bonus one for each
common neighbor belonging to the same community as рќ‘ў and рќ‘Ј. Mathematically,
∑︁
|Γ(𝑢) ∩ Γ(𝑣)| +
𝑓 (𝑤)
𝑤∈Γ(𝑢)∩Γ(𝑣)
where 𝑓 (𝑤) equals 1 if 𝑤 belongs to the same community as 𝑢 and 𝑣 or 0 otherwise and Γ(𝑢) denotes the set of
neighbors of рќ‘ў.
Parameters G : graph
A NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
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The score will be computed for each pair of nodes given in the iterable. The pairs must
be given as 2-tuples (u, v) where u and v are nodes in the graph. If ebunch is None then
all non-existent edges in the graph will be used. Default value: None.
community : string, optional (default = �community’)
Nodes attribute name containing the community information. G[u][community] identifies which community u belongs to. Each node belongs to at most one community.
Default value: �community’.
Returns piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a pair of nodes and p is their
score.
References
[R300]
Examples
>>> import networkx as nx
>>> G = nx.path_graph(3)
>>> G.node[0]['community'] = 0
>>> G.node[1]['community'] = 0
>>> G.node[2]['community'] = 0
>>> preds = nx.cn_soundarajan_hopcroft(G, [(0, 2)])
>>> for u, v, p in preds:
...
'(%d, %d) -> %d' % (u, v, p)
...
'(0, 2) -> 2'
4.28.6 ra_index_soundarajan_hopcroft
ra_index_soundarajan_hopcroft(G, ebunch=None, community=’community’)
Compute the resource allocation index of all node pairs in ebunch using community information.
For two nodes рќ‘ў and рќ‘Ј, this function computes the resource allocation index considering only common neighbors
belonging to the same community as рќ‘ў and рќ‘Ј. Mathematically,
∑︁
𝑤∈Γ(𝑢)∩Γ(𝑣)
𝑓 (𝑤)
|Γ(𝑤)|
where 𝑓 (𝑤) equals 1 if 𝑤 belongs to the same community as 𝑢 and 𝑣 or 0 otherwise and Γ(𝑢) denotes the set of
neighbors of рќ‘ў.
Parameters G : graph
A NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
The score will be computed for each pair of nodes given in the iterable. The pairs must
be given as 2-tuples (u, v) where u and v are nodes in the graph. If ebunch is None then
all non-existent edges in the graph will be used. Default value: None.
community : string, optional (default = �community’)
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Nodes attribute name containing the community information. G[u][community] identifies which community u belongs to. Each node belongs to at most one community.
Default value: �community’.
Returns piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a pair of nodes and p is their
score.
References
[R303]
Examples
>>> import networkx as nx
>>> G = nx.Graph()
>>> G.add_edges_from([(0, 1), (0, 2), (1, 3), (2, 3)])
>>> G.node[0]['community'] = 0
>>> G.node[1]['community'] = 0
>>> G.node[2]['community'] = 1
>>> G.node[3]['community'] = 0
>>> preds = nx.ra_index_soundarajan_hopcroft(G, [(0, 3)])
>>> for u, v, p in preds:
...
'(%d, %d) -> %.8f' % (u, v, p)
...
'(0, 3) -> 0.50000000'
4.28.7 within_inter_cluster
within_inter_cluster(G, ebunch=None, delta=0.001, community=’community’)
Compute the ratio of within- and inter-cluster common neighbors of all node pairs in ebunch.
For two nodes 𝑢 and 𝑣, if a common neighbor 𝑤 belongs to the same community as them, 𝑤 is considered as
within-cluster common neighbor of рќ‘ў and рќ‘Ј. Otherwise, it is considered as inter-cluster common neighbor of
рќ‘ў and рќ‘Ј. The ratio between the size of the set of within- and inter-cluster common neighbors is defined as the
WIC measure. [R305]
Parameters G : graph
A NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
The WIC measure will be computed for each pair of nodes given in the iterable. The
pairs must be given as 2-tuples (u, v) where u and v are nodes in the graph. If ebunch is
None then all non-existent edges in the graph will be used. Default value: None.
delta : float, optional (default = 0.001)
Value to prevent division by zero in case there is no inter-cluster common neighbor
between two nodes. See [R305] for details. Default value: 0.001.
community : string, optional (default = �community’)
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Nodes attribute name containing the community information. G[u][community] identifies which community u belongs to. Each node belongs to at most one community.
Default value: �community’.
Returns piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a pair of nodes and p is their
WIC measure.
References
[R305]
Examples
>>> import networkx as nx
>>> G = nx.Graph()
>>> G.add_edges_from([(0, 1), (0, 2), (0, 3), (1, 4), (2, 4), (3, 4)])
>>> G.node[0]['community'] = 0
>>> G.node[1]['community'] = 1
>>> G.node[2]['community'] = 0
>>> G.node[3]['community'] = 0
>>> G.node[4]['community'] = 0
>>> preds = nx.within_inter_cluster(G, [(0, 4)])
>>> for u, v, p in preds:
...
'(%d, %d) -> %.8f' % (u, v, p)
...
'(0, 4) -> 1.99800200'
>>> preds = nx.within_inter_cluster(G, [(0, 4)], delta=0.5)
>>> for u, v, p in preds:
...
'(%d, %d) -> %.8f' % (u, v, p)
...
'(0, 4) -> 1.33333333'
4.29 Matching
maximal_matching(G)
max_weight_matching(G[, maxcardinality])
Find a maximal cardinality matching in the graph.
Compute a maximum-weighted matching of G.
4.29.1 maximal_matching
maximal_matching(G)
Find a maximal cardinality matching in the graph.
A matching is a subset of edges in which no node occurs more than once. The cardinality of a matching is the
number of matched edges.
Parameters G : NetworkX graph
Undirected graph
Returns matching : set
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A maximal matching of the graph.
Notes
The algorithm greedily selects a maximal matching M of the graph G (i.e. no superset of M exists). It runs in
рќ‘‚(|рќђё|) time.
4.29.2 max_weight_matching
max_weight_matching(G, maxcardinality=False)
Compute a maximum-weighted matching of G.
A matching is a subset of edges in which no node occurs more than once. The cardinality of a matching is the
number of matched edges. The weight of a matching is the sum of the weights of its edges.
Parameters G : NetworkX graph
Undirected graph
maxcardinality: bool, optional
If maxcardinality is True, compute the maximum-cardinality matching with maximum
weight among all maximum-cardinality matchings.
Returns mate : dictionary
The matching is returned as a dictionary, mate, such that mate[v] == w if node v is
matched to node w. Unmatched nodes do not occur as a key in mate.
Notes
If G has edges with �weight’ attribute the edge data are used as weight values else the weights are assumed to
be 1.
This function takes time O(number_of_nodes ** 3).
If all edge weights are integers, the algorithm uses only integer computations. If floating point weights are used,
the algorithm could return a slightly suboptimal matching due to numeric precision errors.
This method is based on the “blossom” method for finding augmenting paths and the “primal-dual” method for
finding a matching of maximum weight, both methods invented by Jack Edmonds [R306].
References
[R306]
4.30 Maximal independent set
Algorithm to find a maximal (not maximum) independent set.
maximal_independent_set(G[, nodes])
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4.30.1 maximal_independent_set
maximal_independent_set(G, nodes=None)
Return a random maximal independent set guaranteed to contain a given set of nodes.
An independent set is a set of nodes such that the subgraph of G induced by these nodes contains no edges. A
maximal independent set is an independent set such that it is not possible to add a new node and still get an
independent set.
Parameters G : NetworkX graph
nodes : list or iterable
Nodes that must be part of the independent set. This set of nodes must be independent.
Returns indep_nodes : list
List of nodes that are part of a maximal independent set.
Raises NetworkXUnfeasible
If the nodes in the provided list are not part of the graph or do not form an independent
set, an exception is raised.
Notes
This algorithm does not solve the maximum independent set problem.
Examples
>>>
>>>
[4,
>>>
[1,
G = nx.path_graph(5)
nx.maximal_independent_set(G)
0, 2]
nx.maximal_independent_set(G, [1])
3]
4.31 Minimum Spanning Tree
Computes minimum spanning tree of a weighted graph.
minimum_spanning_tree(G[, weight])
minimum_spanning_edges(G[, weight, data])
Return a minimum spanning tree or forest of an undirected weighted graph.
Generate edges in a minimum spanning forest of an undirected weighted graph.
4.31.1 minimum_spanning_tree
minimum_spanning_tree(G, weight=’weight’)
Return a minimum spanning tree or forest of an undirected weighted graph.
A minimum spanning tree is a subgraph of the graph (a tree) with the minimum sum of edge weights.
If the graph is not connected a spanning forest is constructed. A spanning forest is a union of the spanning trees
for each connected component of the graph.
Parameters G : NetworkX Graph
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weight : string
Edge data key to use for weight (default �weight’).
Returns G : NetworkX Graph
A minimum spanning tree or forest.
Notes
Uses Kruskal’s algorithm.
If the graph edges do not have a weight attribute a default weight of 1 will be used.
Examples
>>> G=nx.cycle_graph(4)
>>> G.add_edge(0,3,weight=2) # assign weight 2 to edge 0-3
>>> T=nx.minimum_spanning_tree(G)
>>> print(sorted(T.edges(data=True)))
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
4.31.2 minimum_spanning_edges
minimum_spanning_edges(G, weight=’weight’, data=True)
Generate edges in a minimum spanning forest of an undirected weighted graph.
A minimum spanning tree is a subgraph of the graph (a tree) with the minimum sum of edge weights. A spanning
forest is a union of the spanning trees for each connected component of the graph.
Parameters G : NetworkX Graph
weight : string
Edge data key to use for weight (default �weight’).
data : bool, optional
If True yield the edge data along with the edge.
Returns edges : iterator
A generator that produces edges in the minimum spanning tree. The edges are threetuples (u,v,w) where w is the weight.
Notes
Uses Kruskal’s algorithm.
If the graph edges do not have a weight attribute a default weight of 1 will be used.
Modified code from David Eppstein, April 2006 http://www.ics.uci.edu/~eppstein/PADS/
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Examples
>>> G=nx.cycle_graph(4)
>>> G.add_edge(0,3,weight=2) # assign weight 2 to edge 0-3
>>> mst=nx.minimum_spanning_edges(G,data=False) # a generator of MST edges
>>> edgelist=list(mst) # make a list of the edges
>>> print(sorted(edgelist))
[(0, 1), (1, 2), (2, 3)]
4.32 Operators
Unary operations on graphs
complement(G[, name])
reverse(G[, copy])
Return the graph complement of G.
Return the reverse directed graph of G.
4.32.1 complement
complement(G, name=None)
Return the graph complement of G.
Parameters G : graph
A NetworkX graph
name : string
Specify name for new graph
Returns GC : A new graph.
Notes
Note that complement() does not create self-loops and also does not produce parallel edges for MultiGraphs.
Graph, node, and edge data are not propagated to the new graph.
4.32.2 reverse
reverse(G, copy=True)
Return the reverse directed graph of G.
Parameters G : directed graph
A NetworkX directed graph
copy : bool
If True, then a new graph is returned. If False, then the graph is reversed in place.
Returns H : directed graph
The reversed G.
Operations on graphs including union, intersection, difference.
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compose(G, H[, name])
union(G, H[, rename, name])
disjoint_union(G, H)
intersection(G, H)
difference(G, H)
symmetric_difference(G, H)
Return a new graph of G composed with H.
Return the union of graphs G and H.
Return the disjoint union of graphs G and H.
Return a new graph that contains only the edges that exist in both G and H.
Return a new graph that contains the edges that exist in G but not in H.
Return new graph with edges that exist in either G or H but not both.
4.32.3 compose
compose(G, H, name=None)
Return a new graph of G composed with H.
Composition is the simple union of the node sets and edge sets. The node sets of G and H need not be disjoint.
Parameters G,H : graph
A NetworkX graph
name : string
Specify name for new graph
Returns C: A new graph with the same type as G
Notes
It is recommended that G and H be either both directed or both undirected. Attributes from H take precedent
over attributes from G.
4.32.4 union
union(G, H, rename=(None, None), name=None)
Return the union of graphs G and H.
Graphs G and H must be disjoint, otherwise an exception is raised.
Parameters G,H : graph
A NetworkX graph
create_using : NetworkX graph
Use specified graph for result. Otherwise
rename : bool , default=(None, None)
Node names of G and H can be changed by specifying the tuple rename=(�G-�,’H-�) (for
example). Node “u” in G is then renamed “G-u” and “v” in H is renamed “H-v”.
name : string
Specify the name for the union graph
Returns U : A union graph with the same type as G.
See also:
disjoint_union
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Notes
To force a disjoint union with node relabeling, use disjoint_union(G,H) or convert_node_labels_to integers().
Graph, edge, and node attributes are propagated from G and H to the union graph. If a graph attribute is present
in both G and H the value from H is used.
4.32.5 disjoint_union
disjoint_union(G, H)
Return the disjoint union of graphs G and H.
This algorithm forces distinct integer node labels.
Parameters G,H : graph
A NetworkX graph
Returns U : A union graph with the same type as G.
Notes
A new graph is created, of the same class as G. It is recommended that G and H be either both directed or both
undirected.
The nodes of G are relabeled 0 to len(G)-1, and the nodes of H are relabeled len(G) to len(G)+len(H)-1.
Graph, edge, and node attributes are propagated from G and H to the union graph. If a graph attribute is present
in both G and H the value from H is used.
4.32.6 intersection
intersection(G, H)
Return a new graph that contains only the edges that exist in both G and H.
The node sets of H and G must be the same.
Parameters G,H : graph
A NetworkX graph. G and H must have the same node sets.
Returns GH : A new graph with the same type as G.
Notes
Attributes from the graph, nodes, and edges are not copied to the new graph. If you want a new graph of the
intersection of G and H with the attributes (including edge data) from G use remove_nodes_from() as follows
>>>
>>>
>>>
>>>
G=nx.path_graph(3)
H=nx.path_graph(5)
R=G.copy()
R.remove_nodes_from(n for n in G if n not in H)
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4.32.7 difference
difference(G, H)
Return a new graph that contains the edges that exist in G but not in H.
The node sets of H and G must be the same.
Parameters G,H : graph
A NetworkX graph. G and H must have the same node sets.
Returns D : A new graph with the same type as G.
Notes
Attributes from the graph, nodes, and edges are not copied to the new graph. If you want a new graph of
the difference of G and H with with the attributes (including edge data) from G use remove_nodes_from() as
follows:
>>>
>>>
>>>
>>>
G = nx.path_graph(3)
H = nx.path_graph(5)
R = G.copy()
R.remove_nodes_from(n for n in G if n in H)
4.32.8 symmetric_difference
symmetric_difference(G, H)
Return new graph with edges that exist in either G or H but not both.
The node sets of H and G must be the same.
Parameters G,H : graph
A NetworkX graph. G and H must have the same node sets.
Returns D : A new graph with the same type as G.
Notes
Attributes from the graph, nodes, and edges are not copied to the new graph.
Operations on many graphs.
compose_all(graphs[, name])
union_all(graphs[, rename, name])
disjoint_union_all(graphs)
intersection_all(graphs)
Return the composition of all graphs.
Return the union of all graphs.
Return the disjoint union of all graphs.
Return a new graph that contains only the edges that exist in all graphs.
4.32.9 compose_all
compose_all(graphs, name=None)
Return the composition of all graphs.
Composition is the simple union of the node sets and edge sets. The node sets of the supplied graphs need not
be disjoint.
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Parameters graphs : list
List of NetworkX graphs
name : string
Specify name for new graph
Returns C : A graph with the same type as the first graph in list
Notes
It is recommended that the supplied graphs be either all directed or all undirected.
Graph, edge, and node attributes are propagated to the union graph. If a graph attribute is present in multiple
graphs, then the value from the last graph in the list with that attribute is used.
4.32.10 union_all
union_all(graphs, rename=(None, ), name=None)
Return the union of all graphs.
The graphs must be disjoint, otherwise an exception is raised.
Parameters graphs : list of graphs
List of NetworkX graphs
rename : bool , default=(None, None)
Node names of G and H can be changed by specifying the tuple rename=(�G-�,’H-�) (for
example). Node “u” in G is then renamed “G-u” and “v” in H is renamed “H-v”.
name : string
Specify the name for the union [email protected]_implemnted_for(вЂ�direct
Returns U : a graph with the same type as the first graph in list
See also:
union, disjoint_union_all
Notes
To force a disjoint union with node relabeling, use disjoint_union_all(G,H) or convert_node_labels_to integers().
Graph, edge, and node attributes are propagated to the union graph. If a graph attribute is present in multiple
graphs, then the value from the last graph in the list with that attribute is used.
4.32.11 disjoint_union_all
disjoint_union_all(graphs)
Return the disjoint union of all graphs.
This operation forces distinct integer node labels starting with 0 for the first graph in the list and numbering
consecutively.
Parameters graphs : list
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List of NetworkX graphs
Returns U : A graph with the same type as the first graph in list
Notes
It is recommended that the graphs be either all directed or all undirected.
Graph, edge, and node attributes are propagated to the union graph. If a graph attribute is present in multiple
graphs, then the value from the last graph in the list with that attribute is used.
4.32.12 intersection_all
intersection_all(graphs)
Return a new graph that contains only the edges that exist in all graphs.
All supplied graphs must have the same node set.
Parameters graphs_list : list
List of NetworkX graphs
Returns R : A new graph with the same type as the first graph in list
Notes
Attributes from the graph, nodes, and edges are not copied to the new graph.
Graph products.
cartesian_product(G, H)
lexicographic_product(G, H)
strong_product(G, H)
tensor_product(G, H)
Return the Cartesian product of G and H.
Return the lexicographic product of G and H.
Return the strong product of G and H.
Return the tensor product of G and H.
4.32.13 cartesian_product
cartesian_product(G, H)
Return the Cartesian product of G and H.
The tensor product P of the graphs G and H has a node set that is the Cartesian product of the node sets,
$V(P)=V(G) imes V(H)$. P has an edge ((u,v),(x,y)) if and only if (u,v) is an edge in G and x==y or and (x,y)
is an edge in H and u==v. and (x,y) is an edge in H.
Parameters G, H: graphs
Networkx graphs.
Returns P: NetworkX graph
The Cartesian product of G and H. P will be a multi-graph if either G or H is a multigraph. Will be a directed if G and H are directed, and undirected if G and H are undirected.
Raises NetworkXError
If G and H are not both directed or both undirected.
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Notes
Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None.
For example >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0,a1=True) >>>
H.add_node(�a’,a2=’Spam’) >>> P = nx.cartesian_product(G,H) >>> P.nodes() [(0, �a’)]
Edge attributes and edge keys (for multigraphs) are also copied to the new product graph
4.32.14 lexicographic_product
lexicographic_product(G, H)
Return the lexicographic product of G and H.
The lexicographical product P of the graphs G and H has a node set that is the Cartesian product of the node
sets, $V(P)=V(G) imes V(H)$. P has an edge ((u,v),(x,y)) if and only if (u,v) is an edge in G or u==v and (x,y)
is an edge in H.
Parameters G, H: graphs
Networkx graphs.
Returns P: NetworkX graph
The Cartesian product of G and H. P will be a multi-graph if either G or H is a multigraph. Will be a directed if G and H are directed, and undirected if G and H are undirected.
Raises NetworkXError
If G and H are not both directed or both undirected.
Notes
Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None.
For example >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0,a1=True) >>>
H.add_node(�a’,a2=’Spam’) >>> P = nx.lexicographic_product(G,H) >>> P.nodes() [(0, �a’)]
Edge attributes and edge keys (for multigraphs) are also copied to the new product graph
4.32.15 strong_product
strong_product(G, H)
Return the strong product of G and H.
The strong product P of the graphs G and H has a node set that is the Cartesian product of the node sets,
$V(P)=V(G) imes V(H)$. P has an edge ((u,v),(x,y)) if and only if u==v and (x,y) is an edge in H, or x==y and
(u,v) is an edge in G, or (u,v) is an edge in G and (x,y) is an edge in H.
Parameters G, H: graphs
Networkx graphs.
Returns P: NetworkX graph
The Cartesian product of G and H. P will be a multi-graph if either G or H is a multigraph. Will be a directed if G and H are directed, and undirected if G and H are undirected.
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Raises NetworkXError
If G and H are not both directed or both undirected.
Notes
Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None.
For example >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0,a1=True) >>>
H.add_node(�a’,a2=’Spam’) >>> P = nx.strong_product(G,H) >>> P.nodes() [(0, �a’)]
Edge attributes and edge keys (for multigraphs) are also copied to the new product graph
4.32.16 tensor_product
tensor_product(G, H)
Return the tensor product of G and H.
The tensor product P of the graphs G and H has a node set that is the Cartesian product of the node sets,
$V(P)=V(G) times V(H)$. P has an edge ((u,v),(x,y)) if and only if (u,v) is an edge in G and (x,y) is an edge in
H.
Sometimes referred to as the categorical product.
Parameters G, H: graphs
Networkx graphs.
Returns P: NetworkX graph
The tensor product of G and H. P will be a multi-graph if either G or H is a multi-graph.
Will be a directed if G and H are directed, and undirected if G and H are undirected.
Raises NetworkXError
If G and H are not both directed or both undirected.
Notes
Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None.
For example >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0,a1=True) >>>
H.add_node(�a’,a2=’Spam’) >>> P = nx.tensor_product(G,H) >>> P.nodes() [(0, �a’)]
Edge attributes and edge keys (for multigraphs) are also copied to the new product graph
4.33 Rich Club
rich_club_coefficient(G[, normalized, Q])
Return the rich-club coefficient of the graph G.
4.33.1 rich_club_coefficient
rich_club_coefficient(G, normalized=True, Q=100)
Return the rich-club coefficient of the graph G.
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The rich-club coefficient is the ratio, for every degree k, of the number of actual to the number of potential edges
for nodes with degree greater than k:
𝜑(�) =
2𝐸�
𝑁 �(𝑁 � − 1)
where Nk is the number of nodes with degree larger than k, and Ek be the number of edges among those nodes.
Parameters G : NetworkX graph
normalized : bool (optional)
Normalize using randomized network (see [R307])
Q : float (optional, default=100)
If normalized=True build a random network by performing Q*M double-edge swaps,
where M is the number of edges in G, to use as a null-model for normalization.
Returns rc : dictionary
A dictionary, keyed by degree, with rich club coefficient values.
Notes
The rich club definition and algorithm are found in [R307]. This algorithm ignores any edge weights and is not
defined for directed graphs or graphs with parallel edges or self loops.
Estimates for appropriate values of Q are found in [R308].
References
[R307], [R308]
Examples
>>> G = nx.Graph([(0,1),(0,2),(1,2),(1,3),(1,4),(4,5)])
>>> rc = nx.rich_club_coefficient(G,normalized=False)
>>> rc[0]
0.4
4.34 Shortest Paths
Compute the shortest paths and path lengths between nodes in the graph.
These algorithms work with undirected and directed graphs.
shortest_path(G[, source, target, weight])
all_shortest_paths(G, source, target[, weight])
shortest_path_length(G[, source, target, weight])
average_shortest_path_length(G[, weight])
has_path(G, source, target)
4.34. Shortest Paths
Compute shortest paths in the graph.
Compute all shortest paths in the graph.
Compute shortest path lengths in the graph.
Return the average shortest path length.
Return True if G has a path from source to target, False otherwise.
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4.34.1 shortest_path
shortest_path(G, source=None, target=None, weight=None)
Compute shortest paths in the graph.
Parameters G : NetworkX graph
source : node, optional
Starting node for path. If not specified, compute shortest paths using all nodes as source
nodes.
target : node, optional
Ending node for path. If not specified, compute shortest paths using all nodes as target
nodes.
weight : None or string, optional (default = None)
If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the
edge weight. Any edge attribute not present defaults to 1.
Returns path: list or dictionary
All returned paths include both the source and target in the path.
If the source and target are both specified, return a single list of nodes in a shortest path
from the source to the target.
If only the source is specified, return a dictionary keyed by targets with a list of nodes
in a shortest path from the source to one of the targets.
If only the target is specified, return a dictionary keyed by sources with a list of nodes
in a shortest path from one of the sources to the target.
If neither the source nor target are specified return a dictionary of dictionaries with
path[source][target]=[list of nodes in path].
See also:
all_pairs_shortest_path, all_pairs_dijkstra_path, single_source_shortest_path,
single_source_dijkstra_path
Notes
There may be more than one shortest path between a source and target. This returns only one of them.
Examples
>>>
>>>
[0,
>>>
>>>
[0,
>>>
>>>
[0,
>>>
312
G=nx.path_graph(5)
print(nx.shortest_path(G,source=0,target=4))
1, 2, 3, 4]
p=nx.shortest_path(G,source=0) # target not specified
p[4]
1, 2, 3, 4]
p=nx.shortest_path(G,target=4) # source not specified
p[0]
1, 2, 3, 4]
p=nx.shortest_path(G) # source,target not specified
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>>> p[0][4]
[0, 1, 2, 3, 4]
4.34.2 all_shortest_paths
all_shortest_paths(G, source, target, weight=None)
Compute all shortest paths in the graph.
Parameters G : NetworkX graph
source : node
Starting node for path.
target : node
Ending node for path.
weight : None or string, optional (default = None)
If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the
edge weight. Any edge attribute not present defaults to 1.
Returns paths: generator of lists
A generator of all paths between source and target.
See also:
shortest_path, single_source_shortest_path, all_pairs_shortest_path
Notes
There may be many shortest paths between the source and target.
Examples
>>> G=nx.Graph()
>>> G.add_path([0,1,2])
>>> G.add_path([0,10,2])
>>> print([p for p in nx.all_shortest_paths(G,source=0,target=2)])
[[0, 1, 2], [0, 10, 2]]
4.34.3 shortest_path_length
shortest_path_length(G, source=None, target=None, weight=None)
Compute shortest path lengths in the graph.
Parameters G : NetworkX graph
source : node, optional
Starting node for path. If not specified, compute shortest path lengths using all nodes as
source nodes.
target : node, optional
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Ending node for path. If not specified, compute shortest path lengths using all nodes as
target nodes.
weight : None or string, optional (default = None)
If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the
edge weight. Any edge attribute not present defaults to 1.
Returns length: int or dictionary
If the source and target are both specified, return the length of the shortest path from the
source to the target.
If only the source is specified, return a dictionary keyed by targets whose values are the
lengths of the shortest path from the source to one of the targets.
If only the target is specified, return a dictionary keyed by sources whose values are the
lengths of the shortest path from one of the sources to the target.
If neither the source nor target are specified return a dictionary of dictionaries with
path[source][target]=L, where L is the length of the shortest path from source to target.
Raises NetworkXNoPath
If no path exists between source and target.
See also:
all_pairs_shortest_path_length,
all_pairs_dijkstra_path_length,
single_source_shortest_path_length, single_source_dijkstra_path_length
Notes
The length of the path is always 1 less than the number of nodes involved in the path since the length measures
the number of edges followed.
For digraphs this returns the shortest directed path length. To find path lengths in the reverse direction use
G.reverse(copy=False) first to flip the edge orientation.
Examples
>>>
>>>
4
>>>
>>>
4
>>>
>>>
4
>>>
>>>
4
G=nx.path_graph(5)
print(nx.shortest_path_length(G,source=0,target=4))
p=nx.shortest_path_length(G,source=0) # target not specified
p[4]
p=nx.shortest_path_length(G,target=4) # source not specified
p[0]
p=nx.shortest_path_length(G) # source,target not specified
p[0][4]
4.34.4 average_shortest_path_length
average_shortest_path_length(G, weight=None)
Return the average shortest path length.
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The average shortest path length is
рќ‘Ћ=
∑︁
𝑠,𝑡∈𝑉
рќ‘‘(рќ‘ , рќ‘Ў)
рќ‘›(рќ‘› в€’ 1)
where 𝑉 is the set of nodes in 𝐺, 𝑑(𝑠, 𝑡) is the shortest path from 𝑠to 𝑡, and 𝑛 is the number of nodes in 𝐺.
Parameters G : NetworkX graph
weight : None or string, optional (default = None)
If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the
edge weight. Any edge attribute not present defaults to 1.
Raises NetworkXError:
if the graph is not connected.
Examples
>>> G=nx.path_graph(5)
>>> print(nx.average_shortest_path_length(G))
2.0
For disconnected graphs you can compute the average shortest path length for each component:
>>> G=nx.Graph([(1,2),(3,4)]) >>> for g in nx.connected_component_subgraphs(G): ...
print(nx.average_shortest_path_length(g)) 1.0 1.0
4.34.5 has_path
has_path(G, source, target)
Return True if G has a path from source to target, False otherwise.
Parameters G : NetworkX graph
source : node
Starting node for path
target : node
Ending node for path
4.34.6 Advanced Interface
Shortest path algorithms for unweighted graphs.
single_source_shortest_path(G, source[, cutoff])
single_source_shortest_path_length(G, source)
all_pairs_shortest_path(G[, cutoff])
all_pairs_shortest_path_length(G[, cutoff])
predecessor(G, source[, target, cutoff, ...])
4.34. Shortest Paths
Compute shortest path between source and all other nodes reachable
Compute the shortest path lengths from source to all reachable nodes
Compute shortest paths between all nodes.
Compute the shortest path lengths between all nodes in G.
Returns dictionary of predecessors for the path from source to all nod
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single_source_shortest_path
single_source_shortest_path(G, source, cutoff=None)
Compute shortest path between source and all other nodes reachable from source.
Parameters G : NetworkX graph
source : node label
Starting node for path
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns lengths : dictionary
Dictionary, keyed by target, of shortest paths.
See also:
shortest_path
Notes
The shortest path is not necessarily unique. So there can be multiple paths between the source and each target
node, all of which have the same �shortest’ length. For each target node, this function returns only one of those
paths.
Examples
>>>
>>>
>>>
[0,
G=nx.path_graph(5)
path=nx.single_source_shortest_path(G,0)
path[4]
1, 2, 3, 4]
single_source_shortest_path_length
single_source_shortest_path_length(G, source, cutoff=None)
Compute the shortest path lengths from source to all reachable nodes.
Parameters G : NetworkX graph
source : node
Starting node for path
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns lengths : dictionary
Dictionary of shortest path lengths keyed by target.
See also:
shortest_path_length
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Examples
>>>
>>>
>>>
4
>>>
{0:
G=nx.path_graph(5)
length=nx.single_source_shortest_path_length(G,0)
length[4]
print(length)
0, 1: 1, 2: 2, 3: 3, 4: 4}
all_pairs_shortest_path
all_pairs_shortest_path(G, cutoff=None)
Compute shortest paths between all nodes.
Parameters G : NetworkX graph
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns lengths : dictionary
Dictionary, keyed by source and target, of shortest paths.
See also:
floyd_warshall
Examples
>>>
>>>
>>>
[0,
G=nx.path_graph(5)
path=nx.all_pairs_shortest_path(G)
print(path[0][4])
1, 2, 3, 4]
all_pairs_shortest_path_length
all_pairs_shortest_path_length(G, cutoff=None)
Compute the shortest path lengths between all nodes in G.
Parameters G : NetworkX graph
cutoff : integer, optional
depth to stop the search. Only paths of length <= cutoff are returned.
Returns lengths : dictionary
Dictionary of shortest path lengths keyed by source and target.
Notes
The dictionary returned only has keys for reachable node pairs.
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Examples
>>>
>>>
>>>
3
>>>
{0:
G=nx.path_graph(5)
length=nx.all_pairs_shortest_path_length(G)
print(length[1][4])
length[1]
1, 1: 0, 2: 1, 3: 2, 4: 3}
predecessor
predecessor(G, source, target=None, cutoff=None, return_seen=None)
Returns dictionary of predecessors for the path from source to all nodes in G.
Parameters G : NetworkX graph
source : node label
Starting node for path
target : node label, optional
Ending node for path. If provided only predecessors between source and target are
returned
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns pred : dictionary
Dictionary, keyed by node, of predecessors in the shortest path.
Examples
>>>
>>>
[0,
>>>
{0:
G=nx.path_graph(4)
print(G.nodes())
1, 2, 3]
nx.predecessor(G,0)
[], 1: [0], 2: [1], 3: [2]}
Shortest path algorithms for weighed graphs.
dijkstra_path(G, source, target[, weight])
dijkstra_path_length(G, source, target[, weight])
single_source_dijkstra_path(G, source[, ...])
single_source_dijkstra_path_length(G, source)
all_pairs_dijkstra_path(G[, cutoff, weight])
all_pairs_dijkstra_path_length(G[, cutoff, ...])
single_source_dijkstra(G, source[, target, ...])
bidirectional_dijkstra(G, source, target[, ...])
dijkstra_predecessor_and_distance(G, source)
bellman_ford(G, source[, weight])
negative_edge_cycle(G[, weight])
318
Returns the shortest path from source to target in a weighted graph G
Returns the shortest path length from source to target in a weighted g
Compute shortest path between source and all other reachable nodes
Compute the shortest path length between source and all other reacha
Compute shortest paths between all nodes in a weighted graph.
Compute shortest path lengths between all nodes in a weighted graph
Compute shortest paths and lengths in a weighted graph G.
Dijkstra’s algorithm for shortest paths using bidirectional search.
Compute shortest path length and predecessors on shortest paths in w
Compute shortest path lengths and predecessors on shortest paths in w
Return True if there exists a negative edge cycle anywhere in G.
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dijkstra_path
dijkstra_path(G, source, target, weight=’weight’)
Returns the shortest path from source to target in a weighted graph G.
Parameters G : NetworkX graph
source : node
Starting node
target : node
Ending node
weight: string, optional (default=’weight’)
Edge data key corresponding to the edge weight
Returns path : list
List of nodes in a shortest path.
Raises NetworkXNoPath
If no path exists between source and target.
See also:
bidirectional_dijkstra
Notes
Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.
Examples
>>> G=nx.path_graph(5)
>>> print(nx.dijkstra_path(G,0,4))
[0, 1, 2, 3, 4]
dijkstra_path_length
dijkstra_path_length(G, source, target, weight=’weight’)
Returns the shortest path length from source to target in a weighted graph.
Parameters G : NetworkX graph
source : node label
starting node for path
target : node label
ending node for path
weight: string, optional (default=’weight’)
Edge data key corresponding to the edge weight
Returns length : number
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Shortest path length.
Raises NetworkXNoPath
If no path exists between source and target.
See also:
bidirectional_dijkstra
Notes
Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.
Examples
>>> G=nx.path_graph(5)
>>> print(nx.dijkstra_path_length(G,0,4))
4
single_source_dijkstra_path
single_source_dijkstra_path(G, source, cutoff=None, weight=’weight’)
Compute shortest path between source and all other reachable nodes for a weighted graph.
Parameters G : NetworkX graph
source : node
Starting node for path.
weight: string, optional (default=’weight’)
Edge data key corresponding to the edge weight
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns paths : dictionary
Dictionary of shortest path lengths keyed by target.
See also:
single_source_dijkstra
Notes
Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.
Examples
>>>
>>>
>>>
[0,
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G=nx.path_graph(5)
path=nx.single_source_dijkstra_path(G,0)
path[4]
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single_source_dijkstra_path_length
single_source_dijkstra_path_length(G, source, cutoff=None, weight=’weight’)
Compute the shortest path length between source and all other reachable nodes for a weighted graph.
Parameters G : NetworkX graph
source : node label
Starting node for path
weight: string, optional (default=’weight’)
Edge data key corresponding to the edge weight.
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns length : dictionary
Dictionary of shortest lengths keyed by target.
See also:
single_source_dijkstra
Notes
Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.
Examples
>>>
>>>
>>>
4
>>>
{0:
G=nx.path_graph(5)
length=nx.single_source_dijkstra_path_length(G,0)
length[4]
print(length)
0, 1: 1, 2: 2, 3: 3, 4: 4}
all_pairs_dijkstra_path
all_pairs_dijkstra_path(G, cutoff=None, weight=’weight’)
Compute shortest paths between all nodes in a weighted graph.
Parameters G : NetworkX graph
weight: string, optional (default=’weight’)
Edge data key corresponding to the edge weight
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns distance : dictionary
Dictionary, keyed by source and target, of shortest paths.
See also:
floyd_warshall
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Notes
Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.
Examples
>>>
>>>
>>>
[0,
G=nx.path_graph(5)
path=nx.all_pairs_dijkstra_path(G)
print(path[0][4])
1, 2, 3, 4]
all_pairs_dijkstra_path_length
all_pairs_dijkstra_path_length(G, cutoff=None, weight=’weight’)
Compute shortest path lengths between all nodes in a weighted graph.
Parameters G : NetworkX graph
weight: string, optional (default=’weight’)
Edge data key corresponding to the edge weight
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns distance : dictionary
Dictionary, keyed by source and target, of shortest path lengths.
Notes
Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.
The dictionary returned only has keys for reachable node pairs.
Examples
>>>
>>>
>>>
3
>>>
{0:
G=nx.path_graph(5)
length=nx.all_pairs_dijkstra_path_length(G)
print(length[1][4])
length[1]
1, 1: 0, 2: 1, 3: 2, 4: 3}
single_source_dijkstra
single_source_dijkstra(G, source, target=None, cutoff=None, weight=’weight’)
Compute shortest paths and lengths in a weighted graph G.
Uses Dijkstra’s algorithm for shortest paths.
Parameters G : NetworkX graph
source : node label
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Starting node for path
target : node label, optional
Ending node for path
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns distance,path : dictionaries
Returns a tuple of two dictionaries keyed by node. The first dictionary stores distance
from the source. The second stores the path from the source to that node.
See also:
single_source_dijkstra_path, single_source_dijkstra_path_length
Notes
Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.
Based on the Python cookbook recipe (119466) at http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/119466
This algorithm is not guaranteed to work if edge weights are negative or are floating point numbers (overflows
and roundoff errors can cause problems).
Examples
>>>
>>>
>>>
4
>>>
{0:
>>>
[0,
G=nx.path_graph(5)
length,path=nx.single_source_dijkstra(G,0)
print(length[4])
print(length)
0, 1: 1, 2: 2, 3: 3, 4: 4}
path[4]
1, 2, 3, 4]
bidirectional_dijkstra
bidirectional_dijkstra(G, source, target, weight=’weight’)
Dijkstra’s algorithm for shortest paths using bidirectional search.
Parameters G : NetworkX graph
source : node
Starting node.
target : node
Ending node.
weight: string, optional (default=’weight’)
Edge data key corresponding to the edge weight
Returns length : number
Shortest path length.
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Returns a tuple of two dictionaries keyed by node.
The first dictionary stores distance from the source.
The second stores the path from the source to that node.
Raises NetworkXNoPath
If no path exists between source and target.
See also:
shortest_path, shortest_path_length
Notes
Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.
In practice bidirectional Dijkstra is much more than twice as fast as ordinary Dijkstra.
Ordinary Dijkstra expands nodes in a sphere-like manner from the source. The radius of this sphere will eventually be the length of the shortest path. Bidirectional Dijkstra will expand nodes from both the source and
the target, making two spheres of half this radius. Volume of the first sphere is pi*r*r while the others are
2*pi*r/2*r/2, making up half the volume.
This algorithm is not guaranteed to work if edge weights are negative or are floating point numbers (overflows
and roundoff errors can cause problems).
Examples
>>>
>>>
>>>
4
>>>
[0,
G=nx.path_graph(5)
length,path=nx.bidirectional_dijkstra(G,0,4)
print(length)
print(path)
1, 2, 3, 4]
dijkstra_predecessor_and_distance
dijkstra_predecessor_and_distance(G, source, cutoff=None, weight=’weight’)
Compute shortest path length and predecessors on shortest paths in weighted graphs.
Parameters G : NetworkX graph
source : node label
Starting node for path
weight: string, optional (default=’weight’)
Edge data key corresponding to the edge weight
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns pred,distance : dictionaries
Returns two dictionaries representing a list of predecessors of a node and the distance
to each node.
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Notes
Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.
The list of predecessors contains more than one element only when there are more than one shortest paths to the
key node.
bellman_ford
bellman_ford(G, source, weight=’weight’)
Compute shortest path lengths and predecessors on shortest paths in weighted graphs.
The algorithm has a running time of O(mn) where n is the number of nodes and m is the number of edges. It is
slower than Dijkstra but can handle negative edge weights.
Parameters G : NetworkX graph
The algorithm works for all types of graphs, including directed graphs and multigraphs.
source: node label
Starting node for path
weight: string, optional (default=’weight’)
Edge data key corresponding to the edge weight
Returns pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the path and to the distance
from the source respectively.
Raises NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the algorithm raises an exception to
indicate the presence of the negative cost (di)cycle. Note: any negative weight edge in
an undirected graph is a negative cost cycle.
Notes
Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.
The dictionaries returned only have keys for nodes reachable from the source.
In the case where the (di)graph is not connected, if a component not containing the source contains a negative
cost (di)cycle, it will not be detected.
Examples
>>> import networkx as nx
>>> G = nx.path_graph(5, create_using = nx.DiGraph())
>>> pred, dist = nx.bellman_ford(G, 0)
>>> sorted(pred.items())
[(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
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>>>
>>>
>>>
>>>
from nose.tools import assert_raises
G = nx.cycle_graph(5, create_using = nx.DiGraph())
G[1][2]['weight'] = -7
assert_raises(nx.NetworkXUnbounded, nx.bellman_ford, G, 0)
negative_edge_cycle
negative_edge_cycle(G, weight=’weight’)
Return True if there exists a negative edge cycle anywhere in G.
Parameters G : NetworkX graph
weight: string, optional (default=’weight’)
Edge data key corresponding to the edge weight
Returns negative_cycle : bool
True if a negative edge cycle exists, otherwise False.
Notes
Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.
This algorithm uses bellman_ford() but finds negative cycles on any component by first adding a new node
connected to every node, and starting bellman_ford on that node. It then removes that extra node.
Examples
>>> import networkx as nx
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> print(nx.negative_edge_cycle(G))
False
>>> G[1][2]['weight'] = -7
>>> print(nx.negative_edge_cycle(G))
True
4.34.7 Dense Graphs
Floyd-Warshall algorithm for shortest paths.
floyd_warshall(G[, weight])
floyd_warshall_predecessor_and_distance(G[, ...])
floyd_warshall_numpy(G[, nodelist, weight])
Find all-pairs shortest path lengths using Floyd’s algorithm.
Find all-pairs shortest path lengths using Floyd’s algorithm.
Find all-pairs shortest path lengths using Floyd’s algorithm.
floyd_warshall
floyd_warshall(G, weight=’weight’)
Find all-pairs shortest path lengths using Floyd’s algorithm.
Parameters G : NetworkX graph
weight: string, optional (default= �weight’)
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Edge data key corresponding to the edge weight.
Returns distance : dict
A dictionary, keyed by source and target, of shortest paths distances between nodes.
See also:
floyd_warshall_predecessor_and_distance,
floyd_warshall_numpy,
all_pairs_shortest_path, all_pairs_shortest_path_length
Notes
Floyd’s algorithm is appropriate for finding shortest paths in dense graphs or graphs with negative weights when
Dijkstra’s algorithm fails. This algorithm can still fail if there are negative cycles. It has running time O(n^3)
with running space of O(n^2).
floyd_warshall_predecessor_and_distance
floyd_warshall_predecessor_and_distance(G, weight=’weight’)
Find all-pairs shortest path lengths using Floyd’s algorithm.
Parameters G : NetworkX graph
weight: string, optional (default= �weight’)
Edge data key corresponding to the edge weight.
Returns predecessor,distance : dictionaries
Dictionaries, keyed by source and target, of predecessors and distances in the shortest
path.
See also:
floyd_warshall,
floyd_warshall_numpy,
all_pairs_shortest_path_length
all_pairs_shortest_path,
Notes
Floyd’s algorithm is appropriate for finding shortest paths in dense graphs or graphs with negative weights when
Dijkstra’s algorithm fails. This algorithm can still fail if there are negative cycles. It has running time O(n^3)
with running space of O(n^2).
floyd_warshall_numpy
floyd_warshall_numpy(G, nodelist=None, weight=’weight’)
Find all-pairs shortest path lengths using Floyd’s algorithm.
Parameters G : NetworkX graph
nodelist : list, optional
The rows and columns are ordered by the nodes in nodelist. If nodelist is None then the
ordering is produced by G.nodes().
weight: string, optional (default= �weight’)
Edge data key corresponding to the edge weight.
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Returns distance : NumPy matrix
A matrix of shortest path distances between nodes. If there is no path between to nodes
the corresponding matrix entry will be Inf.
Notes
Floyd’s algorithm is appropriate for finding shortest paths in dense graphs or graphs with negative weights when
Dijkstra’s algorithm fails. This algorithm can still fail if there are negative cycles. It has running time O(n^3)
with running space of O(n^2).
4.34.8 A* Algorithm
Shortest paths and path lengths using A* (“A star”) algorithm.
astar_path(G, source, target[, heuristic, ...])
astar_path_length(G, source, target[, ...])
Return a list of nodes in a shortest path between source and target using the A* (“A
Return the length of the shortest path between source and target using the A* (“A-s
astar_path
astar_path(G, source, target, heuristic=None, weight=’weight’)
Return a list of nodes in a shortest path between source and target using the A* (“A-star”) algorithm.
There may be more than one shortest path. This returns only one.
Parameters G : NetworkX graph
source : node
Starting node for path
target : node
Ending node for path
heuristic : function
A function to evaluate the estimate of the distance from the a node to the target. The
function takes two nodes arguments and must return a number.
weight: string, optional (default=’weight’)
Edge data key corresponding to the edge weight.
Raises NetworkXNoPath
If no path exists between source and target.
See also:
shortest_path, dijkstra_path
Examples
>>>
>>>
[0,
>>>
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G=nx.path_graph(5)
print(nx.astar_path(G,0,4))
1, 2, 3, 4]
G=nx.grid_graph(dim=[3,3]) # nodes are two-tuples (x,y)
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>>> def dist(a, b):
...
(x1, y1) = a
...
(x2, y2) = b
...
return ((x1 - x2) ** 2 + (y1 - y2) ** 2) ** 0.5
>>> print(nx.astar_path(G,(0,0),(2,2),dist))
[(0, 0), (0, 1), (1, 1), (1, 2), (2, 2)]
astar_path_length
astar_path_length(G, source, target, heuristic=None, weight=’weight’)
Return the length of the shortest path between source and target using the A* (“A-star”) algorithm.
Parameters G : NetworkX graph
source : node
Starting node for path
target : node
Ending node for path
heuristic : function
A function to evaluate the estimate of the distance from the a node to the target. The
function takes two nodes arguments and must return a number.
Raises NetworkXNoPath
If no path exists between source and target.
See also:
astar_path
4.35 Simple Paths
all_simple_paths(G, source, target[, cutoff])
Generate all simple paths in the graph G from source to target.
4.35.1 all_simple_paths
all_simple_paths(G, source, target, cutoff=None)
Generate all simple paths in the graph G from source to target.
A simple path is a path with no repeated nodes.
Parameters G : NetworkX graph
source : node
Starting node for path
target : node
Ending node for path
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
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Returns path_generator: generator
A generator that produces lists of simple paths. If there are no paths between the source
and target within the given cutoff the generator produces no output.
See also:
all_shortest_paths, shortest_path
Notes
This algorithm uses a modified depth-first search to generate the paths [R309]. A single path can be found in
𝑂(𝑉 + 𝐸) time but the number of simple paths in a graph can be very large, e.g. 𝑂(𝑛!) in the complete graph
of order n.
References
[R309]
Examples
>>> G = nx.complete_graph(4)
>>> for path in nx.all_simple_paths(G, source=0, target=3):
...
print(path)
...
[0, 1, 2, 3]
[0, 1, 3]
[0, 2, 1, 3]
[0, 2, 3]
[0, 3]
>>> paths = nx.all_simple_paths(G, source=0, target=3, cutoff=2)
>>> print(list(paths))
[[0, 1, 3], [0, 2, 3], [0, 3]]
4.36 Swap
Swap edges in a graph.
double_edge_swap(G[, nswap, max_tries])
connected_double_edge_swap(G[, nswap])
Swap two edges in the graph while keeping the node degrees fixed.
Attempt nswap double-edge swaps in the graph G.
4.36.1 double_edge_swap
double_edge_swap(G, nswap=1, max_tries=100)
Swap two edges in the graph while keeping the node degrees fixed.
A double-edge swap removes two randomly chosen edges u-v and x-y and creates the new edges u-x and v-y:
u--v
becomes
x--y
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If either the edge u-x or v-y already exist no swap is performed and another attempt is made to find a suitable
edge pair.
Parameters G : graph
An undirected graph
nswap : integer (optional, default=1)
Number of double-edge swaps to perform
max_tries : integer (optional)
Maximum number of attempts to swap edges
Returns G : graph
The graph after double edge swaps.
Notes
Does not enforce any connectivity constraints.
The graph G is modified in place.
4.36.2 connected_double_edge_swap
connected_double_edge_swap(G, nswap=1)
Attempt nswap double-edge swaps in the graph G.
A double-edge swap removes two randomly chosen edges u-v and x-y and creates the new edges u-x and v-y:
u--v
becomes
x--y
u
|
x
v
|
y
If either the edge u-x or v-y already exist no swap is performed so the actual count of swapped edges is always
<= nswap
Parameters G : graph
An undirected graph
nswap : integer (optional, default=1)
Number of double-edge swaps to perform
Returns G : int
The number of successful swaps
Notes
The initial graph G must be connected, and the resulting graph is connected. The graph G is modified in place.
References
[R310]
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4.37 Traversal
4.37.1 Depth First Search
Basic algorithms for depth-first searching the nodes of a graph.
Based on http://www.ics.uci.edu/~eppstein/PADS/DFS.py by D. Eppstein, July 2004.
dfs_edges(G[, source])
dfs_tree(G, source)
dfs_predecessors(G[, source])
dfs_successors(G[, source])
dfs_preorder_nodes(G[, source])
dfs_postorder_nodes(G[, source])
dfs_labeled_edges(G[, source])
Produce edges in a depth-first-search (DFS).
Return oriented tree constructed from a depth-first-search from source.
Return dictionary of predecessors in depth-first-search from source.
Return dictionary of successors in depth-first-search from source.
Produce nodes in a depth-first-search pre-ordering starting from source.
Produce nodes in a depth-first-search post-ordering starting from source.
Produce edges in a depth-first-search (DFS) labeled by type.
dfs_edges
dfs_edges(G, source=None)
Produce edges in a depth-first-search (DFS).
Parameters G : NetworkX graph
source : node, optional
Specify starting node for depth-first search and return edges in the component reachable
from source.
Returns edges: generator
A generator of edges in the depth-first-search.
Notes
Based on http://www.ics.uci.edu/~eppstein/PADS/DFS.py by D. Eppstein, July 2004.
If a source is not specified then a source is chosen arbitrarily and repeatedly until all components in the graph
are searched.
Examples
>>> G = nx.Graph()
>>> G.add_path([0,1,2])
>>> print(list(nx.dfs_edges(G,0)))
[(0, 1), (1, 2)]
dfs_tree
dfs_tree(G, source)
Return oriented tree constructed from a depth-first-search from source.
Parameters G : NetworkX graph
source : node, optional
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Specify starting node for depth-first search.
Returns T : NetworkX DiGraph
An oriented tree
Examples
>>> G = nx.Graph()
>>> G.add_path([0,1,2])
>>> T = nx.dfs_tree(G,0)
>>> print(T.edges())
[(0, 1), (1, 2)]
dfs_predecessors
dfs_predecessors(G, source=None)
Return dictionary of predecessors in depth-first-search from source.
Parameters G : NetworkX graph
source : node, optional
Specify starting node for depth-first search and return edges in the component reachable
from source.
Returns pred: dict
A dictionary with nodes as keys and predecessor nodes as values.
Notes
Based on http://www.ics.uci.edu/~eppstein/PADS/DFS.py by D. Eppstein, July 2004.
If a source is not specified then a source is chosen arbitrarily and repeatedly until all components in the graph
are searched.
Examples
>>>
>>>
>>>
{1:
G = nx.Graph()
G.add_path([0,1,2])
print(nx.dfs_predecessors(G,0))
0, 2: 1}
dfs_successors
dfs_successors(G, source=None)
Return dictionary of successors in depth-first-search from source.
Parameters G : NetworkX graph
source : node, optional
Specify starting node for depth-first search and return edges in the component reachable
from source.
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Returns succ: dict
A dictionary with nodes as keys and list of successor nodes as values.
Notes
Based on http://www.ics.uci.edu/~eppstein/PADS/DFS.py by D. Eppstein, July 2004.
If a source is not specified then a source is chosen arbitrarily and repeatedly until all components in the graph
are searched.
Examples
>>>
>>>
>>>
{0:
G = nx.Graph()
G.add_path([0,1,2])
print(nx.dfs_successors(G,0))
[1], 1: [2]}
dfs_preorder_nodes
dfs_preorder_nodes(G, source=None)
Produce nodes in a depth-first-search pre-ordering starting from source.
Parameters G : NetworkX graph
source : node, optional
Specify starting node for depth-first search and return edges in the component reachable
from source.
Returns nodes: generator
A generator of nodes in a depth-first-search pre-ordering.
Notes
Based on http://www.ics.uci.edu/~eppstein/PADS/DFS.py by D. Eppstein, July 2004.
If a source is not specified then a source is chosen arbitrarily and repeatedly until all components in the graph
are searched.
Examples
>>>
>>>
>>>
[0,
G = nx.Graph()
G.add_path([0,1,2])
print(list(nx.dfs_preorder_nodes(G,0)))
1, 2]
dfs_postorder_nodes
dfs_postorder_nodes(G, source=None)
Produce nodes in a depth-first-search post-ordering starting from source.
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Parameters G : NetworkX graph
source : node, optional
Specify starting node for depth-first search and return edges in the component reachable
from source.
Returns nodes: generator
A generator of nodes in a depth-first-search post-ordering.
Notes
Based on http://www.ics.uci.edu/~eppstein/PADS/DFS.py by D. Eppstein, July 2004.
If a source is not specified then a source is chosen arbitrarily and repeatedly until all components in the graph
are searched.
Examples
>>>
>>>
>>>
[2,
G = nx.Graph()
G.add_path([0,1,2])
print(list(nx.dfs_postorder_nodes(G,0)))
1, 0]
dfs_labeled_edges
dfs_labeled_edges(G, source=None)
Produce edges in a depth-first-search (DFS) labeled by type.
Parameters G : NetworkX graph
source : node, optional
Specify starting node for depth-first search and return edges in the component reachable
from source.
Returns edges: generator
A generator of edges in the depth-first-search labeled with �forward’, �nontree’, and
�reverse’.
Notes
Based on http://www.ics.uci.edu/~eppstein/PADS/DFS.py by D. Eppstein, July 2004.
If a source is not specified then a source is chosen arbitrarily and repeatedly until all components in the graph
are searched.
Examples
>>> G = nx.Graph()
>>> G.add_path([0,1,2])
>>> edges = (list(nx.dfs_labeled_edges(G,0)))
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4.37.2 Breadth First Search
Basic algorithms for breadth-first searching the nodes of a graph.
bfs_edges(G, source[, reverse])
bfs_tree(G, source[, reverse])
bfs_predecessors(G, source)
bfs_successors(G, source)
Produce edges in a breadth-first-search starting at source.
Return an oriented tree constructed from of a breadth-first-search starting at source.
Return dictionary of predecessors in breadth-first-search from source.
Return dictionary of successors in breadth-first-search from source.
bfs_edges
bfs_edges(G, source, reverse=False)
Produce edges in a breadth-first-search starting at source.
Parameters G : NetworkX graph
source : node, optional
Specify starting node for breadth-first search and return edges in the component reachable from source.
reverse : bool, optional
If True traverse a directed graph in the reverse direction
Returns edges: generator
A generator of edges in the breadth-first-search.
Notes
Based on http://www.ics.uci.edu/~eppstein/PADS/BFS.py by D. Eppstein, July 2004.
If a source is not specified then a source is chosen arbitrarily and repeatedly until all components in the graph
are searched.
Examples
>>> G = nx.Graph()
>>> G.add_path([0,1,2])
>>> print(list(nx.bfs_edges(G,0)))
[(0, 1), (1, 2)]
bfs_tree
bfs_tree(G, source, reverse=False)
Return an oriented tree constructed from of a breadth-first-search starting at source.
Parameters G : NetworkX graph
source : node, optional
Specify starting node for breadth-first search and return edges in the component reachable from source.
reverse : bool, optional
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If True traverse a directed graph in the reverse direction
Returns T: NetworkX DiGraph
An oriented tree
Notes
Based on http://www.ics.uci.edu/~eppstein/PADS/BFS.py by D. Eppstein, July 2004.
If a source is not specified then a source is chosen arbitrarily and repeatedly until all components in the graph
are searched.
Examples
>>> G = nx.Graph()
>>> G.add_path([0,1,2])
>>> print(list(nx.bfs_edges(G,0)))
[(0, 1), (1, 2)]
bfs_predecessors
bfs_predecessors(G, source)
Return dictionary of predecessors in breadth-first-search from source.
Parameters G : NetworkX graph
source : node, optional
Specify starting node for breadth-first search and return edges in the component reachable from source.
Returns pred: dict
A dictionary with nodes as keys and predecessor nodes as values.
Notes
Based on http://www.ics.uci.edu/~eppstein/PADS/BFS.py by D. Eppstein, July 2004.
If a source is not specified then a source is chosen arbitrarily and repeatedly until all components in the graph
are searched.
Examples
>>>
>>>
>>>
{1:
G = nx.Graph()
G.add_path([0,1,2])
print(nx.bfs_predecessors(G,0))
0, 2: 1}
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bfs_successors
bfs_successors(G, source)
Return dictionary of successors in breadth-first-search from source.
Parameters G : NetworkX graph
source : node, optional
Specify starting node for breadth-first search and return edges in the component reachable from source.
Returns succ: dict
A dictionary with nodes as keys and list of succssors nodes as values.
Notes
Based on http://www.ics.uci.edu/~eppstein/PADS/BFS.py by D. Eppstein, July 2004.
If a source is not specified then a source is chosen arbitrarily and repeatedly until all components in the graph
are searched.
Examples
>>>
>>>
>>>
{0:
G = nx.Graph()
G.add_path([0,1,2])
print(nx.bfs_successors(G,0))
[1], 1: [2]}
4.37.3 Depth First Search on Edges
Algorithms for a depth-first traversal of edges in a graph.
edge_dfs(G[, source, orientation])
A directed, depth-first traversal of edges in G, beginning at source.
edge_dfs
edge_dfs(G, source=None, orientation=’original’)
A directed, depth-first traversal of edges in G, beginning at source.
Parameters G : graph
A directed/undirected graph/multigraph.
source : node, list of nodes
The node from which the traversal begins. If None, then a source is chosen arbitrarily
and repeatedly until all edges from each node in the graph are searched.
orientation : �original’ | �reverse’ | �ignore’
For directed graphs and directed multigraphs, edge traversals need not respect the original orientation of the edges. When set to �reverse’, then every edge will be traversed
in the reverse direction. When set to �ignore’, then each directed edge is treated as a
single undirected edge that can be traversed in either direction. For undirected graphs
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and undirected multigraphs, this parameter is meaningless and is not consulted by the
algorithm.
See also:
dfs_edges
Notes
The goal of this function is to visit edges. It differs from the more familiar depth-first traversal of nodes, as
provided by networkx.algorithms.traversal.depth_first_search.dfs_edges(), in that
it does not stop once every node has been visited. In a directed graph with edges [(0, 1), (1, 2), (2, 1)], the edge
(2, 1) would not be visited if not for the functionality provided by this function.
Examples
>>> import networkx as nx
>>> nodes = [0, 1, 2, 3]
>>> edges = [(0, 1), (1, 0), (1, 0), (2, 1), (3, 1)]
>>> list(nx.edge_dfs(nx.Graph(edges), nodes))
[(0, 1), (1, 2), (1, 3)]
>>> list(nx.edge_dfs(nx.DiGraph(edges), nodes))
[(0, 1), (1, 0), (2, 1), (3, 1)]
>>> list(nx.edge_dfs(nx.MultiGraph(edges), nodes))
[(0, 1, 0), (1, 0, 1), (0, 1, 2), (1, 2, 0), (1, 3, 0)]
>>> list(nx.edge_dfs(nx.MultiDiGraph(edges), nodes))
[(0, 1, 0), (1, 0, 0), (1, 0, 1), (2, 1, 0), (3, 1, 0)]
>>> list(nx.edge_dfs(nx.DiGraph(edges), nodes, orientation='ignore'))
[(0, 1, 'forward'), (1, 0, 'forward'), (2, 1, 'reverse'), (3, 1, 'reverse')]
>>> list(nx.edge_dfs(nx.MultiDiGraph(edges), nodes, orientation='ignore'))
[(0, 1, 0, 'forward'), (1, 0, 0, 'forward'), (1, 0, 1, 'reverse'), (2, 1, 0, 'reverse'), (3, 1,
4.38 Tree
4.38.1 Recognition
Recognition Tests
A forest is an acyclic, undirected graph, and a tree is a connected forest. Depending on the subfield, there are various
conventions for generalizing these definitions to directed graphs.
In one convention, directed variants of forest and tree are defined in an identical manner, except that the direction of
the edges is ignored. In effect, each directed edge is treated as a single undirected edge. Then, additional restrictions
are imposed to define branchings and arborescences.
In another convention, directed variants of forest and tree correspond to the previous convention’s branchings and
arborescences, respectively. Then two new terms, polyforest and polytree, are defined to correspond to the other
convention’s forest and tree.
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Summarizing:
+-----------------------------+
| Convention 1 | Convention 2 |
+=============================+
| forest
| polyforest
|
| tree
| polytree
|
| branching
| forest
|
| arborescence | tree
|
+-----------------------------+
Each convention has its reasons. The first convention emphasizes definitional similarity in that directed forests and
trees are only concerned with acyclicity and do not have an in-degree constraint, just as their undirected counterparts do
not. The second convention emphasizes functional similarity in the sense that the directed analog of a spanning tree is
a spanning arborescence. That is, take any spanning tree and choose one node as the root. Then every edge is assigned
a direction such there is a directed path from the root to every other node. The result is a spanning arborescence.
NetworkX follows the first convention. Explicitly, these are:
undirected forest An undirected graph with no undirected cycles.
undirected tree A connected, undirected forest.
directed forest A directed graph with no undirected cycles. Equivalently, the underlying graph structure (which
ignores edge orientations) is an undirected forest. In another convention, this is known as a polyforest.
directed tree A weakly connected, directed forest. Equivalently, the underlying graph structure (which ignores edge
orientations) is an undirected tree. In another convention, this is known as a polytree.
branching A directed forest with each node having, at most, one parent. So the maximum in-degree is equal to 1. In
another convention, this is known as a forest.
arborescence A directed tree with each node having, at most, one parent. So the maximum in-degree is equal to 1.
In another convention, this is known as a tree.
is_tree(G)
is_forest(G)
Returns 𝑇 𝑟𝑢𝑒 if 𝐺 is a tree.
Returns 𝑇 𝑟𝑢𝑒 if G is a forest.
is_tree
is_tree(G)
Returns 𝑇 𝑟𝑢𝑒 if 𝐺 is a tree.
A tree is a connected graph with no undirected cycles.
For directed graphs, рќђє is a tree if the underlying graph is a tree. The underlying graph is obtained by treating
each directed edge as a single undirected edge in a multigraph.
Parameters G : graph
The graph to test.
Returns b : bool
A boolean that is 𝑇 𝑟𝑢𝑒 if 𝐺 is a tree.
See also:
is_arborescence
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Notes
In another convention, a directed tree is known as a polytree and then tree corresponds to an arborescence.
is_forest
is_forest(G)
Returns 𝑇 𝑟𝑢𝑒 if G is a forest.
A forest is a graph with no undirected cycles.
For directed graphs, рќђє is a forest if the underlying graph is a forest. The underlying graph is obtained by treating
each directed edge as a single undirected edge in a multigraph.
Parameters G : graph
The graph to test.
Returns b : bool
A boolean that is 𝑇 𝑟𝑢𝑒 if 𝐺 is a forest.
See also:
is_branching
Notes
In another convention, a directed forest is known as a polyforest and then forest corresponds to a branching.
4.39 Vitality
Vitality measures.
closeness_vitality(G[, weight])
Compute closeness vitality for nodes.
4.39.1 closeness_vitality
closeness_vitality(G, weight=None)
Compute closeness vitality for nodes.
Closeness vitality of a node is the change in the sum of distances between all node pairs when excluding that
node.
Parameters G : graph
weight : None or string (optional)
The name of the edge attribute used as weight. If None the edge weights are ignored.
Returns nodes : dictionary
Dictionary with nodes as keys and closeness vitality as the value.
See also:
closeness_centrality
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References
[R311]
Examples
>>> G=nx.cycle_graph(3)
>>> nx.closeness_vitality(G)
{0: 4.0, 1: 4.0, 2: 4.0}
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CHAPTER
FIVE
FUNCTIONS
Functional interface to graph methods and assorted utilities.
5.1 Graph
degree(G[, nbunch, weight])
degree_histogram(G)
density(G)
info(G[, n])
create_empty_copy(G[, with_nodes])
is_directed(G)
Return degree of single node or of nbunch of nodes.
Return a list of the frequency of each degree value.
Return the density of a graph.
Print short summary of information for the graph G or the node n.
Return a copy of the graph G with all of the edges removed.
Return True if graph is directed.
5.1.1 degree
degree(G, nbunch=None, weight=None)
Return degree of single node or of nbunch of nodes. If nbunch is ommitted, then return degrees of all nodes.
5.1.2 degree_histogram
degree_histogram(G)
Return a list of the frequency of each degree value.
Parameters G : Networkx graph
A graph
Returns hist : list
A list of frequencies of degrees. The degree values are the index in the list.
Notes
Note: the bins are width one, hence len(list) can be large (Order(number_of_edges))
5.1.3 density
density(G)
Return the density of a graph.
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The density for undirected graphs is
рќ‘‘=
2рќ‘љ
,
рќ‘›(рќ‘› в€’ 1)
рќ‘‘=
рќ‘љ
,
рќ‘›(рќ‘› в€’ 1)
and for directed graphs is
where рќ‘› is the number of nodes and рќ‘љ is the number of edges in рќђє.
Notes
The density is 0 for a graph without edges and 1 for a complete graph. The density of multigraphs can be higher
than 1.
Self loops are counted in the total number of edges so graphs with self loops can have density higher than 1.
5.1.4 info
info(G, n=None)
Print short summary of information for the graph G or the node n.
Parameters G : Networkx graph
A graph
n : node (any hashable)
A node in the graph G
5.1.5 create_empty_copy
create_empty_copy(G, with_nodes=True)
Return a copy of the graph G with all of the edges removed.
Parameters G : graph
A NetworkX graph
with_nodes : bool (default=True)
Include nodes.
Notes
Graph, node, and edge data is not propagated to the new graph.
5.1.6 is_directed
is_directed(G)
Return True if graph is directed.
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5.2 Nodes
5.2. Nodes
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nodes(G)
number_of_nodes(G)
nodes_iter(G)
all_neighbors(graph, node)
non_neighbors(graph, node)
common_neighbors(G, u, v)
Return a copy of the graph nodes in a list.
Return the number of nodes in the graph.
Return an iterator over the graph nodes.
Returns all of the neighbors of a node in the graph.
Returns the non-neighbors of the node in the graph.
Return the common neighbors of two nodes in a graph.
5.2.1 nodes
nodes(G)
Return a copy of the graph nodes in a list.
5.2.2 number_of_nodes
number_of_nodes(G)
Return the number of nodes in the graph.
5.2.3 nodes_iter
nodes_iter(G)
Return an iterator over the graph nodes.
5.2.4 all_neighbors
all_neighbors(graph, node)
Returns all of the neighbors of a node in the graph.
If the graph is directed returns predecessors as well as successors.
Parameters graph : NetworkX graph
Graph to find neighbors.
node : node
The node whose neighbors will be returned.
Returns neighbors : iterator
Iterator of neighbors
5.2.5 non_neighbors
non_neighbors(graph, node)
Returns the non-neighbors of the node in the graph.
Parameters graph : NetworkX graph
Graph to find neighbors.
node : node
The node whose neighbors will be returned.
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Returns non_neighbors : iterator
Iterator of nodes in the graph that are not neighbors of the node.
5.2.6 common_neighbors
common_neighbors(G, u, v)
Return the common neighbors of two nodes in a graph.
Parameters G : graph
A NetworkX undirected graph.
u, v : nodes
Nodes in the graph.
Returns cnbors : iterator
Iterator of common neighbors of u and v in the graph.
Raises NetworkXError
If u or v is not a node in the graph.
Examples
>>> G = nx.complete_graph(5)
>>> sorted(nx.common_neighbors(G, 0, 1))
[2, 3, 4]
5.3 Edges
edges(G[, nbunch])
number_of_edges(G)
edges_iter(G[, nbunch])
non_edges(graph)
Return list of edges incident to nodes in nbunch.
Return the number of edges in the graph.
Return iterator over edges incident to nodes in nbunch.
Returns the non-existent edges in the graph.
5.3.1 edges
edges(G, nbunch=None)
Return list of edges incident to nodes in nbunch.
Return all edges if nbunch is unspecified or nbunch=None.
For digraphs, edges=out_edges
5.3.2 number_of_edges
number_of_edges(G)
Return the number of edges in the graph.
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5.3.3 edges_iter
edges_iter(G, nbunch=None)
Return iterator over edges incident to nodes in nbunch.
Return all edges if nbunch is unspecified or nbunch=None.
For digraphs, edges=out_edges
5.3.4 non_edges
non_edges(graph)
Returns the non-existent edges in the graph.
Parameters graph : NetworkX graph.
Graph to find non-existent edges.
Returns non_edges : iterator
Iterator of edges that are not in the graph.
5.4 Attributes
set_node_attributes(G, name, values)
get_node_attributes(G, name)
set_edge_attributes(G, name, values)
get_edge_attributes(G, name)
Set node attributes from dictionary of nodes and values
Get node attributes from graph
Set edge attributes from dictionary of edge tuples and values.
Get edge attributes from graph
5.4.1 set_node_attributes
set_node_attributes(G, name, values)
Set node attributes from dictionary of nodes and values
Parameters G : NetworkX Graph
name : string
Attribute name
values: dict
Dictionary of attribute values keyed by node. If рќ‘Јрќ‘Ћрќ‘™рќ‘ўрќ‘’рќ‘ is not a dictionary, then it is
treated as a single attribute value that is then applied to every node in рќђє.
Examples
>>>
>>>
>>>
>>>
1.0
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G = nx.path_graph(3)
bb = nx.betweenness_centrality(G)
nx.set_node_attributes(G, 'betweenness', bb)
G.node[1]['betweenness']
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5.4.2 get_node_attributes
get_node_attributes(G, name)
Get node attributes from graph
Parameters G : NetworkX Graph
name : string
Attribute name
Returns Dictionary of attributes keyed by node.
Examples
>>> G=nx.Graph()
>>> G.add_nodes_from([1,2,3],color='red')
>>> color=nx.get_node_attributes(G,'color')
>>> color[1]
'red'
5.4.3 set_edge_attributes
set_edge_attributes(G, name, values)
Set edge attributes from dictionary of edge tuples and values.
Parameters G : NetworkX Graph
name : string
Attribute name
values : dict
Dictionary of attribute values keyed by edge (tuple). For multigraphs, the keys tuples
must be of the form (u, v, key). For non-multigraphs, the keys must be tuples of the
form (u, v). If рќ‘Јрќ‘Ћрќ‘™рќ‘ўрќ‘’рќ‘ is not a dictionary, then it is treated as a single attribute value that
is then applied to every edge in рќђє.
Examples
>>>
>>>
>>>
>>>
2.0
G = nx.path_graph(3)
bb = nx.edge_betweenness_centrality(G, normalized=False)
nx.set_edge_attributes(G, 'betweenness', bb)
G[1][2]['betweenness']
5.4.4 get_edge_attributes
get_edge_attributes(G, name)
Get edge attributes from graph
Parameters G : NetworkX Graph
name : string
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Attribute name
Returns Dictionary of attributes keyed by edge. For (di)graphs, the keys are
2-tuples of the form: (u,v). For multi(di)graphs, the keys are 3-tuples of
the form: (u, v, key).
Examples
>>> G=nx.Graph()
>>> G.add_path([1,2,3],color='red')
>>> color=nx.get_edge_attributes(G,'color')
>>> color[(1,2)]
'red'
5.5 Freezing graph structure
freeze(G)
is_frozen(G)
Modify graph to prevent further change by adding or removing nodes or edges.
Return True if graph is frozen.
5.5.1 freeze
freeze(G)
Modify graph to prevent further change by adding or removing nodes or edges.
Node and edge data can still be modified.
Parameters G : graph
A NetworkX graph
See also:
is_frozen
Notes
To “unfreeze” a graph you must make a copy by creating a new graph object:
>>> graph = nx.path_graph(4)
>>> frozen_graph = nx.freeze(graph)
>>> unfrozen_graph = nx.Graph(frozen_graph)
>>> nx.is_frozen(unfrozen_graph)
False
Examples
>>>
>>>
>>>
>>>
...
350
G=nx.Graph()
G.add_path([0,1,2,3])
G=nx.freeze(G)
try:
G.add_edge(4,5)
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... except nx.NetworkXError as e:
...
print(str(e))
Frozen graph can't be modified
5.5.2 is_frozen
is_frozen(G)
Return True if graph is frozen.
Parameters G : graph
A NetworkX graph
See also:
freeze
5.5. Freezing graph structure
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GRAPH GENERATORS
6.1 Atlas
Generators for the small graph atlas.
See “An Atlas of Graphs” by Ronald C. Read and Robin J. Wilson, Oxford University Press, 1998.
Because of its size, this module is not imported by default.
graph_atlas_g()
Return the list [G0,G1,...,G1252] of graphs as named in the Graph Atlas.
6.1.1 graph_atlas_g
graph_atlas_g()
Return the list [G0,G1,...,G1252] of graphs as named in the Graph Atlas. G0,G1,...,G1252 are all graphs with
up to 7 nodes.
The graphs are listed:
1. in increasing order of number of nodes;
2. for a fixed number of nodes, in increasing order of the number of edges;
3. for fixed numbers of nodes and edges, in increasing order of the degree sequence, for example 111223
< 112222;
4. for fixed degree sequence, in increasing number of automorphisms.
Note that indexing is set up so that for GAG=graph_atlas_g(), then G123=GAG[123] and G[0]=empty_graph(0)
6.2 Classic
Generators for some classic graphs.
The typical graph generator is called as follows:
>>> G=nx.complete_graph(100)
returning the complete graph on n nodes labeled 0,..,99 as a simple graph. Except for empty_graph, all the generators
in this module return a Graph class (i.e. a simple, undirected graph).
balanced_tree(r, h[, create_using])
Return the perfectly balanced r-tree of height h.
C
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Table 6.2 – continued from previous page
barbell_graph(m1, m2[, create_using])
Return the Barbell Graph: two complete graphs connected by a path.
complete_graph(n[, create_using])
Return the complete graph K_n with n nodes.
complete_bipartite_graph(n1, n2[, create_using]) Return the complete bipartite graph K_{n1_n2}.
circular_ladder_graph(n[, create_using])
Return the circular ladder graph CL_n of length n.
cycle_graph(n[, create_using])
Return the cycle graph C_n over n nodes.
dorogovtsev_goltsev_mendes_graph(n[, ...])
Return the hierarchically constructed Dorogovtsev-Goltsev-Mendes gra
empty_graph([n, create_using])
Return the empty graph with n nodes and zero edges.
grid_2d_graph(m, n[, periodic, create_using])
Return the 2d grid graph of mxn nodes, each connected to its nearest ne
grid_graph(dim[, periodic])
Return the n-dimensional grid graph.
hypercube_graph(n)
Return the n-dimensional hypercube.
ladder_graph(n[, create_using])
Return the Ladder graph of length n.
lollipop_graph(m, n[, create_using])
Return the Lollipop Graph; рќђѕрќ‘љ connected to рќ‘ѓрќ‘› .
null_graph([create_using])
Return the Null graph with no nodes or edges.
path_graph(n[, create_using])
Return the Path graph P_n of n nodes linearly connected by n-1 edges.
star_graph(n[, create_using])
Return the Star graph with n+1 nodes: one center node, connected to n o
trivial_graph([create_using])
Return the Trivial graph with one node (with integer label 0) and no edg
wheel_graph(n[, create_using])
Return the wheel graph: a single hub node connected to each node of th
6.2.1 balanced_tree
balanced_tree(r, h, create_using=None)
Return the perfectly balanced r-tree of height h.
Parameters r : int
Branching factor of the tree
h : int
Height of the tree
create_using : NetworkX graph type, optional
Use specified type to construct graph (default = networkx.Graph)
Returns G : networkx Graph
A tree with n nodes
Notes
This is the rooted tree where all leaves are at distance h from the root. The root has degree r and all other internal
nodes have degree r+1.
Node labels are the integers 0 (the root) up to number_of_nodes - 1.
Also refered to as a complete r-ary tree.
6.2.2 barbell_graph
barbell_graph(m1, m2, create_using=None)
Return the Barbell Graph: two complete graphs connected by a path.
For m1 > 1 and m2 >= 0.
Two identical complete graphs K_{m1} form the left and right bells, and are connected by a path P_{m2}.
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The 2*m1+m2 nodes are numbered 0,...,m1-1 for the left barbell, m1,...,m1+m2-1 for the path, and
m1+m2,...,2*m1+m2-1 for the right barbell.
The 3 subgraphs are joined via the edges (m1-1,m1) and (m1+m2-1,m1+m2). If m2=0, this is merely two
complete graphs joined together.
This graph is an extremal example in David Aldous and Jim Fill’s etext on Random Walks on Graphs.
6.2.3 complete_graph
complete_graph(n, create_using=None)
Return the complete graph K_n with n nodes.
Node labels are the integers 0 to n-1.
6.2.4 complete_bipartite_graph
complete_bipartite_graph(n1, n2, create_using=None)
Return the complete bipartite graph K_{n1_n2}.
Composed of two partitions with n1 nodes in the first and n2 nodes in the second. Each node in the first is
connected to each node in the second.
Node labels are the integers 0 to n1+n2-1
6.2.5 circular_ladder_graph
circular_ladder_graph(n, create_using=None)
Return the circular ladder graph CL_n of length n.
CL_n consists of two concentric n-cycles in which each of the n pairs of concentric nodes are joined by an edge.
Node labels are the integers 0 to n-1
6.2.6 cycle_graph
cycle_graph(n, create_using=None)
Return the cycle graph C_n over n nodes.
C_n is the n-path with two end-nodes connected.
Node labels are the integers 0 to n-1 If create_using is a DiGraph, the direction is in increasing order.
6.2.7 dorogovtsev_goltsev_mendes_graph
dorogovtsev_goltsev_mendes_graph(n, create_using=None)
Return the hierarchically constructed Dorogovtsev-Goltsev-Mendes graph.
n is the generation. See: arXiv:/cond-mat/0112143 by Dorogovtsev, Goltsev and Mendes.
6.2. Classic
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6.2.8 empty_graph
empty_graph(n=0, create_using=None)
Return the empty graph with n nodes and zero edges.
Node labels are the integers 0 to n-1
For example: >>> G=nx.empty_graph(10) >>> G.number_of_nodes() 10 >>> G.number_of_edges() 0
The variable create_using should point to a “graph”-like object that will be cleaned (nodes and edges will
be removed) and refitted as an empty “graph” with n nodes with integer labels. This capability is useful for
specifying the class-nature of the resulting empty “graph” (i.e. Graph, DiGraph, MyWeirdGraphClass, etc.).
The variable create_using has two main uses: Firstly, the variable create_using can be used to create an empty
digraph, network,etc. For example,
>>> n=10
>>> G=nx.empty_graph(n,create_using=nx.DiGraph())
will create an empty digraph on n nodes.
Secondly, one can pass an existing graph (digraph, pseudograph, etc.) via create_using. For example, if G is
an existing graph (resp. digraph, pseudograph, etc.), then empty_graph(n,create_using=G) will empty G (i.e.
delete all nodes and edges using G.clear() in base) and then add n nodes and zero edges, and return the modified
graph (resp. digraph, pseudograph, etc.).
See also create_empty_copy(G).
6.2.9 grid_2d_graph
grid_2d_graph(m, n, periodic=False, create_using=None)
Return the 2d grid graph of mxn nodes, each connected to its nearest neighbors. Optional argument periodic=True will connect boundary nodes via periodic boundary conditions.
6.2.10 grid_graph
grid_graph(dim, periodic=False)
Return the n-dimensional grid graph.
The dimension is the length of the list �dim’ and the size in each dimension is the value of the list element.
E.g. G=grid_graph(dim=[2,3]) produces a 2x3 grid graph.
If periodic=True then join grid edges with periodic boundary conditions.
6.2.11 hypercube_graph
hypercube_graph(n)
Return the n-dimensional hypercube.
Node labels are the integers 0 to 2**n - 1.
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6.2.12 ladder_graph
ladder_graph(n, create_using=None)
Return the Ladder graph of length n.
This is two rows of n nodes, with each pair connected by a single edge.
Node labels are the integers 0 to 2*n - 1.
6.2.13 lollipop_graph
lollipop_graph(m, n, create_using=None)
Return the Lollipop Graph; рќђѕрќ‘љ connected to рќ‘ѓрќ‘› .
This is the Barbell Graph without the right barbell.
For m>1 and n>=0, the complete graph K_m is connected to the path P_n. The resulting m+n nodes are labelled
0,...,m-1 for the complete graph and m,...,m+n-1 for the path. The 2 subgraphs are joined via the edge (m-1,m).
If n=0, this is merely a complete graph.
Node labels are the integers 0 to number_of_nodes - 1.
(This graph is an extremal example in David Aldous and Jim Fill’s etext on Random Walks on Graphs.)
6.2.14 null_graph
null_graph(create_using=None)
Return the Null graph with no nodes or edges.
See empty_graph for the use of create_using.
6.2.15 path_graph
path_graph(n, create_using=None)
Return the Path graph P_n of n nodes linearly connected by n-1 edges.
Node labels are the integers 0 to n - 1. If create_using is a DiGraph then the edges are directed in increasing
order.
6.2.16 star_graph
star_graph(n, create_using=None)
Return the Star graph with n+1 nodes: one center node, connected to n outer nodes.
Node labels are the integers 0 to n.
6.2.17 trivial_graph
trivial_graph(create_using=None)
Return the Trivial graph with one node (with integer label 0) and no edges.
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6.2.18 wheel_graph
wheel_graph(n, create_using=None)
Return the wheel graph: a single hub node connected to each node of the (n-1)-node cycle graph.
Node labels are the integers 0 to n - 1.
6.3 Expanders
Provides explicit constructions of expander graphs.
margulis_gabber_galil_graph(n[, create_using])
chordal_cycle_graph(p[, create_using])
Return the Margulis-Gabber-Galil undirected MultiGraph on рќ‘›2 nodes.
Return the chordal cycle graph on рќ‘ќ nodes.
6.4 Small
Various small and named graphs, together with some compact generators.
make_small_graph(graph_description[, ...])
LCF_graph(n, shift_list, repeats[, create_using])
bull_graph([create_using])
chvatal_graph([create_using])
cubical_graph([create_using])
desargues_graph([create_using])
diamond_graph([create_using])
dodecahedral_graph([create_using])
frucht_graph([create_using])
heawood_graph([create_using])
house_graph([create_using])
house_x_graph([create_using])
icosahedral_graph([create_using])
krackhardt_kite_graph([create_using])
moebius_kantor_graph([create_using])
octahedral_graph([create_using])
pappus_graph()
petersen_graph([create_using])
sedgewick_maze_graph([create_using])
tetrahedral_graph([create_using])
truncated_cube_graph([create_using])
truncated_tetrahedron_graph([create_using])
tutte_graph([create_using])
Return the small graph described by graph_description.
Return the cubic graph specified in LCF notation.
Return the Bull graph.
Return the ChvГЎtal graph.
Return the 3-regular Platonic Cubical graph.
Return the Desargues graph.
Return the Diamond graph.
Return the Platonic Dodecahedral graph.
Return the Frucht Graph.
Return the Heawood graph, a (3,6) cage.
Return the House graph (square with triangle on top).
Return the House graph with a cross inside the house square.
Return the Platonic Icosahedral graph.
Return the Krackhardt Kite Social Network.
Return the Moebius-Kantor graph.
Return the Platonic Octahedral graph.
Return the Pappus graph.
Return the Petersen graph.
Return a small maze with a cycle.
Return the 3-regular Platonic Tetrahedral graph.
Return the skeleton of the truncated cube.
Return the skeleton of the truncated Platonic tetrahedron.
Return the Tutte graph.
6.4.1 make_small_graph
make_small_graph(graph_description, create_using=None)
Return the small graph described by graph_description.
graph_description is a list of the form [ltype,name,n,xlist]
Here ltype is one of “adjacencylist” or “edgelist”, name is the name of the graph and n the number of nodes.
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This constructs a graph of n nodes with integer labels 0,..,n-1.
If ltype=”adjacencylist” then xlist is an adjacency list with exactly n entries, in with the j’th entry (which can be
empty) specifies the nodes connected to vertex j. e.g. the “square” graph C_4 can be obtained by
>>> G=nx.make_small_graph(["adjacencylist","C_4",4,[[2,4],[1,3],[2,4],[1,3]]])
or, since we do not need to add edges twice,
>>> G=nx.make_small_graph(["adjacencylist","C_4",4,[[2,4],[3],[4],[]]])
If ltype=”edgelist” then xlist is an edge list written as [[v1,w2],[v2,w2],...,[vk,wk]], where vj and wj integers in
the range 1,..,n e.g. the “square” graph C_4 can be obtained by
>>> G=nx.make_small_graph(["edgelist","C_4",4,[[1,2],[3,4],[2,3],[4,1]]])
Use the create_using argument to choose the graph class/type.
6.4.2 LCF_graph
LCF_graph(n, shift_list, repeats, create_using=None)
Return the cubic graph specified in LCF notation.
LCF notation (LCF=Lederberg-Coxeter-Fruchte) is a compressed notation used in the generation of various
cubic Hamiltonian graphs of high symmetry. See, for example, dodecahedral_graph, desargues_graph, heawood_graph and pappus_graph below.
n (number of nodes) The starting graph is the n-cycle with nodes 0,...,n-1. (The null graph is returned if n <
0.)
shift_list = [s1,s2,..,sk], a list of integer shifts mod n,
repeats integer specifying the number of times that shifts in shift_list are successively applied to each v_current
in the n-cycle to generate an edge between v_current and v_current+shift mod n.
For v1 cycling through the n-cycle a total of k*repeats with shift cycling through shiftlist repeats times connect
v1 with v1+shift mod n
The utility graph K_{3,3}
>>> G=nx.LCF_graph(6,[3,-3],3)
The Heawood graph
>>> G=nx.LCF_graph(14,[5,-5],7)
See http://mathworld.wolfram.com/LCFNotation.html for a description and references.
6.4.3 bull_graph
bull_graph(create_using=None)
Return the Bull graph.
6.4.4 chvatal_graph
chvatal_graph(create_using=None)
Return the ChvГЎtal graph.
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6.4.5 cubical_graph
cubical_graph(create_using=None)
Return the 3-regular Platonic Cubical graph.
6.4.6 desargues_graph
desargues_graph(create_using=None)
Return the Desargues graph.
6.4.7 diamond_graph
diamond_graph(create_using=None)
Return the Diamond graph.
6.4.8 dodecahedral_graph
dodecahedral_graph(create_using=None)
Return the Platonic Dodecahedral graph.
6.4.9 frucht_graph
frucht_graph(create_using=None)
Return the Frucht Graph.
The Frucht Graph is the smallest cubical graph whose automorphism group consists only of the identity element.
6.4.10 heawood_graph
heawood_graph(create_using=None)
Return the Heawood graph, a (3,6) cage.
6.4.11 house_graph
house_graph(create_using=None)
Return the House graph (square with triangle on top).
6.4.12 house_x_graph
house_x_graph(create_using=None)
Return the House graph with a cross inside the house square.
6.4.13 icosahedral_graph
icosahedral_graph(create_using=None)
Return the Platonic Icosahedral graph.
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6.4.14 krackhardt_kite_graph
krackhardt_kite_graph(create_using=None)
Return the Krackhardt Kite Social Network.
A 10 actor social network introduced by David Krackhardt to illustrate: degree, betweenness, centrality, closeness, etc. The traditional labeling is: Andre=1, Beverley=2, Carol=3, Diane=4, Ed=5, Fernando=6, Garth=7,
Heather=8, Ike=9, Jane=10.
6.4.15 moebius_kantor_graph
moebius_kantor_graph(create_using=None)
Return the Moebius-Kantor graph.
6.4.16 octahedral_graph
octahedral_graph(create_using=None)
Return the Platonic Octahedral graph.
6.4.17 pappus_graph
pappus_graph()
Return the Pappus graph.
6.4.18 petersen_graph
petersen_graph(create_using=None)
Return the Petersen graph.
6.4.19 sedgewick_maze_graph
sedgewick_maze_graph(create_using=None)
Return a small maze with a cycle.
This is the maze used in Sedgewick,3rd Edition, Part 5, Graph Algorithms, Chapter 18, e.g. Figure 18.2 and
following. Nodes are numbered 0,..,7
6.4.20 tetrahedral_graph
tetrahedral_graph(create_using=None)
Return the 3-regular Platonic Tetrahedral graph.
6.4.21 truncated_cube_graph
truncated_cube_graph(create_using=None)
Return the skeleton of the truncated cube.
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6.4.22 truncated_tetrahedron_graph
truncated_tetrahedron_graph(create_using=None)
Return the skeleton of the truncated Platonic tetrahedron.
6.4.23 tutte_graph
tutte_graph(create_using=None)
Return the Tutte graph.
6.5 Random Graphs
Generators for random graphs.
fast_gnp_random_graph(n, p[, seed, directed])
gnp_random_graph(n, p[, seed, directed])
dense_gnm_random_graph(n, m[, seed])
gnm_random_graph(n, m[, seed, directed])
erdos_renyi_graph(n, p[, seed, directed])
binomial_graph(n, p[, seed, directed])
newman_watts_strogatz_graph(n, k, p[, seed])
watts_strogatz_graph(n, k, p[, seed])
connected_watts_strogatz_graph(n, k, p[, ...])
random_regular_graph(d, n[, seed])
barabasi_albert_graph(n, m[, seed])
powerlaw_cluster_graph(n, m, p[, seed])
random_lobster(n, p1, p2[, seed])
random_shell_graph(constructor[, seed])
random_powerlaw_tree(n[, gamma, seed, tries])
random_powerlaw_tree_sequence(n[, gamma, ...])
Return a random graph G_{n,p} (ErdЛќos-RГ©nyi graph, binomial graph).
Return a random graph G_{n,p} (ErdЛќos-RГ©nyi graph, binomial graph).
Return the random graph G_{n,m}.
Return the random graph G_{n,m}.
Return a random graph G_{n,p} (ErdЛќos-RГ©nyi graph, binomial graph).
Return a random graph G_{n,p} (ErdЛќos-RГ©nyi graph, binomial graph).
Return a Newman-Watts-Strogatz small world graph.
Return a Watts-Strogatz small-world graph.
Return a connected Watts-Strogatz small-world graph.
Return a random regular graph of n nodes each with degree d.
Return random graph using BarabГЎsi-Albert preferential attachment mo
Holme and Kim algorithm for growing graphs with powerlaw degree di
Return a random lobster.
Return a random shell graph for the constructor given.
Return a tree with a powerlaw degree distribution.
Return a degree sequence for a tree with a powerlaw distribution.
6.5.1 fast_gnp_random_graph
fast_gnp_random_graph(n, p, seed=None, directed=False)
Return a random graph G_{n,p} (ErdЛќos-RГ©nyi graph, binomial graph).
Parameters n : int
The number of nodes.
p : float
Probability for edge creation.
seed : int, optional
Seed for random number generator (default=None).
directed : bool, optional (default=False)
If True return a directed graph
See also:
gnp_random_graph
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Notes
The G_{n,p} graph algorithm chooses each of the [n(n-1)]/2 (undirected) or n(n-1) (directed) possible edges
with probability p.
This algorithm is O(n+m) where m is the expected number of edges m=p*n*(n-1)/2.
It should be faster than gnp_random_graph when p is small and the expected number of edges is small (sparse
graph).
References
[R345]
6.5.2 gnp_random_graph
gnp_random_graph(n, p, seed=None, directed=False)
Return a random graph G_{n,p} (ErdЛќos-RГ©nyi graph, binomial graph).
Chooses each of the possible edges with probability p.
This is also called binomial_graph and erdos_renyi_graph.
Parameters n : int
The number of nodes.
p : float
Probability for edge creation.
seed : int, optional
Seed for random number generator (default=None).
directed : bool, optional (default=False)
If True return a directed graph
See also:
fast_gnp_random_graph
Notes
This is an O(n^2) algorithm. For sparse graphs (small p) see fast_gnp_random_graph for a faster algorithm.
References
[R346], [R347]
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6.5.3 dense_gnm_random_graph
dense_gnm_random_graph(n, m, seed=None)
Return the random graph G_{n,m}.
Gives a graph picked randomly out of the set of all graphs with n nodes and m edges. This algorithm should be
faster than gnm_random_graph for dense graphs.
Parameters n : int
The number of nodes.
m : int
The number of edges.
seed : int, optional
Seed for random number generator (default=None).
See also:
gnm_random_graph
Notes
Algorithm by Keith M. Briggs Mar 31, 2006. Inspired by Knuth’s Algorithm S (Selection sampling technique),
in section 3.4.2 of [R342].
References
[R342]
6.5.4 gnm_random_graph
gnm_random_graph(n, m, seed=None, directed=False)
Return the random graph G_{n,m}.
Produces a graph picked randomly out of the set of all graphs with n nodes and m edges.
Parameters n : int
The number of nodes.
m : int
The number of edges.
seed : int, optional
Seed for random number generator (default=None).
directed : bool, optional (default=False)
If True return a directed graph
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6.5.5 erdos_renyi_graph
erdos_renyi_graph(n, p, seed=None, directed=False)
Return a random graph G_{n,p} (ErdЛќos-RГ©nyi graph, binomial graph).
Chooses each of the possible edges with probability p.
This is also called binomial_graph and erdos_renyi_graph.
Parameters n : int
The number of nodes.
p : float
Probability for edge creation.
seed : int, optional
Seed for random number generator (default=None).
directed : bool, optional (default=False)
If True return a directed graph
See also:
fast_gnp_random_graph
Notes
This is an O(n^2) algorithm. For sparse graphs (small p) see fast_gnp_random_graph for a faster algorithm.
References
[R343], [R344]
6.5.6 binomial_graph
binomial_graph(n, p, seed=None, directed=False)
Return a random graph G_{n,p} (ErdЛќos-RГ©nyi graph, binomial graph).
Chooses each of the possible edges with probability p.
This is also called binomial_graph and erdos_renyi_graph.
Parameters n : int
The number of nodes.
p : float
Probability for edge creation.
seed : int, optional
Seed for random number generator (default=None).
directed : bool, optional (default=False)
If True return a directed graph
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See also:
fast_gnp_random_graph
Notes
This is an O(n^2) algorithm. For sparse graphs (small p) see fast_gnp_random_graph for a faster algorithm.
References
[R340], [R341]
6.5.7 newman_watts_strogatz_graph
newman_watts_strogatz_graph(n, k, p, seed=None)
Return a Newman-Watts-Strogatz small world graph.
Parameters n : int
The number of nodes
k : int
Each node is connected to k nearest neighbors in ring topology
p : float
The probability of adding a new edge for each edge
seed : int, optional
seed for random number generator (default=None)
See also:
watts_strogatz_graph
Notes
First create a ring over n nodes. Then each node in the ring is connected with its k nearest neighbors (k-1
neighbors if k is odd). Then shortcuts are created by adding new edges as follows: for each edge u-v in the
underlying “n-ring with k nearest neighbors” with probability p add a new edge u-w with randomly-chosen
existing node w. In contrast with watts_strogatz_graph(), no edges are removed.
References
[R348]
6.5.8 watts_strogatz_graph
watts_strogatz_graph(n, k, p, seed=None)
Return a Watts-Strogatz small-world graph.
Parameters n : int
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The number of nodes
k : int
Each node is connected to k nearest neighbors in ring topology
p : float
The probability of rewiring each edge
seed : int, optional
Seed for random number generator (default=None)
See also:
newman_watts_strogatz_graph, connected_watts_strogatz_graph
Notes
First create a ring over n nodes. Then each node in the ring is connected with its k nearest neighbors (k-1
neighbors if k is odd). Then shortcuts are created by replacing some edges as follows: for each edge u-v in the
underlying “n-ring with k nearest neighbors” with probability p replace it with a new edge u-w with uniformly
random choice of existing node w.
In contrast with newman_watts_strogatz_graph(), the random rewiring does not increase the number of edges.
The rewired graph is not guaranteed to be connected as in connected_watts_strogatz_graph().
References
[R352]
6.5.9 connected_watts_strogatz_graph
connected_watts_strogatz_graph(n, k, p, tries=100, seed=None)
Return a connected Watts-Strogatz small-world graph.
Attempt to generate a connected realization by repeated generation of Watts-Strogatz small-world graphs. An
exception is raised if the maximum number of tries is exceeded.
Parameters n : int
The number of nodes
k : int
Each node is connected to k nearest neighbors in ring topology
p : float
The probability of rewiring each edge
tries : int
Number of attempts to generate a connected graph.
seed : int, optional
The seed for random number generator.
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See also:
newman_watts_strogatz_graph, watts_strogatz_graph
6.5.10 random_regular_graph
random_regular_graph(d, n, seed=None)
Return a random regular graph of n nodes each with degree d.
The resulting graph G has no self-loops or parallel edges.
Parameters d : int
Degree
n : integer
Number of nodes. The value of n*d must be even.
seed : hashable object
The seed for random number generator.
Notes
The nodes are numbered form 0 to n-1.
Kim and Vu’s paper [R351] shows that this algorithm samples in an asymptotically uniform way from the space
of random graphs when d = O(n**(1/3-epsilon)).
References
[R350], [R351]
6.5.11 barabasi_albert_graph
barabasi_albert_graph(n, m, seed=None)
Return random graph using BarabГЎsi-Albert preferential attachment model.
A graph of n nodes is grown by attaching new nodes each with m edges that are preferentially attached to
existing nodes with high degree.
Parameters n : int
Number of nodes
m : int
Number of edges to attach from a new node to existing nodes
seed : int, optional
Seed for random number generator (default=None).
Returns G : Graph
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Notes
The initialization is a graph with with m nodes and no edges.
References
[R339]
6.5.12 powerlaw_cluster_graph
powerlaw_cluster_graph(n, m, p, seed=None)
Holme and Kim algorithm for growing graphs with powerlaw degree distribution and approximate average
clustering.
Parameters n : int
the number of nodes
m : int
the number of random edges to add for each new node
p : float,
Probability of adding a triangle after adding a random edge
seed : int, optional
Seed for random number generator (default=None).
Notes
The average clustering has a hard time getting above a certain cutoff that depends on m. This cutoff is often
quite low. Note that the transitivity (fraction of triangles to possible triangles) seems to go down with network
size.
It is essentially the BarabГЎsi-Albert (B-A) growth model with an extra step that each random edge is followed
by a chance of making an edge to one of its neighbors too (and thus a triangle).
This algorithm improves on B-A in the sense that it enables a higher average clustering to be attained if desired.
It seems possible to have a disconnected graph with this algorithm since the initial m nodes may not be all linked
to a new node on the first iteration like the B-A model.
References
[R349]
6.5.13 random_lobster
random_lobster(n, p1, p2, seed=None)
Return a random lobster.
A lobster is a tree that reduces to a caterpillar when pruning all leaf nodes.
A caterpillar is a tree that reduces to a path graph when pruning all leaf nodes (p2=0).
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Parameters n : int
The expected number of nodes in the backbone
p1 : float
Probability of adding an edge to the backbone
p2 : float
Probability of adding an edge one level beyond backbone
seed : int, optional
Seed for random number generator (default=None).
6.5.14 random_shell_graph
random_shell_graph(constructor, seed=None)
Return a random shell graph for the constructor given.
Parameters constructor: a list of three-tuples
(n,m,d) for each shell starting at the center shell.
n : int
The number of nodes in the shell
m : int
The number or edges in the shell
d : float
The ratio of inter-shell (next) edges to intra-shell edges. d=0 means no intra shell edges,
d=1 for the last shell
seed : int, optional
Seed for random number generator (default=None).
Examples
>>> constructor=[(10,20,0.8),(20,40,0.8)]
>>> G=nx.random_shell_graph(constructor)
6.5.15 random_powerlaw_tree
random_powerlaw_tree(n, gamma=3, seed=None, tries=100)
Return a tree with a powerlaw degree distribution.
Parameters n : int,
The number of nodes
gamma : float
Exponent of the power-law
seed : int, optional
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Seed for random number generator (default=None).
tries : int
Number of attempts to adjust sequence to make a tree
Notes
A trial powerlaw degree sequence is chosen and then elements are swapped with new elements from a powerlaw
distribution until the sequence makes a tree (#edges=#nodes-1).
6.5.16 random_powerlaw_tree_sequence
random_powerlaw_tree_sequence(n, gamma=3, seed=None, tries=100)
Return a degree sequence for a tree with a powerlaw distribution.
Parameters n : int,
The number of nodes
gamma : float
Exponent of the power-law
seed : int, optional
Seed for random number generator (default=None).
tries : int
Number of attempts to adjust sequence to make a tree
Notes
A trial powerlaw degree sequence is chosen and then elements are swapped with new elements from a powerlaw
distribution until the sequence makes a tree (#edges=#nodes-1).
6.6 Degree Sequence
Generate graphs with a given degree sequence or expected degree sequence.
configuration_model(deg_sequence[, ...])
directed_configuration_model(...[, ...])
expected_degree_graph(w[, seed, selfloops])
havel_hakimi_graph(deg_sequence[, create_using])
directed_havel_hakimi_graph(in_deg_sequence, ...)
degree_sequence_tree(deg_sequence[, ...])
random_degree_sequence_graph(sequence[, ...])
Return a random graph with the given degree sequence.
Return a directed_random graph with the given degree sequences.
Return a random graph with given expected degrees.
Return a simple graph with given degree sequence constructed using
Return a directed graph with the given degree sequences.
Make a tree for the given degree sequence.
Return a simple random graph with the given degree sequence.
6.6.1 configuration_model
configuration_model(deg_sequence, create_using=None, seed=None)
Return a random graph with the given degree sequence.
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The configuration model generates a random pseudograph (graph with parallel edges and self loops) by randomly assigning edges to match the given degree sequence.
Parameters deg_sequence : list of integers
Each list entry corresponds to the degree of a node.
create_using : graph, optional (default MultiGraph)
Return graph of this type. The instance will be cleared.
seed : hashable object, optional
Seed for random number generator.
Returns G : MultiGraph
A graph with the specified degree sequence. Nodes are labeled starting at 0 with an
index corresponding to the position in deg_sequence.
Raises NetworkXError
If the degree sequence does not have an even sum.
See also:
is_valid_degree_sequence
Notes
As described by Newman [R315].
A non-graphical degree sequence (not realizable by some simple graph) is allowed since this function returns
graphs with self loops and parallel edges. An exception is raised if the degree sequence does not have an even
sum.
This configuration model construction process can lead to duplicate edges and loops. You can remove the
self-loops and parallel edges (see below) which will likely result in a graph that doesn’t have the exact degree
sequence specified.
The density of self-loops and parallel edges tends to decrease as the number of nodes increases. However,
typically the number of self-loops will approach a Poisson distribution with a nonzero mean, and similarly for
the number of parallel edges. Consider a node with k stubs. The probability of being joined to another stub of
the same node is basically (k-1)/N where k is the degree and N is the number of nodes. So the probability of
a self-loop scales like c/N for some constant c. As N grows, this means we expect c self-loops. Similarly for
parallel edges.
References
[R315]
Examples
>>> from networkx.utils import powerlaw_sequence
>>> z=nx.utils.create_degree_sequence(100,powerlaw_sequence)
>>> G=nx.configuration_model(z)
To remove parallel edges:
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>>> G=nx.Graph(G)
To remove self loops:
>>> G.remove_edges_from(G.selfloop_edges())
6.6.2 directed_configuration_model
directed_configuration_model(in_degree_sequence, out_degree_sequence, create_using=None,
seed=None)
Return a directed_random graph with the given degree sequences.
The configuration model generates a random directed pseudograph (graph with parallel edges and self loops) by
randomly assigning edges to match the given degree sequences.
Parameters in_degree_sequence : list of integers
Each list entry corresponds to the in-degree of a node.
out_degree_sequence : list of integers
Each list entry corresponds to the out-degree of a node.
create_using : graph, optional (default MultiDiGraph)
Return graph of this type. The instance will be cleared.
seed : hashable object, optional
Seed for random number generator.
Returns G : MultiDiGraph
A graph with the specified degree sequences. Nodes are labeled starting at 0 with an
index corresponding to the position in deg_sequence.
Raises NetworkXError
If the degree sequences do not have the same sum.
See also:
configuration_model
Notes
Algorithm as described by Newman [R316].
A non-graphical degree sequence (not realizable by some simple graph) is allowed since this function returns
graphs with self loops and parallel edges. An exception is raised if the degree sequences does not have the same
sum.
This configuration model construction process can lead to duplicate edges and loops. You can remove the
self-loops and parallel edges (see below) which will likely result in a graph that doesn’t have the exact degree
sequence specified. This “finite-size effect” decreases as the size of the graph increases.
References
[R316]
6.6. Degree Sequence
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Examples
>>>
>>>
>>>
>>>
>>>
>>>
D=nx.DiGraph([(0,1),(1,2),(2,3)]) # directed path graph
din=list(D.in_degree().values())
dout=list(D.out_degree().values())
din.append(1)
dout[0]=2
D=nx.directed_configuration_model(din,dout)
To remove parallel edges:
>>> D=nx.DiGraph(D)
To remove self loops:
>>> D.remove_edges_from(D.selfloop_edges())
6.6.3 expected_degree_graph
expected_degree_graph(w, seed=None, selfloops=True)
Return a random graph with given expected degrees.
Given a sequence of expected degrees 𝑊 = (𝑤0 , 𝑤1 , . . . , 𝑤𝑛−1 ) of length 𝑛 this algorithm assigns an edge
between node рќ‘ў and node рќ‘Ј with probability
𝑤𝑢 𝑤𝑣
𝑝𝑢𝑣 = ∑︀
.
� 𝑤�
Parameters w : list
The list of expected degrees.
selfloops: bool (default=True)
Set to False to remove the possibility of self-loop edges.
seed : hashable object, optional
The seed for the random number generator.
Returns Graph
Notes
The nodes have integer labels corresponding to index of expected degrees input sequence.
The complexity of this algorithm is рќ’Є(рќ‘› + рќ‘љ) where рќ‘› is the number of nodes and рќ‘љ is the expected number
of edges.
The model in [R318] includes the possibility of self-loop edges. Set selfloops=False to produce a graph without
self loops.
For finite graphs this model doesn’t produce exactly the given expected degree sequence. Instead the expected
degrees are as follows.
For the case without self loops (selfloops=False),
рќђё[рќ‘‘рќ‘’рќ‘”(рќ‘ў)] =
∑︁
рќ‘ЈМё=рќ‘ў
374
рќ‘ќрќ‘ўрќ‘Ј
)пё‚
(пё‚
𝑤𝑢
∑︀
.
= 𝑤𝑢 1 −
� 𝑤�
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NetworkX uses the standard convention that a self-loop edge counts 2 in the degree of a node, so with self loops
(selfloops=True),
)пё‚
(пё‚
∑︁
𝑤𝑢
∑︀
.
рќђё[рќ‘‘рќ‘’рќ‘”(рќ‘ў)] =
𝑝𝑢𝑣 + 2𝑝𝑢𝑢 = 𝑤𝑢 1 +
� 𝑤�
рќ‘ЈМё=рќ‘ў
References
[R318], [R319]
Examples
>>> z=[10 for i in range(100)]
>>> G=nx.expected_degree_graph(z)
6.6.4 havel_hakimi_graph
havel_hakimi_graph(deg_sequence, create_using=None)
Return a simple graph with given degree sequence constructed using the Havel-Hakimi algorithm.
Parameters deg_sequence: list of integers
Each integer corresponds to the degree of a node (need not be sorted).
create_using : graph, optional (default Graph)
Return graph of this type. The instance will be cleared. Directed graphs are not allowed.
Raises NetworkXException
For a non-graphical degree sequence (i.e. one not realizable by some simple graph).
Notes
The Havel-Hakimi algorithm constructs a simple graph by successively connecting the node of highest degree
to other nodes of highest degree, resorting remaining nodes by degree, and repeating the process. The resulting
graph has a high degree-associativity. Nodes are labeled 1,.., len(deg_sequence), corresponding to their position
in deg_sequence.
The basic algorithm is from Hakimi [R320] and was generalized by Kleitman and Wang [R321].
References
[R320], [R321]
6.6.5 directed_havel_hakimi_graph
directed_havel_hakimi_graph(in_deg_sequence, out_deg_sequence, create_using=None)
Return a directed graph with the given degree sequences.
Parameters in_deg_sequence : list of integers
Each list entry corresponds to the in-degree of a node.
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out_deg_sequence : list of integers
Each list entry corresponds to the out-degree of a node.
create_using : graph, optional (default DiGraph)
Return graph of this type. The instance will be cleared.
Returns G : DiGraph
A graph with the specified degree sequences. Nodes are labeled starting at 0 with an
index corresponding to the position in deg_sequence
Raises NetworkXError
If the degree sequences are not digraphical.
See also:
configuration_model
Notes
Algorithm as described by Kleitman and Wang [R317].
References
[R317]
6.6.6 degree_sequence_tree
degree_sequence_tree(deg_sequence, create_using=None)
Make a tree for the given degree sequence.
A tree has #nodes-#edges=1 so the degree sequence must have len(deg_sequence)-sum(deg_sequence)/2=1
6.6.7 random_degree_sequence_graph
random_degree_sequence_graph(sequence, seed=None, tries=10)
Return a simple random graph with the given degree sequence.
If the maximum degree рќ‘‘рќ‘љ in the sequence is рќ‘‚(рќ‘љ1/4 ) then the algorithm produces almost uniform random
graphs in рќ‘‚(рќ‘љрќ‘‘рќ‘љ ) time where рќ‘љ is the number of edges.
Parameters sequence : list of integers
Sequence of degrees
seed : hashable object, optional
Seed for random number generator
tries : int, optional
Maximum number of tries to create a graph
Returns G : Graph
A graph with the specified degree sequence. Nodes are labeled starting at 0 with an
index corresponding to the position in the sequence.
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Raises NetworkXUnfeasible
If the degree sequence is not graphical.
NetworkXError
If a graph is not produced in specified number of tries
See also:
is_valid_degree_sequence, configuration_model
Notes
The generator algorithm [R322] is not guaranteed to produce a graph.
References
[R322]
Examples
>>>
>>>
>>>
[1,
sequence = [1, 2, 2, 3]
G = nx.random_degree_sequence_graph(sequence)
sorted(G.degree().values())
2, 2, 3]
6.7 Random Clustered
Generate graphs with given degree and triangle sequence.
random_clustered_graph(joint_degree_sequence)
Generate a random graph with the given joint degree and triangle degree s
6.7.1 random_clustered_graph
random_clustered_graph(joint_degree_sequence, create_using=None, seed=None)
Generate a random graph with the given joint degree and triangle degree sequence.
This uses a configuration model-like approach to generate a random pseudograph (graph with parallel edges and
self loops) by randomly assigning edges to match the given indepdenent edge and triangle degree sequence.
Parameters joint_degree_sequence : list of integer pairs
Each list entry corresponds to the independent edge degree and triangle degree of a
node.
create_using : graph, optional (default MultiGraph)
Return graph of this type. The instance will be cleared.
seed : hashable object, optional
The seed for the random number generator.
Returns G : MultiGraph
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A graph with the specified degree sequence. Nodes are labeled starting at 0 with an
index corresponding to the position in deg_sequence.
Raises NetworkXError
If the independent edge degree sequence sum is not even or the triangle degree sequence
sum is not divisible by 3.
Notes
As described by Miller [R337] (see also Newman [R338] for an equivalent description).
A non-graphical degree sequence (not realizable by some simple graph) is allowed since this function returns
graphs with self loops and parallel edges. An exception is raised if the independent degree sequence does not
have an even sum or the triangle degree sequence sum is not divisible by 3.
This configuration model-like construction process can lead to duplicate edges and loops. You can remove the
self-loops and parallel edges (see below) which will likely result in a graph that doesn’t have the exact degree
sequence specified. This “finite-size effect” decreases as the size of the graph increases.
References
[R337], [R338]
Examples
>>> deg_tri=[[1,0],[1,0],[1,0],[2,0],[1,0],[2,1],[0,1],[0,1]]
>>> G = nx.random_clustered_graph(deg_tri)
To remove parallel edges:
>>> G=nx.Graph(G)
To remove self loops:
>>> G.remove_edges_from(G.selfloop_edges())
6.8 Directed
Generators for some directed graphs.
gn_graph: growing network gnc_graph: growing network with copying gnr_graph: growing network with redirection
scale_free_graph: scale free directed graph
gn_graph(n[, kernel, create_using, seed])
gnr_graph(n, p[, create_using, seed])
gnc_graph(n[, create_using, seed])
scale_free_graph(n[, alpha, beta, gamma, ...])
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Return the GN digraph with n nodes.
Return the GNR digraph with n nodes and redirection probability p.
Return the GNC digraph with n nodes.
Return a scale free directed graph.
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6.8.1 gn_graph
gn_graph(n, kernel=None, create_using=None, seed=None)
Return the GN digraph with n nodes.
The GN (growing network) graph is built by adding nodes one at a time with a link to one previously added
node. The target node for the link is chosen with probability based on degree. The default attachment kernel is
a linear function of degree.
The graph is always a (directed) tree.
Parameters n : int
The number of nodes for the generated graph.
kernel : function
The attachment kernel.
create_using : graph, optional (default DiGraph)
Return graph of this type. The instance will be cleared.
seed : hashable object, optional
The seed for the random number generator.
References
[R323]
Examples
>>> D=nx.gn_graph(10)
>>> G=D.to_undirected()
# the GN graph
# the undirected version
To specify an attachment kernel use the kernel keyword
>>> D=nx.gn_graph(10,kernel=lambda x:x**1.5) # A_k=k^1.5
6.8.2 gnr_graph
gnr_graph(n, p, create_using=None, seed=None)
Return the GNR digraph with n nodes and redirection probability p.
The GNR (growing network with redirection) graph is built by adding nodes one at a time with a link to one
previously added node. The previous target node is chosen uniformly at random. With probabiliy p the link is
instead “redirected” to the successor node of the target. The graph is always a (directed) tree.
Parameters n : int
The number of nodes for the generated graph.
p : float
The redirection probability.
create_using : graph, optional (default DiGraph)
Return graph of this type. The instance will be cleared.
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seed : hashable object, optional
The seed for the random number generator.
References
[R325]
Examples
>>> D=nx.gnr_graph(10,0.5) # the GNR graph
>>> G=D.to_undirected() # the undirected version
6.8.3 gnc_graph
gnc_graph(n, create_using=None, seed=None)
Return the GNC digraph with n nodes.
The GNC (growing network with copying) graph is built by adding nodes one at a time with a links to one
previously added node (chosen uniformly at random) and to all of that node’s successors.
Parameters n : int
The number of nodes for the generated graph.
create_using : graph, optional (default DiGraph)
Return graph of this type. The instance will be cleared.
seed : hashable object, optional
The seed for the random number generator.
References
[R324]
6.8.4 scale_free_graph
scale_free_graph(n, alpha=0.41, beta=0.54, gamma=0.05,
ate_using=None, seed=None)
Return a scale free directed graph.
delta_in=0.2,
delta_out=0,
cre-
Parameters n : integer
Number of nodes in graph
alpha : float
Probability for adding a new node connected to an existing node chosen randomly according to the in-degree distribution.
beta : float
Probability for adding an edge between two existing nodes. One existing node is chosen
randomly according the in-degree distribution and the other chosen randomly according
to the out-degree distribution.
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gamma : float
Probability for adding a new node conecgted to an existing node chosen randomly according to the out-degree distribution.
delta_in : float
Bias for choosing ndoes from in-degree distribution.
delta_out : float
Bias for choosing ndoes from out-degree distribution.
create_using : graph, optional (default MultiDiGraph)
Use this graph instance to start the process (default=3-cycle).
seed : integer, optional
Seed for random number generator
Notes
The sum of alpha, beta, and gamma must be 1.
References
[R326]
Examples
>>> G=nx.scale_free_graph(100)
6.9 Geometric
Generators for geometric graphs.
random_geometric_graph(n, radius[, dim, pos])
geographical_threshold_graph(n, theta[, ...])
waxman_graph(n[, alpha, beta, L, domain])
navigable_small_world_graph(n[, p, q, r, ...])
Return the random geometric graph in the unit cube.
Return a geographical threshold graph.
Return a Waxman random graph.
Return a navigable small-world graph.
6.9.1 random_geometric_graph
random_geometric_graph(n, radius, dim=2, pos=None)
Return the random geometric graph in the unit cube.
The random geometric graph model places n nodes uniformly at random in the unit cube Two nodes рќ‘ў, рќ‘Ј are
connected with an edge if рќ‘‘(рќ‘ў, рќ‘Ј) <= рќ‘џ where рќ‘‘ is the Euclidean distance and рќ‘џ is a radius threshold.
Parameters n : int
Number of nodes
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radius: float
Distance threshold value
dim : int, optional
Dimension of graph
pos : dict, optional
A dictionary keyed by node with node positions as values.
Returns Graph
Notes
This uses an рќ‘›2 algorithm to build the graph. A faster algorithm is possible using k-d trees.
The pos keyword can be used to specify node positions so you can create an arbitrary distribution and domain
for positions. If you need a distance function other than Euclidean you’ll have to hack the algorithm.
E.g to use a 2d Gaussian distribution of node positions with mean (0,0) and std. dev. 2
>>>
>>>
>>>
>>>
import random
n=20
p=dict((i,(random.gauss(0,2),random.gauss(0,2))) for i in range(n))
G = nx.random_geometric_graph(n,0.2,pos=p)
References
[R331]
Examples
>>> G = nx.random_geometric_graph(20,0.1)
6.9.2 geographical_threshold_graph
geographical_threshold_graph(n, theta, alpha=2, dim=2, pos=None, weight=None)
Return a geographical threshold graph.
The geographical threshold graph model places n nodes uniformly at random in a rectangular domain. Each
node 𝑢 is assigned a weight 𝑤𝑢 . Two nodes 𝑢, 𝑣 are connected with an edge if
𝑤𝑢 + 𝑤𝑣 ≥ 𝜃𝑟𝛼
where рќ‘џ is the Euclidean distance between рќ‘ў and рќ‘Ј, and рќњѓ, рќ›ј are parameters.
Parameters n : int
Number of nodes
theta: float
Threshold value
alpha: float, optional
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Exponent of distance function
dim : int, optional
Dimension of graph
pos : dict
Node positions as a dictionary of tuples keyed by node.
weight : dict
Node weights as a dictionary of numbers keyed by node.
Returns Graph
Notes
If weights are not specified they are assigned to nodes by drawing randomly from an the exponential distribution
with rate parameter рќњ† = 1. To specify a weights from a different distribution assign them to a dictionary and
pass it as the weight= keyword
>>>
>>>
>>>
>>>
import random
n = 20
w=dict((i,random.expovariate(5.0)) for i in range(n))
G = nx.geographical_threshold_graph(20,50,weight=w)
If node positions are not specified they are randomly assigned from the uniform distribution.
References
[R328], [R329]
Examples
>>> G = nx.geographical_threshold_graph(20,50)
6.9.3 waxman_graph
waxman_graph(n, alpha=0.4, beta=0.1, L=None, domain=(0, 0, 1, 1))
Return a Waxman random graph.
The Waxman random graph models place n nodes uniformly at random in a rectangular domain. Two nodes u,v
are connected with an edge with probability
рќ‘ќ = рќ›ј * рќ‘’рќ‘Ґрќ‘ќ(в€’рќ‘‘/(рќ›Ѕ * рќђї)).
This function implements both Waxman models.
Waxman-1: рќђї not specified The distance рќ‘‘ is the Euclidean distance between the nodes u and v. рќђї is the
maximum distance between all nodes in the graph.
Waxman-2: рќђї specified The distance рќ‘‘ is chosen randomly in [0, рќђї].
Parameters n : int
Number of nodes
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alpha: float
Model parameter
beta: float
Model parameter
L : float, optional
Maximum distance between nodes. If not specified the actual distance is calculated.
domain : tuple of numbers, optional
Domain size (xmin, ymin, xmax, ymax)
Returns G: Graph
References
[R332]
6.9.4 navigable_small_world_graph
navigable_small_world_graph(n, p=1, q=1, r=2, dim=2, seed=None)
Return a navigable small-world graph.
A navigable small-world graph is a directed grid with additional long-range connections that are chosen randomly. From [R330]:
Begin with a set of nodes that are identified with the set of lattice points in an рќ‘› Г— рќ‘› square,
(𝑖, 𝑗) : 𝑖 ∈ 1, 2, . . . , 𝑛, 𝑗 ∈ 1, 2, . . . , 𝑛 and define the lattice distance between two nodes (𝑖, 𝑗) and (�, 𝑙) to be
the number of “lattice steps” separating them: 𝑑((𝑖, 𝑗), (�, 𝑙)) = |� − 𝑖| + |𝑙 − 𝑗|.
For a universal constant рќ‘ќ, the node рќ‘ў has a directed edge to every other node within lattice distance рќ‘ќ (local
contacts) .
For universal constants 𝑞 ≥ 0 and 𝑟 ≥ 0 construct directed edges from 𝑢 to 𝑞 other nodes (long-range contacts)
using independent random trials; the i’th directed edge from 𝑢 has endpoint 𝑣 with probability proportional to
рќ‘‘(рќ‘ў, рќ‘Ј)в€’рќ‘џ .
Parameters n : int
The number of nodes.
p : int
The diameter of short range connections. Each node is connected to every other node
within lattice distance p.
q : int
The number of long-range connections for each node.
r : float
Exponent for decaying probability of connections. The probability of connecting to a
node at lattice distance d is 1/d^r.
dim : int
Dimension of grid
seed : int, optional
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Seed for random number generator (default=None).
References
[R330]
6.10 Hybrid
Hybrid
kl_connected_subgraph(G, k, l[, low_memory, ...])
is_kl_connected(G, k, l[, low_memory])
Returns the maximum locally (k,l) connected subgraph of G.
Returns True if G is kl connected.
6.10.1 kl_connected_subgraph
kl_connected_subgraph(G, k, l, low_memory=False, same_as_graph=False)
Returns the maximum locally (k,l) connected subgraph of G.
(k,l)-connected subgraphs are presented by Fan Chung and Li in “The Small World Phenomenon in hybrid
power law graphs” to appear in “Complex Networks” (Ed. E. Ben-Naim) Lecture Notes in Physics, Springer
(2004)
low_memory=True then use a slightly slower, but lower memory version same_as_graph=True then return a
tuple with subgraph and pflag for if G is kl-connected
6.10.2 is_kl_connected
is_kl_connected(G, k, l, low_memory=False)
Returns True if G is kl connected.
6.11 Bipartite
Generators and functions for bipartite graphs.
bipartite_configuration_model(aseq, bseq[, ...])
bipartite_havel_hakimi_graph(aseq, bseq[, ...])
bipartite_reverse_havel_hakimi_graph(aseq, bseq)
bipartite_alternating_havel_hakimi_graph(...)
bipartite_preferential_attachment_graph(aseq, p)
bipartite_random_graph(n, m, p[, seed, directed])
bipartite_gnmk_random_graph(n, m, k[, seed, ...])
Return a random bipartite graph from two given degree sequenc
Return a bipartite graph from two given degree sequences using
Return a bipartite graph from two given degree sequences using
Return a bipartite graph from two given degree sequences using
Create a bipartite graph with a preferential attachment model fro
Return a bipartite random graph.
Return a random bipartite graph G_{n,m,k}.
6.11.1 bipartite_configuration_model
bipartite_configuration_model(aseq, bseq, create_using=None, seed=None)
Return a random bipartite graph from two given degree sequences.
Parameters aseq : list
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Degree sequence for node set A.
bseq : list
Degree sequence for node set B.
create_using : NetworkX graph instance, optional
Return graph of this type.
seed : integer, optional
Seed for random number generator.
Nodes from the set A are connected to nodes in the set B by
choosing randomly from the possible free stubs, one in A and
one in B.
Notes
The sum of the two sequences must be equal: sum(aseq)=sum(bseq) If no graph type is specified use MultiGraph
with parallel edges. If you want a graph with no parallel edges use create_using=Graph() but then the resulting
degree sequences might not be exact.
The nodes are assigned the attribute �bipartite’ with the value 0 or 1 to indicate which bipartite set the node
belongs to.
6.11.2 bipartite_havel_hakimi_graph
bipartite_havel_hakimi_graph(aseq, bseq, create_using=None)
Return a bipartite graph from two given degree sequences using a Havel-Hakimi style construction.
Nodes from the set A are connected to nodes in the set B by connecting the highest degree nodes in set A to the
highest degree nodes in set B until all stubs are connected.
Parameters aseq : list
Degree sequence for node set A.
bseq : list
Degree sequence for node set B.
create_using : NetworkX graph instance, optional
Return graph of this type.
Notes
The sum of the two sequences must be equal: sum(aseq)=sum(bseq) If no graph type is specified use MultiGraph
with parallel edges. If you want a graph with no parallel edges use create_using=Graph() but then the resulting
degree sequences might not be exact.
The nodes are assigned the attribute �bipartite’ with the value 0 or 1 to indicate which bipartite set the node
belongs to.
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6.11.3 bipartite_reverse_havel_hakimi_graph
bipartite_reverse_havel_hakimi_graph(aseq, bseq, create_using=None)
Return a bipartite graph from two given degree sequences using a Havel-Hakimi style construction.
Nodes from set A are connected to nodes in the set B by connecting the highest degree nodes in set A to the
lowest degree nodes in set B until all stubs are connected.
Parameters aseq : list
Degree sequence for node set A.
bseq : list
Degree sequence for node set B.
create_using : NetworkX graph instance, optional
Return graph of this type.
Notes
The sum of the two sequences must be equal: sum(aseq)=sum(bseq) If no graph type is specified use MultiGraph
with parallel edges. If you want a graph with no parallel edges use create_using=Graph() but then the resulting
degree sequences might not be exact.
The nodes are assigned the attribute �bipartite’ with the value 0 or 1 to indicate which bipartite set the node
belongs to.
6.11.4 bipartite_alternating_havel_hakimi_graph
bipartite_alternating_havel_hakimi_graph(aseq, bseq, create_using=None)
Return a bipartite graph from two given degree sequences using an alternating Havel-Hakimi style construction.
Nodes from the set A are connected to nodes in the set B by connecting the highest degree nodes in set A to
alternatively the highest and the lowest degree nodes in set B until all stubs are connected.
Parameters aseq : list
Degree sequence for node set A.
bseq : list
Degree sequence for node set B.
create_using : NetworkX graph instance, optional
Return graph of this type.
Notes
The sum of the two sequences must be equal: sum(aseq)=sum(bseq) If no graph type is specified use MultiGraph
with parallel edges. If you want a graph with no parallel edges use create_using=Graph() but then the resulting
degree sequences might not be exact.
The nodes are assigned the attribute �bipartite’ with the value 0 or 1 to indicate which bipartite set the node
belongs to.
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6.11.5 bipartite_preferential_attachment_graph
bipartite_preferential_attachment_graph(aseq, p, create_using=None, seed=None)
Create a bipartite graph with a preferential attachment model from a given single degree sequence.
Parameters aseq : list
Degree sequence for node set A.
p : float
Probability that a new bottom node is added.
create_using : NetworkX graph instance, optional
Return graph of this type.
seed : integer, optional
Seed for random number generator.
References
[R313]
6.11.6 bipartite_random_graph
bipartite_random_graph(n, m, p, seed=None, directed=False)
Return a bipartite random graph.
This is a bipartite version of the binomial (ErdЛќos-RГ©nyi) graph.
Parameters n : int
The number of nodes in the first bipartite set.
m : int
The number of nodes in the second bipartite set.
p : float
Probability for edge creation.
seed : int, optional
Seed for random number generator (default=None).
directed : bool, optional (default=False)
If True return a directed graph
See also:
gnp_random_graph, bipartite_configuration_model
Notes
The bipartite random graph algorithm chooses each of the n*m (undirected) or 2*nm (directed) possible edges
with probability p.
This algorithm is O(n+m) where m is the expected number of edges.
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The nodes are assigned the attribute �bipartite’ with the value 0 or 1 to indicate which bipartite set the node
belongs to.
References
[R314]
6.11.7 bipartite_gnmk_random_graph
bipartite_gnmk_random_graph(n, m, k, seed=None, directed=False)
Return a random bipartite graph G_{n,m,k}.
Produces a bipartite graph chosen randomly out of the set of all graphs with n top nodes, m bottom nodes, and
k edges.
Parameters n : int
The number of nodes in the first bipartite set.
m : int
The number of nodes in the second bipartite set.
k : int
The number of edges
seed : int, optional
Seed for random number generator (default=None).
directed : bool, optional (default=False)
If True return a directed graph
See also:
gnm_random_graph
Notes
If k > m * n then a complete bipartite graph is returned.
This graph is a bipartite version of the рќђєрќ‘›рќ‘љ random graph model.
Examples
G = nx.bipartite_gnmk_random_graph(10,20,50)
6.12 Line Graph
Line graph algorithms.
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6.12.1 Undirected Graphs
For an undirected graph G without multiple edges, each edge can be written as a set {u,v}. Its line graph L has the
edges of G as its nodes. If x and y are two nodes in L, then {x,y} is an edge in L if and only if the intersection of x and
y is nonempty. Thus, the set of all edges is determined by the set of all pair-wise intersections of edges in G.
Trivially, every edge x={u,v} in G would have a nonzero intersection with itself, and so every node in L should have
a self-loop. This is not so interesting, and the original context of line graphs was with simple graphs, which had no
self-loops or multiple edges. The line graph was also meant to be simple graph and thus, self-loops in L are not part
of the standard definition of a line graph. In a pair-wise intersection matrix, this is analogous to not including the
diagonal as part of the line graph definition.
Self-loops and multiple edges in G add nodes to L in a natural way, and do not require any fundamental changes to the
definition. It might be argued that the self-loops we excluded before should now be included. However, the self-loops
are still “trivial” in some sense and thus, are usually excluded.
6.12.2 Directed Graphs
For a directed graph G without multiple edges, each edge can be written as a tuple (u,v). Its line graph L has the edges
of G as its nodes. If x=(a,b) and y=(c,d) are two nodes in L, then (x,y) is an edge in L if and only if the tail of x matches
the head of y—e.g., b=c.
Due to the directed nature of the edges, it is no longer the case that every edge x=(u,v) should be connected to itself
with a self-loop in L. Now, the only time self-loops arise is if G itself has a self-loop. So such self-loops are no longer
“trivial” but instead, represent essential features of the topology of G. For this reason, the historical development of
line digraphs is such that self-loops are included. When the graph G has multiple edges, once again only superficial
changes are required to the definition.
6.12.3 References
Harary, Frank, and Norman, Robert Z., “Some properties of line digraphs”, Rend. Circ. Mat. Palermo, II. Ser.
9 (1960), 161–168.
Hemminger, R. L.; Beineke, L. W. (1978), “Line graphs and line digraphs”, in Beineke, L. W.; Wilson, R. J., Selected Topics in Graph Theory, Academic Press Inc., pp. 271–305.
line_graph(G[, create_using])
Return the line graph of the graph or digraph G.
6.12.4 line_graph
line_graph(G, create_using=None)
Return the line graph of the graph or digraph G.
The line graph of a graph G has a node for each edge in G and an edge between those nodes if the two edges in
G share a common node. For directed graphs, nodes are connected only if they form a directed path of length 2.
The nodes of the line graph are 2-tuples of nodes in the original graph (or 3-tuples for multigraphs, with the key
of the edge as the 3rd element).
For more discussion, see the docstring in networkx.generators.line.
Parameters G : graph
A NetworkX Graph, DiGraph, MultiGraph, or MultiDigraph.
Returns L : graph
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The line graph of G.
Notes
Graph, node, and edge data are not propagated to the new graph. For undirected graphs, the nodes in G must be
sortable—otherwise, the constructed line graph may not be correct.
Examples
>>> G = nx.star_graph(3)
>>> L = nx.line_graph(G)
>>> print(sorted(map(sorted, L.edges()))) # makes a clique, K3
[[(0, 1), (0, 2)], [(0, 1), (0, 3)], [(0, 2), (0, 3)]]
6.13 Ego Graph
Ego graph.
ego_graph(G, n[, radius, center, ...])
Returns induced subgraph of neighbors centered at node n within a given radius.
6.13.1 ego_graph
ego_graph(G, n, radius=1, center=True, undirected=False, distance=None)
Returns induced subgraph of neighbors centered at node n within a given radius.
Parameters G : graph
A NetworkX Graph or DiGraph
n : node
A single node
radius : number, optional
Include all neighbors of distance<=radius from n.
center : bool, optional
If False, do not include center node in graph
undirected : bool, optional
If True use both in- and out-neighbors of directed graphs.
distance : key, optional
Use specified edge data key as distance. For example, setting distance=’weight’ will
use the edge weight to measure the distance from the node n.
Notes
For directed graphs D this produces the “out” neighborhood or successors. If you want the neighborhood of
predecessors first reverse the graph with D.reverse(). If you want both directions use the keyword argument
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undirected=True.
Node, edge, and graph attributes are copied to the returned subgraph.
6.14 Stochastic
Stocastic graph.
stochastic_graph(G[, copy, weight])
Return a right-stochastic representation of G.
6.14.1 stochastic_graph
stochastic_graph(G, copy=True, weight=’weight’)
Return a right-stochastic representation of G.
A right-stochastic graph is a weighted digraph in which all of the node (out) neighbors edge weights sum to 1.
Parameters G : directed graph
A NetworkX DiGraph
copy : boolean, optional
If True make a copy of the graph, otherwise modify the original graph
weight : edge attribute key (optional, default=’weight’)
Edge data key used for weight. If no attribute is found for an edge the edge weight is
set to 1. Weights must be positive numbers.
6.15 Intersection
Generators for random intersection graphs.
uniform_random_intersection_graph(n, m, p[, ...])
k_random_intersection_graph(n, m, k)
general_random_intersection_graph(n, m, p)
Return a uniform random intersection graph.
Return a intersection graph with randomly chosen attribute sets for e
Return a random intersection graph with independent probabilities fo
6.15.1 uniform_random_intersection_graph
uniform_random_intersection_graph(n, m, p, seed=None)
Return a uniform random intersection graph.
Parameters n : int
The number of nodes in the first bipartite set (nodes)
m : int
The number of nodes in the second bipartite set (attributes)
p : float
Probability of connecting nodes between bipartite sets
seed : int, optional
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Seed for random number generator (default=None).
See also:
gnp_random_graph
References
[R335], [R336]
6.15.2 k_random_intersection_graph
k_random_intersection_graph(n, m, k)
Return a intersection graph with randomly chosen attribute sets for each node that are of equal size (k).
Parameters n : int
The number of nodes in the first bipartite set (nodes)
m : int
The number of nodes in the second bipartite set (attributes)
k : float
Size of attribute set to assign to each node.
seed : int, optional
Seed for random number generator (default=None).
See also:
gnp_random_graph, uniform_random_intersection_graph
References
[R334]
6.15.3 general_random_intersection_graph
general_random_intersection_graph(n, m, p)
Return a random intersection graph with independent probabilities for connections between node and attribute
sets.
Parameters n : int
The number of nodes in the first bipartite set (nodes)
m : int
The number of nodes in the second bipartite set (attributes)
p : list of floats of length m
Probabilities for connecting nodes to each attribute
seed : int, optional
Seed for random number generator (default=None).
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See also:
gnp_random_graph, uniform_random_intersection_graph
References
[R333]
6.16 Social Networks
Famous social networks.
karate_club_graph()
davis_southern_women_graph()
florentine_families_graph()
Return Zachary’s Karate club graph.
Return Davis Southern women social network.
Return Florentine families graph.
6.16.1 karate_club_graph
karate_club_graph()
Return Zachary’s Karate club graph.
References
[R355], [R356]
6.16.2 davis_southern_women_graph
davis_southern_women_graph()
Return Davis Southern women social network.
This is a bipartite graph.
References
[R353]
6.16.3 florentine_families_graph
florentine_families_graph()
Return Florentine families graph.
References
[R354]
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SEVEN
LINEAR ALGEBRA
7.1 Graph Matrix
Adjacency matrix and incidence matrix of graphs.
adjacency_matrix(G[, nodelist, weight])
incidence_matrix(G[, nodelist, edgelist, ...])
Return adjacency matrix of G.
Return incidence matrix of G.
7.1.1 adjacency_matrix
adjacency_matrix(G, nodelist=None, weight=’weight’)
Return adjacency matrix of G.
Parameters G : graph
A NetworkX graph
nodelist : list, optional
The rows and columns are ordered according to the nodes in nodelist. If nodelist is
None, then the ordering is produced by G.nodes().
weight : string or None, optional (default=’weight’)
The edge data key used to provide each value in the matrix. If None, then each edge has
weight 1.
Returns A : SciPy sparse matrix
Adjacency matrix representation of G.
See also:
to_numpy_matrix, to_scipy_sparse_matrix, to_dict_of_dicts
Notes
If you want a pure Python adjacency matrix representation try networkx.convert.to_dict_of_dicts which will
return a dictionary-of-dictionaries format that can be addressed as a sparse matrix.
For MultiGraph/MultiDiGraph with parallel edges the weights are summed. See to_numpy_matrix for other
options.
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The convention used for self-loop edges in graphs is to assign the diagonal matrix entry value to the edge weight
attribute (or the number 1 if the edge has no weight attribute). If the alternate convention of doubling the edge
weight is desired the resulting Scipy sparse matrix can be modified as follows:
>>> import scipy as sp
>>> G = nx.Graph([(1,1)])
>>> A = nx.adjacency_matrix(G)
>>> print(A.todense())
[[1]]
>>> A.setdiag(A.diagonal()*2)
>>> print(A.todense())
[[2]]
7.1.2 incidence_matrix
incidence_matrix(G, nodelist=None, edgelist=None, oriented=False, weight=None)
Return incidence matrix of G.
The incidence matrix assigns each row to a node and each column to an edge. For a standard incidence matrix a
1 appears wherever a row’s node is incident on the column’s edge. For an oriented incidence matrix each edge
is assigned an orientation (arbitrarily for undirected and aligning to direction for directed). A -1 appears for the
tail of an edge and 1 for the head of the edge. The elements are zero otherwise.
Parameters G : graph
A NetworkX graph
nodelist : list, optional (default= all nodes in G)
The rows are ordered according to the nodes in nodelist. If nodelist is None, then the
ordering is produced by G.nodes().
edgelist : list, optional (default= all edges in G)
The columns are ordered according to the edges in edgelist. If edgelist is None, then the
ordering is produced by G.edges().
oriented: bool, optional (default=False)
If True, matrix elements are +1 or -1 for the head or tail node respectively of each edge.
If False, +1 occurs at both nodes.
weight : string or None, optional (default=None)
The edge data key used to provide each value in the matrix. If None, then each edge has
weight 1. Edge weights, if used, should be positive so that the orientation can provide
the sign.
Returns A : SciPy sparse matrix
The incidence matrix of G.
Notes
For MultiGraph/MultiDiGraph, the edges in edgelist should be (u,v,key) 3-tuples.
“Networks are the best discrete model for so many problems in applied mathematics” [R357].
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References
[R357]
7.2 Laplacian Matrix
Laplacian matrix of graphs.
laplacian_matrix(G[, nodelist, weight])
normalized_laplacian_matrix(G[, nodelist, ...])
directed_laplacian_matrix(G[, nodelist, ...])
Return the Laplacian matrix of G.
Return the normalized Laplacian matrix of G.
Return the directed Laplacian matrix of G.
7.2.1 laplacian_matrix
laplacian_matrix(G, nodelist=None, weight=’weight’)
Return the Laplacian matrix of G.
The graph Laplacian is the matrix L = D - A, where A is the adjacency matrix and D is the diagonal matrix of
node degrees.
Parameters G : graph
A NetworkX graph
nodelist : list, optional
The rows and columns are ordered according to the nodes in nodelist. If nodelist is
None, then the ordering is produced by G.nodes().
weight : string or None, optional (default=’weight’)
The edge data key used to compute each value in the matrix. If None, then each edge
has weight 1.
Returns L : SciPy sparse matrix
The Laplacian matrix of G.
See also:
to_numpy_matrix, normalized_laplacian_matrix
Notes
For MultiGraph/MultiDiGraph, the edges weights are summed.
7.2.2 normalized_laplacian_matrix
normalized_laplacian_matrix(G, nodelist=None, weight=’weight’)
Return the normalized Laplacian matrix of G.
The normalized graph Laplacian is the matrix
рќ‘Ѓ рќђї = рќђ·в€’1/2 рќђїрќђ·в€’1/2
where рќђї is the graph Laplacian and рќђ· is the diagonal matrix of node degrees.
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Parameters G : graph
A NetworkX graph
nodelist : list, optional
The rows and columns are ordered according to the nodes in nodelist. If nodelist is
None, then the ordering is produced by G.nodes().
weight : string or None, optional (default=’weight’)
The edge data key used to compute each value in the matrix. If None, then each edge
has weight 1.
Returns L : NumPy matrix
The normalized Laplacian matrix of G.
See also:
laplacian_matrix
Notes
For MultiGraph/MultiDiGraph, the edges weights are summed. See to_numpy_matrix for other options.
If the Graph contains selfloops, D is defined as diag(sum(A,1)), where A is the adjencency matrix [R360].
References
[R359], [R360]
7.2.3 directed_laplacian_matrix
directed_laplacian_matrix(G, nodelist=None, weight=’weight’, walk_type=None, alpha=0.95)
Return the directed Laplacian matrix of G.
The graph directed Laplacian is the matrix
𝐿 = 𝐼 − (Φ1/2 𝑃 Φ−1/2 + Φ−1/2 𝑃 𝑇 Φ1/2 )/2
where рќђј is the identity matrix, рќ‘ѓ is the transition matrix of the graph, and О¦ a matrix with the Perron vector of
рќ‘ѓ in the diagonal and zeros elsewhere.
Depending on the value of walk_type, рќ‘ѓ can be the transition matrix induced by a random walk, a lazy random
walk, or a random walk with teleportation (PageRank).
Parameters G : DiGraph
A NetworkX graph
nodelist : list, optional
The rows and columns are ordered according to the nodes in nodelist. If nodelist is
None, then the ordering is produced by G.nodes().
weight : string or None, optional (default=’weight’)
The edge data key used to compute each value in the matrix. If None, then each edge
has weight 1.
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walk_type : string or None, optional (default=None)
If None, рќ‘ѓ is selected depending on the properties of the graph. Otherwise is one of
�random’, �lazy’, or �pagerank’
alpha : real
(1 - alpha) is the teleportation probability used with pagerank
Returns L : NumPy array
Normalized Laplacian of G.
Raises NetworkXError
If NumPy cannot be imported
NetworkXNotImplemnted
If G is not a DiGraph
See also:
laplacian_matrix
Notes
Only implemented for DiGraphs
References
[R358]
7.3 Spectrum
Eigenvalue spectrum of graphs.
laplacian_spectrum(G[, weight])
adjacency_spectrum(G[, weight])
Return eigenvalues of the Laplacian of G
Return eigenvalues of the adjacency matrix of G.
7.3.1 laplacian_spectrum
laplacian_spectrum(G, weight=’weight’)
Return eigenvalues of the Laplacian of G
Parameters G : graph
A NetworkX graph
weight : string or None, optional (default=’weight’)
The edge data key used to compute each value in the matrix. If None, then each edge
has weight 1.
Returns evals : NumPy array
Eigenvalues
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See also:
laplacian_matrix
Notes
For MultiGraph/MultiDiGraph, the edges weights are summed. See to_numpy_matrix for other options.
7.3.2 adjacency_spectrum
adjacency_spectrum(G, weight=’weight’)
Return eigenvalues of the adjacency matrix of G.
Parameters G : graph
A NetworkX graph
weight : string or None, optional (default=’weight’)
The edge data key used to compute each value in the matrix. If None, then each edge
has weight 1.
Returns evals : NumPy array
Eigenvalues
See also:
adjacency_matrix
Notes
For MultiGraph/MultiDiGraph, the edges weights are summed. See to_numpy_matrix for other options.
7.4 Algebraic Connectivity
Algebraic connectivity and Fiedler vectors of undirected graphs.
algebraic_connectivity(G[, weight, ...])
fiedler_vector(G[, weight, normalized, tol, ...])
spectral_ordering(G[, weight, normalized, ...])
Return the algebraic connectivity of an undirected graph.
Return the Fiedler vector of a connected undirected graph.
Compute the spectral_ordering of a graph.
7.4.1 algebraic_connectivity
algebraic_connectivity(G, weight=’weight’, normalized=False, tol=1e-08, method=’tracemin’)
Return the algebraic connectivity of an undirected graph.
The algebraic connectivity of a connected undirected graph is the second smallest eigenvalue of its Laplacian
matrix.
Parameters G : NetworkX graph
An undirected graph.
weight : object, optional
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The data key used to determine the weight of each edge. If None, then each edge has
unit weight. Default value: None.
normalized : bool, optional
Whether the normalized Laplacian matrix is used. Default value: False.
tol : float, optional
Tolerance of relative residual in eigenvalue computation. Default value: 1e-8.
method : string, optional
Method of eigenvalue computation. It should be one of �tracemin’ (TraceMIN), �lanczos’ (Lanczos iteration) and �lobpcg’ (LOBPCG). Default value: �tracemin’.
The TraceMIN algorithm uses a linear system solver. The following values allow specifying the solver to be used.
Value
�tracemin_pcg’
�tracemin_chol’
�tracemin_lu’
Solver
Preconditioned conjugate gradient method
Cholesky factorization
LU factorization
Returns algebraic_connectivity : float
Algebraic connectivity.
Raises NetworkXNotImplemented
If G is directed.
NetworkXError
If G has less than two nodes.
See also:
laplacian_matrix
Notes
Edge weights are interpreted by their absolute values. For MultiGraph’s, weights of parallel edges are summed.
Zero-weighted edges are ignored.
To use Cholesky factorization in the TraceMIN algorithm, the scikits.sparse package must be installed.
7.4.2 fiedler_vector
fiedler_vector(G, weight=’weight’, normalized=False, tol=1e-08, method=’tracemin’)
Return the Fiedler vector of a connected undirected graph.
The Fiedler vector of a connected undirected graph is the eigenvector corresponding to the second smallest
eigenvalue of the Laplacian matrix of of the graph.
Parameters G : NetworkX graph
An undirected graph.
weight : object, optional
The data key used to determine the weight of each edge. If None, then each edge has
unit weight. Default value: None.
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normalized : bool, optional
Whether the normalized Laplacian matrix is used. Default value: False.
tol : float, optional
Tolerance of relative residual in eigenvalue computation. Default value: 1e-8.
method : string, optional
Method of eigenvalue computation. It should be one of �tracemin’ (TraceMIN), �lanczos’ (Lanczos iteration) and �lobpcg’ (LOBPCG). Default value: �tracemin’.
The TraceMIN algorithm uses a linear system solver. The following values allow specifying the solver to be used.
Value
�tracemin_pcg’
�tracemin_chol’
�tracemin_lu’
Solver
Preconditioned conjugate gradient method
Cholesky factorization
LU factorization
Returns fiedler_vector : NumPy array of floats.
Fiedler vector.
Raises NetworkXNotImplemented
If G is directed.
NetworkXError
If G has less than two nodes or is not connected.
See also:
laplacian_matrix
Notes
Edge weights are interpreted by their absolute values. For MultiGraph’s, weights of parallel edges are summed.
Zero-weighted edges are ignored.
To use Cholesky factorization in the TraceMIN algorithm, the scikits.sparse package must be installed.
7.4.3 spectral_ordering
spectral_ordering(G, weight=’weight’, normalized=False, tol=1e-08, method=’tracemin’)
Compute the spectral_ordering of a graph.
The spectral ordering of a graph is an ordering of its nodes where nodes in the same weakly connected components appear contiguous and ordered by their corresponding elements in the Fiedler vector of the component.
Parameters G : NetworkX graph
A graph.
weight : object, optional
The data key used to determine the weight of each edge. If None, then each edge has
unit weight. Default value: None.
normalized : bool, optional
Whether the normalized Laplacian matrix is used. Default value: False.
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tol : float, optional
Tolerance of relative residual in eigenvalue computation. Default value: 1e-8.
method : string, optional
Method of eigenvalue computation. It should be one of �tracemin’ (TraceMIN), �lanczos’ (Lanczos iteration) and �lobpcg’ (LOBPCG). Default value: �tracemin’.
The TraceMIN algorithm uses a linear system solver. The following values allow specifying the solver to be used.
Value
�tracemin_pcg’
�tracemin_chol’
�tracemin_lu’
Solver
Preconditioned conjugate gradient method
Cholesky factorization
LU factorization
Returns spectral_ordering : NumPy array of floats.
Spectral ordering of nodes.
Raises NetworkXError
If G is empty.
See also:
laplacian_matrix
Notes
Edge weights are interpreted by their absolute values. For MultiGraph’s, weights of parallel edges are summed.
Zero-weighted edges are ignored.
To use Cholesky factorization in the TraceMIN algorithm, the scikits.sparse package must be installed.
7.5 Attribute Matrices
Functions for constructing matrix-like objects from graph attributes.
attr_matrix(G[, edge_attr, node_attr, ...])
attr_sparse_matrix(G[, edge_attr, ...])
Returns a NumPy matrix using attributes from G.
Returns a SciPy sparse matrix using attributes from G.
7.5.1 attr_matrix
attr_matrix(G, edge_attr=None, node_attr=None, normalized=False, rc_order=None, dtype=None, order=None)
Returns a NumPy matrix using attributes from G.
If only рќђє is passed in, then the adjacency matrix is constructed.
Let A be a discrete set of values for the node attribute рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘Ћ рќ‘Ўрќ‘Ўрќ‘џ. Then the elements of A represent the rows and
columns of the constructed matrix. Now, iterate through every edge e=(u,v) in рќђє and consider the value of the
edge attribute рќ‘’рќ‘‘рќ‘”рќ‘’рќ‘Ћ рќ‘Ўрќ‘Ўрќ‘џ. If ua and va are the values of the node attribute рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘Ћ рќ‘Ўрќ‘Ўрќ‘џ for u and v, respectively, then
the value of the edge attribute is added to the matrix element at (ua, va).
Parameters G : graph
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The NetworkX graph used to construct the NumPy matrix.
edge_attr : str, optional
Each element of the matrix represents a running total of the specified edge attribute for
edges whose node attributes correspond to the rows/cols of the matirx. The attribute
must be present for all edges in the graph. If no attribute is specified, then we just count
the number of edges whose node attributes correspond to the matrix element.
node_attr : str, optional
Each row and column in the matrix represents a particular value of the node attribute.
The attribute must be present for all nodes in the graph. Note, the values of this attribute
should be reliably hashable. So, float values are not recommended. If no attribute is
specified, then the rows and columns will be the nodes of the graph.
normalized : bool, optional
If True, then each row is normalized by the summation of its values.
rc_order : list, optional
A list of the node attribute values. This list specifies the ordering of rows and columns
of the array. If no ordering is provided, then the ordering will be random (and also, a
return value).
Returns M : NumPy matrix
The attribute matrix.
ordering : list
If рќ‘џрќ‘ђрќ‘њ рќ‘џрќ‘‘рќ‘’рќ‘џ was specified, then only the matrix is returned. However, if рќ‘џрќ‘ђрќ‘њ рќ‘џрќ‘‘рќ‘’рќ‘џ was
None, then the ordering used to construct the matrix is returned as well.
Other Parameters dtype : NumPy data-type, optional
A valid NumPy dtype used to initialize the array. Keep in mind certain dtypes can
yield unexpected results if the array is to be normalized. The parameter is passed to
numpy.zeros(). If unspecified, the NumPy default is used.
order : {�C’, �F’}, optional
Whether to store multidimensional data in C- or Fortran-contiguous (row- or columnwise) order in memory. This parameter is passed to numpy.zeros(). If unspecified, the
NumPy default is used.
Examples
Construct an adjacency matrix:
>>> G = nx.Graph()
>>> G.add_edge(0,1,thickness=1,weight=3)
>>> G.add_edge(0,2,thickness=2)
>>> G.add_edge(1,2,thickness=3)
>>> nx.attr_matrix(G, rc_order=[0,1,2])
matrix([[ 0., 1., 1.],
[ 1., 0., 1.],
[ 1., 1., 0.]])
Alternatively, we can obtain the matrix describing edge thickness.
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>>> nx.attr_matrix(G, edge_attr='thickness', rc_order=[0,1,2])
matrix([[ 0., 1., 2.],
[ 1., 0., 3.],
[ 2., 3., 0.]])
We can also color the nodes and ask for the probability distribution over all edges (u,v) describing:
Pr(v has color Y | u has color X)
>>> G.node[0]['color'] = 'red'
>>> G.node[1]['color'] = 'red'
>>> G.node[2]['color'] = 'blue'
>>> rc = ['red', 'blue']
>>> nx.attr_matrix(G, node_attr='color', normalized=True, rc_order=rc)
matrix([[ 0.33333333, 0.66666667],
[ 1.
, 0.
]])
For example, the above tells us that for all edges (u,v):
Pr( v is red | u is red) = 1/3 Pr( v is blue | u is red) = 2/3
Pr( v is red | u is blue) = 1 Pr( v is blue | u is blue) = 0
Finally, we can obtain the total weights listed by the node colors.
>>> nx.attr_matrix(G, edge_attr='weight', node_attr='color', rc_order=rc)
matrix([[ 3., 2.],
[ 2., 0.]])
Thus, the total weight over all edges (u,v) with u and v having colors:
(red, red) is 3 # the sole contribution is from edge (0,1) (red, blue) is 2 # contributions from edges
(0,2) and (1,2) (blue, red) is 2 # same as (red, blue) since graph is undirected (blue, blue) is 0 # there
are no edges with blue endpoints
7.5.2 attr_sparse_matrix
attr_sparse_matrix(G, edge_attr=None, node_attr=None,
dtype=None)
Returns a SciPy sparse matrix using attributes from G.
normalized=False,
rc_order=None,
If only рќђє is passed in, then the adjacency matrix is constructed.
Let A be a discrete set of values for the node attribute рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘Ћ рќ‘Ўрќ‘Ўрќ‘џ. Then the elements of A represent the rows and
columns of the constructed matrix. Now, iterate through every edge e=(u,v) in рќђє and consider the value of the
edge attribute рќ‘’рќ‘‘рќ‘”рќ‘’рќ‘Ћ рќ‘Ўрќ‘Ўрќ‘џ. If ua and va are the values of the node attribute рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘Ћ рќ‘Ўрќ‘Ўрќ‘џ for u and v, respectively, then
the value of the edge attribute is added to the matrix element at (ua, va).
Parameters G : graph
The NetworkX graph used to construct the NumPy matrix.
edge_attr : str, optional
Each element of the matrix represents a running total of the specified edge attribute for
edges whose node attributes correspond to the rows/cols of the matirx. The attribute
must be present for all edges in the graph. If no attribute is specified, then we just count
the number of edges whose node attributes correspond to the matrix element.
node_attr : str, optional
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Each row and column in the matrix represents a particular value of the node attribute.
The attribute must be present for all nodes in the graph. Note, the values of this attribute
should be reliably hashable. So, float values are not recommended. If no attribute is
specified, then the rows and columns will be the nodes of the graph.
normalized : bool, optional
If True, then each row is normalized by the summation of its values.
rc_order : list, optional
A list of the node attribute values. This list specifies the ordering of rows and columns
of the array. If no ordering is provided, then the ordering will be random (and also, a
return value).
Returns M : SciPy sparse matrix
The attribute matrix.
ordering : list
If рќ‘џрќ‘ђрќ‘њ рќ‘џрќ‘‘рќ‘’рќ‘џ was specified, then only the matrix is returned. However, if рќ‘џрќ‘ђрќ‘њ рќ‘џрќ‘‘рќ‘’рќ‘џ was
None, then the ordering used to construct the matrix is returned as well.
Other Parameters dtype : NumPy data-type, optional
A valid NumPy dtype used to initialize the array. Keep in mind certain dtypes can
yield unexpected results if the array is to be normalized. The parameter is passed to
numpy.zeros(). If unspecified, the NumPy default is used.
Examples
Construct an adjacency matrix:
>>> G = nx.Graph()
>>> G.add_edge(0,1,thickness=1,weight=3)
>>> G.add_edge(0,2,thickness=2)
>>> G.add_edge(1,2,thickness=3)
>>> M = nx.attr_sparse_matrix(G, rc_order=[0,1,2])
>>> M.todense()
matrix([[ 0., 1., 1.],
[ 1., 0., 1.],
[ 1., 1., 0.]])
Alternatively, we can obtain the matrix describing edge thickness.
>>> M = nx.attr_sparse_matrix(G, edge_attr='thickness', rc_order=[0,1,2])
>>> M.todense()
matrix([[ 0., 1., 2.],
[ 1., 0., 3.],
[ 2., 3., 0.]])
We can also color the nodes and ask for the probability distribution over all edges (u,v) describing:
Pr(v has color Y | u has color X)
>>>
>>>
>>>
>>>
>>>
406
G.node[0]['color'] = 'red'
G.node[1]['color'] = 'red'
G.node[2]['color'] = 'blue'
rc = ['red', 'blue']
M = nx.attr_sparse_matrix(G, node_attr='color',
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>>> M.todense()
matrix([[ 0.33333333,
[ 1.
,
0.66666667],
0.
]])
For example, the above tells us that for all edges (u,v):
Pr( v is red | u is red) = 1/3 Pr( v is blue | u is red) = 2/3
Pr( v is red | u is blue) = 1 Pr( v is blue | u is blue) = 0
Finally, we can obtain the total weights listed by the node colors.
>>> M = nx.attr_sparse_matrix(G, edge_attr='weight',
>>> M.todense()
matrix([[ 3., 2.],
[ 2., 0.]])
node_attr=
Thus, the total weight over all edges (u,v) with u and v having colors:
(red, red) is 3 # the sole contribution is from edge (0,1) (red, blue) is 2 # contributions from edges
(0,2) and (1,2) (blue, red) is 2 # same as (red, blue) since graph is undirected (blue, blue) is 0 # there
are no edges with blue endpoints
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EIGHT
CONVERTING TO AND FROM OTHER DATA FORMATS
8.1 To NetworkX Graph
Functions to convert NetworkX graphs to and from other formats.
The preferred way of converting data to a NetworkX graph is through the graph constuctor. The constructor calls the
to_networkx_graph() function which attempts to guess the input type and convert it automatically.
8.1.1 Examples
Create a graph with a single edge from a dictionary of dictionaries
>>> d={0: {1: 1}} # dict-of-dicts single edge (0,1)
>>> G=nx.Graph(d)
8.1.2 See Also
nx_pygraphviz, nx_pydot
to_networkx_graph(data[, create_using, ...])
Make a NetworkX graph from a known data structure.
8.1.3 to_networkx_graph
to_networkx_graph(data, create_using=None, multigraph_input=False)
Make a NetworkX graph from a known data structure.
The preferred way to call this is automatically from the class constructor
>>> d={0: {1: {'weight':1}}} # dict-of-dicts single edge (0,1)
>>> G=nx.Graph(d)
instead of the equivalent
>>> G=nx.from_dict_of_dicts(d)
Parameters data : a object to be converted
Current known types are: any NetworkX graph dict-of-dicts dist-of-lists list of edges
numpy matrix numpy ndarray scipy sparse matrix pygraphviz agraph
create_using : NetworkX graph
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Use specified graph for result. Otherwise a new graph is created.
multigraph_input : bool (default False)
If True and data is a dict_of_dicts, try to create a multigraph assuming
dict_of_dict_of_lists. If data and create_using are both multigraphs then create a multigraph from a multigraph.
8.2 Dictionaries
to_dict_of_dicts(G[, nodelist, edge_data])
from_dict_of_dicts(d[, create_using, ...])
Return adjacency representation of graph as a dictionary of dictionaries.
Return a graph from a dictionary of dictionaries.
8.2.1 to_dict_of_dicts
to_dict_of_dicts(G, nodelist=None, edge_data=None)
Return adjacency representation of graph as a dictionary of dictionaries.
Parameters G : graph
A NetworkX graph
nodelist : list
Use only nodes specified in nodelist
edge_data : list, optional
If provided, the value of the dictionary will be set to edge_data for all edges. This
is useful to make an adjacency matrix type representation with 1 as the edge data. If
edgedata is None, the edgedata in G is used to fill the values. If G is a multigraph, the
edgedata is a dict for each pair (u,v).
8.2.2 from_dict_of_dicts
from_dict_of_dicts(d, create_using=None, multigraph_input=False)
Return a graph from a dictionary of dictionaries.
Parameters d : dictionary of dictionaries
A dictionary of dictionaries adjacency representation.
create_using : NetworkX graph
Use specified graph for result. Otherwise a new graph is created.
multigraph_input : bool (default False)
When True, the values of the inner dict are assumed to be containers of edge data for
multiple edges. Otherwise this routine assumes the edge data are singletons.
Examples
>>> dod= {0: {1:{'weight':1}}} # single edge (0,1)
>>> G=nx.from_dict_of_dicts(dod)
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or >>> G=nx.Graph(dod) # use Graph constructor
8.3 Lists
to_dict_of_lists(G[, nodelist])
from_dict_of_lists(d[, create_using])
to_edgelist(G[, nodelist])
from_edgelist(edgelist[, create_using])
Return adjacency representation of graph as a dictionary of lists.
Return a graph from a dictionary of lists.
Return a list of edges in the graph.
Return a graph from a list of edges.
8.3.1 to_dict_of_lists
to_dict_of_lists(G, nodelist=None)
Return adjacency representation of graph as a dictionary of lists.
Parameters G : graph
A NetworkX graph
nodelist : list
Use only nodes specified in nodelist
Notes
Completely ignores edge data for MultiGraph and MultiDiGraph.
8.3.2 from_dict_of_lists
from_dict_of_lists(d, create_using=None)
Return a graph from a dictionary of lists.
Parameters d : dictionary of lists
A dictionary of lists adjacency representation.
create_using : NetworkX graph
Use specified graph for result. Otherwise a new graph is created.
Examples
>>> dol= {0:[1]} # single edge (0,1)
>>> G=nx.from_dict_of_lists(dol)
or >>> G=nx.Graph(dol) # use Graph constructor
8.3.3 to_edgelist
to_edgelist(G, nodelist=None)
Return a list of edges in the graph.
Parameters G : graph
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A NetworkX graph
nodelist : list
Use only nodes specified in nodelist
8.3.4 from_edgelist
from_edgelist(edgelist, create_using=None)
Return a graph from a list of edges.
Parameters edgelist : list or iterator
Edge tuples
create_using : NetworkX graph
Use specified graph for result. Otherwise a new graph is created.
Examples
>>> edgelist= [(0,1)] # single edge (0,1)
>>> G=nx.from_edgelist(edgelist)
or >>> G=nx.Graph(edgelist) # use Graph constructor
8.4 Numpy
Functions to convert NetworkX graphs to and from numpy/scipy matrices.
The preferred way of converting data to a NetworkX graph is through the graph constuctor. The constructor calls the
to_networkx_graph() function which attempts to guess the input type and convert it automatically.
8.4.1 Examples
Create a 10 node random graph from a numpy matrix
>>> import numpy
>>> a = numpy.reshape(numpy.random.random_integers(0,1,size=100),(10,10))
>>> D = nx.DiGraph(a)
or equivalently
>>> D = nx.to_networkx_graph(a,create_using=nx.DiGraph())
8.4.2 See Also
nx_pygraphviz, nx_pydot
to_numpy_matrix(G[, nodelist, dtype, order, ...])
to_numpy_recarray(G[, nodelist, dtype, order])
from_numpy_matrix(A[, create_using])
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Return the graph adjacency matrix as a NumPy matrix.
Return the graph adjacency matrix as a NumPy recarray.
Return a graph from numpy matrix.
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8.4.3 to_numpy_matrix
to_numpy_matrix(G, nodelist=None, dtype=None, order=None, multigraph_weight=<built-in function
sum>, weight=’weight’, nonedge=0.0)
Return the graph adjacency matrix as a NumPy matrix.
Parameters G : graph
The NetworkX graph used to construct the NumPy matrix.
nodelist : list, optional
The rows and columns are ordered according to the nodes in рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў. If рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў is
None, then the ordering is produced by G.nodes().
dtype : NumPy data type, optional
A valid single NumPy data type used to initialize the array. This must be a simple type
such as int or numpy.float64 and not a compound data type (see to_numpy_recarray) If
None, then the NumPy default is used.
order : {�C’, �F’}, optional
Whether to store multidimensional data in C- or Fortran-contiguous (row- or columnwise) order in memory. If None, then the NumPy default is used.
multigraph_weight : {sum, min, max}, optional
An operator that determines how weights in multigraphs are handled. The default is to
sum the weights of the multiple edges.
weight : string or None optional (default=’weight’)
The edge attribute that holds the numerical value used for the edge weight. If an edge
does not have that attribute, then the value 1 is used instead.
nonedge : float (default=0.0)
The matrix values corresponding to nonedges are typically set to zero. However, this
could be undesirable if there are matrix values corresponding to actual edges that also
have the value zero. If so, one might prefer nonedges to have some other value, such as
nan.
Returns M : NumPy matrix
Graph adjacency matrix
See also:
to_numpy_recarray, from_numpy_matrix
Notes
The matrix entries are assigned to the weight edge attribute. When an edge does not have a weight attribute, the
value of the entry is set to the number 1. For multiple (parallel) edges, the values of the entries are determined
by the �multigraph_weight’ paramter. The default is to sum the weight attributes for each of the parallel edges.
When рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў does not contain every node in рќђє, the matrix is built from the subgraph of рќђє that is induced by
the nodes in рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў.
The convention used for self-loop edges in graphs is to assign the diagonal matrix entry value to the weight
attributr of the edge (or the number 1 if the edge has no weight attribute). If the alternate convention of doubling
the edge weight is desired the resulting Numpy matrix can be modified as follows:
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>>> import numpy as np
>>> G = nx.Graph([(1,1)])
>>> A = nx.to_numpy_matrix(G)
>>> A
matrix([[ 1.]])
>>> A.A[np.diag_indices_from(A)] *= 2
>>> A
matrix([[ 2.]])
Examples
>>> G = nx.MultiDiGraph()
>>> G.add_edge(0,1,weight=2)
>>> G.add_edge(1,0)
>>> G.add_edge(2,2,weight=3)
>>> G.add_edge(2,2)
>>> nx.to_numpy_matrix(G, nodelist=[0,1,2])
matrix([[ 0., 2., 0.],
[ 1., 0., 0.],
[ 0., 0., 4.]])
8.4.4 to_numpy_recarray
to_numpy_recarray(G, nodelist=None, dtype=[(�weight’, <type �float’>)], order=None)
Return the graph adjacency matrix as a NumPy recarray.
Parameters G : graph
The NetworkX graph used to construct the NumPy matrix.
nodelist : list, optional
The rows and columns are ordered according to the nodes in рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў. If рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў is
None, then the ordering is produced by G.nodes().
dtype : NumPy data-type, optional
A valid NumPy named dtype used to initialize the NumPy recarray. The data type
names are assumed to be keys in the graph edge attribute dictionary.
order : {�C’, �F’}, optional
Whether to store multidimensional data in C- or Fortran-contiguous (row- or columnwise) order in memory. If None, then the NumPy default is used.
Returns M : NumPy recarray
The graph with specified edge data as a Numpy recarray
Notes
When рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў does not contain every node in рќђє, the matrix is built from the subgraph of рќђє that is induced by
the nodes in рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў.
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Examples
>>> G = nx.Graph()
>>> G.add_edge(1,2,weight=7.0,cost=5)
>>> A=nx.to_numpy_recarray(G,dtype=[('weight',float),('cost',int)])
>>> print(A.weight)
[[ 0. 7.]
[ 7. 0.]]
>>> print(A.cost)
[[0 5]
[5 0]]
8.4.5 from_numpy_matrix
from_numpy_matrix(A, create_using=None)
Return a graph from numpy matrix.
The numpy matrix is interpreted as an adjacency matrix for the graph.
Parameters A : numpy matrix
An adjacency matrix representation of a graph
create_using : NetworkX graph
Use specified graph for result. The default is Graph()
See also:
to_numpy_matrix, to_numpy_recarray
Notes
If the numpy matrix has a single data type for each matrix entry it will be converted to an appropriate Python
data type.
If the numpy matrix has a user-specified compound data type the names of the data fields will be used as attribute
keys in the resulting NetworkX graph.
Examples
Simple integer weights on edges:
>>> import numpy
>>> A=numpy.matrix([[1,1],[2,1]])
>>> G=nx.from_numpy_matrix(A)
User defined compound data type on edges:
>>> import numpy
>>> dt=[('weight',float),('cost',int)]
>>> A=numpy.matrix([[(1.0,2)]],dtype=dt)
>>> G=nx.from_numpy_matrix(A)
>>> G.edges()
[(0, 0)]
>>> G[0][0]['cost']
2
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>>> G[0][0]['weight']
1.0
8.5 Scipy
to_scipy_sparse_matrix(G[, nodelist, dtype, ...])
from_scipy_sparse_matrix(A[, create_using, ...])
Return the graph adjacency matrix as a SciPy sparse matrix.
Return a graph from scipy sparse matrix adjacency list.
8.5.1 to_scipy_sparse_matrix
to_scipy_sparse_matrix(G, nodelist=None, dtype=None, weight=’weight’, format=’csr’)
Return the graph adjacency matrix as a SciPy sparse matrix.
Parameters G : graph
The NetworkX graph used to construct the NumPy matrix.
nodelist : list, optional
The rows and columns are ordered according to the nodes in рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў. If рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў is
None, then the ordering is produced by G.nodes().
dtype : NumPy data-type, optional
A valid NumPy dtype used to initialize the array. If None, then the NumPy default is
used.
weight : string or None optional (default=’weight’)
The edge attribute that holds the numerical value used for the edge weight. If None then
all edge weights are 1.
format : str in {�bsr’, �csr’, �csc’, �coo’, �lil’, �dia’, �dok’}
The type of the matrix to be returned (default �csr’). For some algorithms different
implementations of sparse matrices can perform better. See [R312] for details.
Returns M : SciPy sparse matrix
Graph adjacency matrix.
Notes
The matrix entries are populated using the edge attribute held in parameter weight. When an edge does not have
that attribute, the value of the entry is 1.
For multiple edges the matrix values are the sums of the edge weights.
When рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў does not contain every node in рќђє, the matrix is built from the subgraph of рќђє that is induced by
the nodes in рќ‘›рќ‘њрќ‘‘рќ‘’рќ‘™рќ‘–рќ‘ рќ‘Ў.
Uses coo_matrix format. To convert to other formats specify the format= keyword.
The convention used for self-loop edges in graphs is to assign the diagonal matrix entry value to the weight
attribute of the edge (or the number 1 if the edge has no weight attribute). If the alternate convention of doubling
the edge weight is desired the resulting Scipy sparse matrix can be modified as follows:
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>>> import scipy as sp
>>> G = nx.Graph([(1,1)])
>>> A = nx.to_scipy_sparse_matrix(G)
>>> print(A.todense())
[[1]]
>>> A.setdiag(A.diagonal()*2)
>>> print(A.todense())
[[2]]
References
[R312]
Examples
>>>
>>>
>>>
>>>
>>>
>>>
>>>
[[0
[1
[0
G = nx.MultiDiGraph()
G.add_edge(0,1,weight=2)
G.add_edge(1,0)
G.add_edge(2,2,weight=3)
G.add_edge(2,2)
S = nx.to_scipy_sparse_matrix(G, nodelist=[0,1,2])
print(S.todense())
2 0]
0 0]
0 4]]
8.5.2 from_scipy_sparse_matrix
from_scipy_sparse_matrix(A, create_using=None, edge_attribute=’weight’)
Return a graph from scipy sparse matrix adjacency list.
Parameters A: scipy sparse matrix
An adjacency matrix representation of a graph
create_using: NetworkX graph
Use specified graph for result. The default is Graph()
edge_attribute: string
Name of edge attribute to store matrix numeric value. The data will have the same type
as the matrix entry (int, float, (real,imag)).
Examples
>>> import scipy.sparse
>>> A = scipy.sparse.eye(2,2,1)
>>> G = nx.from_scipy_sparse_matrix(A)
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NINE
READING AND WRITING GRAPHS
9.1 Adjacency List
Read and write NetworkX graphs as adjacency lists.
Adjacency list format is useful for graphs without data associated with nodes or edges and for nodes that can be
meaningfully represented as strings.
9.1.1 Format
The adjacency list format consists of lines with node labels. The first label in a line is the source node. Further labels
in the line are considered target nodes and are added to the graph along with an edge between the source node and
target node.
The graph with edges a-b, a-c, d-e can be represented as the following adjacency list (anything following the # in a
line is a comment):
a b c # source target target
d e
read_adjlist(path[, comments, delimiter, ...])
write_adjlist(G, path[, comments, ...])
parse_adjlist(lines[, comments, delimiter, ...])
generate_adjlist(G[, delimiter])
Read graph in adjacency list format from path.
Write graph G in single-line adjacency-list format to path.
Parse lines of a graph adjacency list representation.
Generate a single line of the graph G in adjacency list format.
9.1.2 read_adjlist
read_adjlist(path, comments=’#’, delimiter=None, create_using=None, nodetype=None, encoding=’utf8’)
Read graph in adjacency list format from path.
Parameters path : string or file
Filename or file handle to read. Filenames ending in .gz or .bz2 will be uncompressed.
create_using: NetworkX graph container
Use given NetworkX graph for holding nodes or edges.
nodetype : Python type, optional
Convert nodes to this type.
comments : string, optional
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Marker for comment lines
delimiter : string, optional
Separator for node labels. The default is whitespace.
create_using: NetworkX graph container
Use given NetworkX graph for holding nodes or edges.
Returns G: NetworkX graph
The graph corresponding to the lines in adjacency list format.
See also:
write_adjlist
Notes
This format does not store graph or node data.
Examples
>>> G=nx.path_graph(4)
>>> nx.write_adjlist(G, "test.adjlist")
>>> G=nx.read_adjlist("test.adjlist")
The path can be a filehandle or a string with the name of the file. If a filehandle is provided, it has to be opened
in �rb’ mode.
>>> fh=open("test.adjlist", 'rb')
>>> G=nx.read_adjlist(fh)
Filenames ending in .gz or .bz2 will be compressed.
>>> nx.write_adjlist(G,"test.adjlist.gz")
>>> G=nx.read_adjlist("test.adjlist.gz")
The optional nodetype is a function to convert node strings to nodetype.
For example
>>> G=nx.read_adjlist("test.adjlist", nodetype=int)
will attempt to convert all nodes to integer type.
Since nodes must be hashable, the function nodetype must return hashable types (e.g. int, float, str, frozenset or tuples of those, etc.)
The optional create_using parameter is a NetworkX graph container. The default is Graph(), an undirected
graph. To read the data as a directed graph use
>>> G=nx.read_adjlist("test.adjlist", create_using=nx.DiGraph())
9.1.3 write_adjlist
write_adjlist(G, path, comments=’#’, delimiter=’ �, encoding=’utf-8’)
Write graph G in single-line adjacency-list format to path.
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Parameters G : NetworkX graph
path : string or file
Filename or file handle for data output. Filenames ending in .gz or .bz2 will be compressed.
comments : string, optional
Marker for comment lines
delimiter : string, optional
Separator for node labels
encoding : string, optional
Text encoding.
See also:
read_adjlist, generate_adjlist
Notes
This format does not store graph, node, or edge data.
Examples
>>> G=nx.path_graph(4)
>>> nx.write_adjlist(G,"test.adjlist")
The path can be a filehandle or a string with the name of the file. If a filehandle is provided, it has to be opened
in �wb’ mode.
>>> fh=open("test.adjlist",'wb')
>>> nx.write_adjlist(G, fh)
9.1.4 parse_adjlist
parse_adjlist(lines, comments=’#’, delimiter=None, create_using=None, nodetype=None)
Parse lines of a graph adjacency list representation.
Parameters lines : list or iterator of strings
Input data in adjlist format
create_using: NetworkX graph container
Use given NetworkX graph for holding nodes or edges.
nodetype : Python type, optional
Convert nodes to this type.
comments : string, optional
Marker for comment lines
delimiter : string, optional
Separator for node labels. The default is whitespace.
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create_using: NetworkX graph container
Use given NetworkX graph for holding nodes or edges.
Returns G: NetworkX graph
The graph corresponding to the lines in adjacency list format.
See also:
read_adjlist
Examples
>>> lines = ['1 2 5',
...
'2 3 4',
...
'3 5',
...
'4',
...
'5']
>>> G = nx.parse_adjlist(lines, nodetype = int)
>>> G.nodes()
[1, 2, 3, 4, 5]
>>> G.edges()
[(1, 2), (1, 5), (2, 3), (2, 4), (3, 5)]
9.1.5 generate_adjlist
generate_adjlist(G, delimiter=’ �)
Generate a single line of the graph G in adjacency list format.
Parameters G : NetworkX graph
delimiter : string, optional
Separator for node labels
Returns lines : string
Lines of data in adjlist format.
See also:
write_adjlist, read_adjlist
Examples
>>>
>>>
...
0 1
1 2
2 3
3 4
4 5
5 6
6
422
G = nx.lollipop_graph(4, 3)
for line in nx.generate_adjlist(G):
print(line)
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9.2 Multiline Adjacency List
Read and write NetworkX graphs as multi-line adjacency lists.
The multi-line adjacency list format is useful for graphs with nodes that can be meaningfully represented as strings.
With this format simple edge data can be stored but node or graph data is not.
9.2.1 Format
The first label in a line is the source node label followed by the node degree d. The next d lines are target node labels
and optional edge data. That pattern repeats for all nodes in the graph.
The graph with edges a-b, a-c, d-e can be represented as the following adjacency list (anything following the # in a
line is a comment):
# example.multiline-adjlist
a 2
b
c
d 1
e
read_multiline_adjlist(path[, comments, ...])
write_multiline_adjlist(G, path[, ...])
parse_multiline_adjlist(lines[, comments, ...])
generate_multiline_adjlist(G[, delimiter])
Read graph in multi-line adjacency list format from path.
Write the graph G in multiline adjacency list format to path
Parse lines of a multiline adjacency list representation of a graph.
Generate a single line of the graph G in multiline adjacency list format.
9.2.2 read_multiline_adjlist
read_multiline_adjlist(path, comments=’#’, delimiter=None, create_using=None, nodetype=None,
edgetype=None, encoding=’utf-8’)
Read graph in multi-line adjacency list format from path.
Parameters path : string or file
Filename or file handle to read. Filenames ending in .gz or .bz2 will be uncompressed.
create_using: NetworkX graph container
Use given NetworkX graph for holding nodes or edges.
nodetype : Python type, optional
Convert nodes to this type.
edgetype : Python type, optional
Convert edge data to this type.
comments : string, optional
Marker for comment lines
delimiter : string, optional
Separator for node labels. The default is whitespace.
create_using: NetworkX graph container
Use given NetworkX graph for holding nodes or edges.
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Returns G: NetworkX graph
See also:
write_multiline_adjlist
Notes
This format does not store graph, node, or edge data.
Examples
>>> G=nx.path_graph(4)
>>> nx.write_multiline_adjlist(G,"test.adjlist")
>>> G=nx.read_multiline_adjlist("test.adjlist")
The path can be a file or a string with the name of the file. If a file s provided, it has to be opened in �rb’ mode.
>>> fh=open("test.adjlist", 'rb')
>>> G=nx.read_multiline_adjlist(fh)
Filenames ending in .gz or .bz2 will be compressed.
>>> nx.write_multiline_adjlist(G,"test.adjlist.gz")
>>> G=nx.read_multiline_adjlist("test.adjlist.gz")
The optional nodetype is a function to convert node strings to nodetype.
For example
>>> G=nx.read_multiline_adjlist("test.adjlist", nodetype=int)
will attempt to convert all nodes to integer type.
The optional edgetype is a function to convert edge data strings to edgetype.
>>> G=nx.read_multiline_adjlist("test.adjlist")
The optional create_using parameter is a NetworkX graph container. The default is Graph(), an undirected
graph. To read the data as a directed graph use
>>> G=nx.read_multiline_adjlist("test.adjlist", create_using=nx.DiGraph())
9.2.3 write_multiline_adjlist
write_multiline_adjlist(G, path, delimiter=’ �, comments=’#’, encoding=’utf-8’)
Write the graph G in multiline adjacency list format to path
Parameters G : NetworkX graph
comments : string, optional
Marker for comment lines
delimiter : string, optional
Separator for node labels
encoding : string, optional
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Text encoding.
See also:
read_multiline_adjlist
Examples
>>> G=nx.path_graph(4)
>>> nx.write_multiline_adjlist(G,"test.adjlist")
The path can be a file handle or a string with the name of the file. If a file handle is provided, it has to be opened
in �wb’ mode.
>>> fh=open("test.adjlist",'wb')
>>> nx.write_multiline_adjlist(G,fh)
Filenames ending in .gz or .bz2 will be compressed.
>>> nx.write_multiline_adjlist(G,"test.adjlist.gz")
9.2.4 parse_multiline_adjlist
parse_multiline_adjlist(lines, comments=’#’, delimiter=None,
type=None, edgetype=None)
Parse lines of a multiline adjacency list representation of a graph.
create_using=None,
node-
Parameters lines : list or iterator of strings
Input data in multiline adjlist format
create_using: NetworkX graph container
Use given NetworkX graph for holding nodes or edges.
nodetype : Python type, optional
Convert nodes to this type.
comments : string, optional
Marker for comment lines
delimiter : string, optional
Separator for node labels. The default is whitespace.
create_using: NetworkX graph container
Use given NetworkX graph for holding nodes or edges.
Returns G: NetworkX graph
The graph corresponding to the lines in multiline adjacency list format.
Examples
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>>>
...
...
...
...
>>>
>>>
[1,
lines = ['1 2',
"2 {'weight':3, 'name': 'Frodo'}",
"3 {}",
"2 1",
"5 {'weight':6, 'name': 'Saruman'}"]
G = nx.parse_multiline_adjlist(iter(lines), nodetype = int)
G.nodes()
2, 3, 5]
9.2.5 generate_multiline_adjlist
generate_multiline_adjlist(G, delimiter=’ �)
Generate a single line of the graph G in multiline adjacency list format.
Parameters G : NetworkX graph
delimiter : string, optional
Separator for node labels
Returns lines : string
Lines of data in multiline adjlist format.
See also:
write_multiline_adjlist, read_multiline_adjlist
Examples
>>> G = nx.lollipop_graph(4, 3)
>>> for line in nx.generate_multiline_adjlist(G):
...
print(line)
0 3
1 {}
2 {}
3 {}
1 2
2 {}
3 {}
2 1
3 {}
3 1
4 {}
4 1
5 {}
5 1
6 {}
6 0
9.3 Edge List
Read and write NetworkX graphs as edge lists.
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The multi-line adjacency list format is useful for graphs with nodes that can be meaningfully represented as strings.
With the edgelist format simple edge data can be stored but node or graph data is not. There is no way of representing
isolated nodes unless the node has a self-loop edge.
9.3.1 Format
You can read or write three formats of edge lists with these functions.
Node pairs with no data:
1 2
Python dictionary as data:
1 2 {'weight':7, 'color':'green'}
Arbitrary data:
1 2 7 green
read_edgelist(path[, comments, delimiter, ...])
write_edgelist(G, path[, comments, ...])
read_weighted_edgelist(path[, comments, ...])
write_weighted_edgelist(G, path[, comments, ...])
generate_edgelist(G[, delimiter, data])
parse_edgelist(lines[, comments, delimiter, ...])
Read a graph from a list of edges.
Write graph as a list of edges.
Read a graph as list of edges with numeric weights.
Write graph G as a list of edges with numeric weights.
Generate a single line of the graph G in edge list format.
Parse lines of an edge list representation of a graph.
9.3.2 read_edgelist
read_edgelist(path, comments=’#’, delimiter=None, create_using=None, nodetype=None, data=True,
edgetype=None, encoding=’utf-8’)
Read a graph from a list of edges.
Parameters path : file or string
File or filename to read. If a file is provided, it must be opened in �rb’ mode. Filenames
ending in .gz or .bz2 will be uncompressed.
comments : string, optional
The character used to indicate the start of a comment.
delimiter : string, optional
The string used to separate values. The default is whitespace.
create_using : Graph container, optional,
Use specified container to build graph. The default is networkx.Graph, an undirected
graph.
nodetype : int, float, str, Python type, optional
Convert node data from strings to specified type
data : bool or list of (label,type) tuples
Tuples specifying dictionary key names and types for edge data
edgetype : int, float, str, Python type, optional OBSOLETE
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Convert edge data from strings to specified type and use as �weight’
encoding: string, optional
Specify which encoding to use when reading file.
Returns G : graph
A networkx Graph or other type specified with create_using
See also:
parse_edgelist
Notes
Since nodes must be hashable, the function nodetype must return hashable types (e.g. int, float, str, frozenset or tuples of those, etc.)
Examples
>>> nx.write_edgelist(nx.path_graph(4), "test.edgelist")
>>> G=nx.read_edgelist("test.edgelist")
>>> fh=open("test.edgelist", 'rb')
>>> G=nx.read_edgelist(fh)
>>> fh.close()
>>> G=nx.read_edgelist("test.edgelist", nodetype=int)
>>> G=nx.read_edgelist("test.edgelist",create_using=nx.DiGraph())
Edgelist with data in a list:
>>> textline = '1 2 3'
>>> fh = open('test.edgelist','w')
>>> d = fh.write(textline)
>>> fh.close()
>>> G = nx.read_edgelist('test.edgelist', nodetype=int, data=(('weight',float),))
>>> G.nodes()
[1, 2]
>>> G.edges(data = True)
[(1, 2, {'weight': 3.0})]
See parse_edgelist() for more examples of formatting.
9.3.3 write_edgelist
write_edgelist(G, path, comments=’#’, delimiter=’ �, data=True, encoding=’utf-8’)
Write graph as a list of edges.
Parameters G : graph
A NetworkX graph
path : file or string
File or filename to write. If a file is provided, it must be opened in �wb’ mode. Filenames
ending in .gz or .bz2 will be compressed.
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comments : string, optional
The character used to indicate the start of a comment
delimiter : string, optional
The string used to separate values. The default is whitespace.
data : bool or list, optional
If False write no edge data. If True write a string representation of the edge data dictionary.. If a list (or other iterable) is provided, write the keys specified in the list.
encoding: string, optional
Specify which encoding to use when writing file.
See also:
write_edgelist, write_weighted_edgelist
Examples
>>>
>>>
>>>
>>>
>>>
>>>
>>>
G=nx.path_graph(4)
nx.write_edgelist(G, "test.edgelist")
G=nx.path_graph(4)
fh=open("test.edgelist",'wb')
nx.write_edgelist(G, fh)
nx.write_edgelist(G, "test.edgelist.gz")
nx.write_edgelist(G, "test.edgelist.gz", data=False)
>>>
>>>
>>>
>>>
>>>
G=nx.Graph()
G.add_edge(1,2,weight=7,color='red')
nx.write_edgelist(G,'test.edgelist',data=False)
nx.write_edgelist(G,'test.edgelist',data=['color'])
nx.write_edgelist(G,'test.edgelist',data=['color','weight'])
9.3.4 read_weighted_edgelist
read_weighted_edgelist(path, comments=’#’, delimiter=None, create_using=None, nodetype=None,
encoding=’utf-8’)
Read a graph as list of edges with numeric weights.
Parameters path : file or string
File or filename to read. If a file is provided, it must be opened in �rb’ mode. Filenames
ending in .gz or .bz2 will be uncompressed.
comments : string, optional
The character used to indicate the start of a comment.
delimiter : string, optional
The string used to separate values. The default is whitespace.
create_using : Graph container, optional,
Use specified container to build graph. The default is networkx.Graph, an undirected
graph.
nodetype : int, float, str, Python type, optional
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Convert node data from strings to specified type
encoding: string, optional
Specify which encoding to use when reading file.
Returns G : graph
A networkx Graph or other type specified with create_using
Notes
Since nodes must be hashable, the function nodetype must return hashable types (e.g. int, float, str, frozenset or tuples of those, etc.)
Example edgelist file format.
With numeric edge data:
#
#
#
a
a
d
read with
>>> G=nx.read_weighted_edgelist(fh)
source target data
b 1
c 3.14159
e 42
9.3.5 write_weighted_edgelist
write_weighted_edgelist(G, path, comments=’#’, delimiter=’ �, encoding=’utf-8’)
Write graph G as a list of edges with numeric weights.
Parameters G : graph
A NetworkX graph
path : file or string
File or filename to write. If a file is provided, it must be opened in �wb’ mode. Filenames
ending in .gz or .bz2 will be compressed.
comments : string, optional
The character used to indicate the start of a comment
delimiter : string, optional
The string used to separate values. The default is whitespace.
encoding: string, optional
Specify which encoding to use when writing file.
See also:
read_edgelist, write_edgelist, write_weighted_edgelist
Examples
>>> G=nx.Graph()
>>> G.add_edge(1,2,weight=7)
>>> nx.write_weighted_edgelist(G, 'test.weighted.edgelist')
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9.3.6 generate_edgelist
generate_edgelist(G, delimiter=’ �, data=True)
Generate a single line of the graph G in edge list format.
Parameters G : NetworkX graph
delimiter : string, optional
Separator for node labels
data : bool or list of keys
If False generate no edge data. If True use a dictionary representation of edge data. If a
list of keys use a list of data values corresponding to the keys.
Returns lines : string
Lines of data in adjlist format.
See also:
write_adjlist, read_adjlist
Examples
>>>
>>>
>>>
>>>
...
0 1
0 2
0 3
1 2
1 3
2 3
3 4
4 5
5 6
G = nx.lollipop_graph(4, 3)
G[1][2]['weight'] = 3
G[3][4]['capacity'] = 12
for line in nx.generate_edgelist(G, data=False):
print(line)
>>>
...
0 1
0 2
0 3
1 2
1 3
2 3
3 4
4 5
5 6
for line in nx.generate_edgelist(G):
print(line)
{}
{}
{}
{'weight': 3}
{}
{}
{'capacity': 12}
{}
{}
>>> for line in nx.generate_edgelist(G,data=['weight']):
...
print(line)
0 1
0 2
0 3
1 2 3
1 3
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2
3
4
5
3
4
5
6
9.3.7 parse_edgelist
parse_edgelist(lines, comments=’#’, delimiter=None,
data=True)
Parse lines of an edge list representation of a graph.
create_using=None,
nodetype=None,
Parameters lines : list or iterator of strings
Input data in edgelist format
comments : string, optional
Marker for comment lines
delimiter : string, optional
Separator for node labels
create_using: NetworkX graph container, optional
Use given NetworkX graph for holding nodes or edges.
nodetype : Python type, optional
Convert nodes to this type.
data : bool or list of (label,type) tuples
If False generate no edge data or if True use a dictionary representation of edge data or
a list tuples specifying dictionary key names and types for edge data.
Returns G: NetworkX Graph
The graph corresponding to lines
See also:
read_weighted_edgelist
Examples
Edgelist with no data:
>>> lines = ["1 2",
...
"2 3",
...
"3 4"]
>>> G = nx.parse_edgelist(lines, nodetype = int)
>>> G.nodes()
[1, 2, 3, 4]
>>> G.edges()
[(1, 2), (2, 3), (3, 4)]
Edgelist with data in Python dictionary representation:
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>>> lines = ["1 2 {'weight':3}",
...
"2 3 {'weight':27}",
...
"3 4 {'weight':3.0}"]
>>> G = nx.parse_edgelist(lines, nodetype = int)
>>> G.nodes()
[1, 2, 3, 4]
>>> G.edges(data = True)
[(1, 2, {'weight': 3}), (2, 3, {'weight': 27}), (3, 4, {'weight': 3.0})]
Edgelist with data in a list:
>>> lines = ["1 2 3",
...
"2 3 27",
...
"3 4 3.0"]
>>> G = nx.parse_edgelist(lines, nodetype = int, data=(('weight',float),))
>>> G.nodes()
[1, 2, 3, 4]
>>> G.edges(data = True)
[(1, 2, {'weight': 3.0}), (2, 3, {'weight': 27.0}), (3, 4, {'weight': 3.0})]
9.4 GEXF
Read and write graphs in GEXF format.
GEXF (Graph Exchange XML Format) is a language for describing complex network structures, their associated data
and dynamics.
This implementation does not support mixed graphs (directed and undirected edges together).
9.4.1 Format
GEXF is an XML format.
See http://gexf.net/format/schema.html
http://gexf.net/format/basic.html for examples.
read_gexf(path[, node_type, relabel, version])
write_gexf(G, path[, encoding, prettyprint, ...])
relabel_gexf_graph(G)
for
the
specification
and
Read graph in GEXF format from path.
Write G in GEXF format to path.
Relabel graph using “label” node keyword for node label.
9.4.2 read_gexf
read_gexf(path, node_type=None, relabel=False, version=�1.1draft’)
Read graph in GEXF format from path.
“GEXF (Graph Exchange XML Format) is a language for describing complex networks structures, their associated data and dynamics” [R361].
Parameters path : file or string
File or file name to write. File names ending in .gz or .bz2 will be compressed.
node_type: Python type (default: None)
Convert node ids to this type if not None.
relabel : bool (default: False)
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If True relabel the nodes to use the GEXF node “label” attribute instead of the node “id”
attribute as the NetworkX node label.
Returns graph: NetworkX graph
If no parallel edges are found a Graph or DiGraph is returned. Otherwise a MultiGraph
or MultiDiGraph is returned.
Notes
This implementation does not support mixed graphs (directed and undirected edges together).
References
[R361]
9.4.3 write_gexf
write_gexf(G, path, encoding=’utf-8’, prettyprint=True, version=�1.1draft’)
Write G in GEXF format to path.
“GEXF (Graph Exchange XML Format) is a language for describing complex networks structures, their associated data and dynamics” [R362].
Parameters G : graph
A NetworkX graph
path : file or string
File or file name to write. File names ending in .gz or .bz2 will be compressed.
encoding : string (optional)
Encoding for text data.
prettyprint : bool (optional)
If True use line breaks and indenting in output XML.
Notes
This implementation does not support mixed graphs (directed and undirected edges together).
The node id attribute is set to be the string of the node label. If you want to specify an id use set it as node data,
e.g. node[’a’][’id’]=1 to set the id of node �a’ to 1.
References
[R362]
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Examples
>>> G=nx.path_graph(4)
>>> nx.write_gexf(G, "test.gexf")
9.4.4 relabel_gexf_graph
relabel_gexf_graph(G)
Relabel graph using “label” node keyword for node label.
Parameters G : graph
A NetworkX graph read from GEXF data
Returns H : graph
A NetworkX graph with relabed nodes
Notes
This function relabels the nodes in a NetworkX graph with the “label” attribute. It also handles relabeling the
specific GEXF node attributes “parents”, and “pid”.
9.5 GML
Read graphs in GML format.
“GML, the G>raph Modelling Language, is our proposal for a portable file format for graphs. GML’s key features are
portability, simple syntax, extensibility and flexibility. A GML file consists of a hierarchical key-value lists. Graphs
can be annotated with arbitrary data structures. The idea for a common file format was born at the GD�95; this proposal
is the outcome of many discussions. GML is the standard file format in the Graphlet graph editor system. It has been
overtaken and adapted by several other systems for drawing graphs.”
See http://www.infosun.fim.uni-passau.de/Graphlet/GML/gml-tr.html
9.5.1 Format
See http://www.infosun.fim.uni-passau.de/Graphlet/GML/gml-tr.html for format specification.
Example graphs in GML format: http://www-personal.umich.edu/~mejn/netdata/
read_gml(path[, label, destringizer])
write_gml(G, path[, stringizer])
parse_gml(lines[, label, destringizer])
generate_gml(G[, stringizer])
literal_destringizer(rep)
literal_stringizer(value)
9.5. GML
Read graph in GML format from path.
Write a graph G in GML format to the file or file handle path.
Parse GML graph from a string or iterable.
Generate a single entry of the graph G in GML format.
Convert a Python literal to the value it represents.
Convert a value to a Python literal in GML representation.
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9.5.2 read_gml
read_gml(path, label=’label’, destringizer=None)
Read graph in GML format from path.
Parameters path : filename or filehandle
The filename or filehandle to read from.
label : string, optional
If not None, the parsed nodes will be renamed according to node attributes indicated by
label. Default value: ’label’.
destringizer : callable, optional
A destringizer that recovers values stored as strings in GML. If it cannot convert a string
to a value, a ValueError is raised. Default value : None.
Returns G : NetworkX graph
The parsed graph.
Raises NetworkXError
If the input cannot be parsed.
See also:
write_gml, parse_gml
Notes
The GML specification says that files should be ASCII encoded, with any extended ASCII characters (iso88591) appearing as HTML character entities.
References
GML specification: http://www.infosun.fim.uni-passau.de/Graphlet/GML/gml-tr.html
Examples
>>> G = nx.path_graph(4)
>>> nx.write_gml(G, 'test.gml')
>>> H = nx.read_gml('test.gml')
9.5.3 write_gml
write_gml(G, path, stringizer=None)
Write a graph G in GML format to the file or file handle path.
Parameters G : NetworkX graph
The graph to be converted to GML.
path : filename or filehandle
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The filename or filehandle to write. Files whose names end with .gz or .bz2 will be
compressed.
stringizer : callable, optional
A stringizer which converts non-int/float/dict values into strings. If it cannot convert a
value into a string, it should raise a ValueError raised to indicate that. Default value:
None.
Raises NetworkXError
If stringizer cannot convert a value into a string, or the value to convert is not a
string while stringizer is None.
See also:
read_gml, generate_gml
Notes
Graph attributes named ’directed’, ’multigraph’, ’node’ or ’edge’,node attributes named ’id’
or ’label’, edge attributes named ’source’ or ’target’ (or ’key’ if G is a multigraph) are ignored
because these attribute names are used to encode the graph structure.
Examples
>>> G = nx.path_graph(4)
>>> nx.write_gml(G, "test.gml")
Filenames ending in .gz or .bz2 will be compressed.
>>> nx.write_gml(G, "test.gml.gz")
9.5.4 parse_gml
parse_gml(lines, label=’label’, destringizer=None)
Parse GML graph from a string or iterable.
Parameters lines : string or iterable of strings
Data in GML format.
label : string, optional
If not None, the parsed nodes will be renamed according to node attributes indicated by
label. Default value: ’label’.
destringizer : callable, optional
A destringizer that recovers values stored as strings in GML. If it cannot convert a string
to a value, a ValueError is raised. Default value : None.
Returns G : NetworkX graph
The parsed graph.
Raises NetworkXError
If the input cannot be parsed.
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See also:
write_gml, read_gml
Notes
This stores nested GML attributes as dictionaries in the NetworkX graph, node, and edge attribute structures.
References
GML specification: http://www.infosun.fim.uni-passau.de/Graphlet/GML/gml-tr.html
9.5.5 generate_gml
generate_gml(G, stringizer=None)
Generate a single entry of the graph G in GML format.
Parameters G : NetworkX graph
The graph to be converted to GML.
stringizer : callable, optional
A stringizer which converts non-int/float/dict values into strings. If it cannot convert a
value into a string, it should raise a ValueError raised to indicate that. Default value:
None.
Returns lines: generator of strings
Lines of GML data. Newlines are not appended.
Raises NetworkXError
If stringizer cannot convert a value into a string, or the value to convert is not a
string while stringizer is None.
Notes
Graph attributes named ’directed’, ’multigraph’, ’node’ or ’edge’,node attributes named ’id’
or ’label’, edge attributes named ’source’ or ’target’ (or ’key’ if G is a multigraph) are ignored
because these attribute names are used to encode the graph structure.
9.5.6 literal_destringizer
literal_destringizer(rep)
Convert a Python literal to the value it represents.
Parameters rep : string
A Python literal.
Returns value : object
The value of the Python literal.
Raises ValueError
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If rep is not a Python literal.
9.5.7 literal_stringizer
literal_stringizer(value)
Convert a value to a Python literal in GML representation.
Parameters value : object
The value to be converted to GML representation.
Returns rep : string
A double-quoted Python literal representing value. Unprintable characters are replaced
by XML character references.
Raises ValueError
If value cannot be converted to GML.
Notes
literal_stringizer is largely the same as repr in terms of functionality but attempts prefix unicode
and bytes literals with u and b to provide better interoperability of data generated by Python 2 and Python 3.
The original value can be recovered using the networkx.readwrite.gml.literal_destringizer
function.
9.6 Pickle
Read and write NetworkX graphs as Python pickles.
“The pickle module implements a fundamental, but powerful algorithm for serializing and de-serializing a Python
object structure. “Pickling” is the process whereby a Python object hierarchy is converted into a byte stream, and
“unpickling” is the inverse operation, whereby a byte stream is converted back into an object hierarchy.”
Note that NetworkX graphs can contain any hashable Python object as node (not just integers and strings). For arbitrary
data types it may be difficult to represent the data as text. In that case using Python pickles to store the graph data can
be used.
9.6.1 Format
See http://docs.python.org/library/pickle.html
read_gpickle(path)
write_gpickle(G, path[, protocol])
Read graph object in Python pickle format.
Write graph in Python pickle format.
9.6.2 read_gpickle
read_gpickle(path)
Read graph object in Python pickle format.
Pickles are a serialized byte stream of a Python object [R363]. This format will preserve Python objects used as
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nodes or edges.
Parameters path : file or string
File or filename to write. Filenames ending in .gz or .bz2 will be uncompressed.
Returns G : graph
A NetworkX graph
References
[R363]
Examples
>>> G = nx.path_graph(4)
>>> nx.write_gpickle(G, "test.gpickle")
>>> G = nx.read_gpickle("test.gpickle")
9.6.3 write_gpickle
write_gpickle(G, path, protocol=2)
Write graph in Python pickle format.
Pickles are a serialized byte stream of a Python object [R364]. This format will preserve Python objects used as
nodes or edges.
Parameters G : graph
A NetworkX graph
path : file or string
File or filename to write. Filenames ending in .gz or .bz2 will be compressed.
protocol : integer
Pickling protocol to use. Default value: pickle.HIGHEST_PROTOCOL.
References
[R364]
Examples
>>> G = nx.path_graph(4)
>>> nx.write_gpickle(G, "test.gpickle")
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9.7 GraphML
Read and write graphs in GraphML format.
This implementation does not support mixed graphs (directed and unidirected edges together), hyperedges, nested
graphs, or ports.
“GraphML is a comprehensive and easy-to-use file format for graphs. It consists of a language core to describe the
structural properties of a graph and a flexible extension mechanism to add application-specific data. Its main features
include support of
• directed, undirected, and mixed graphs,
• hypergraphs,
• hierarchical graphs,
• graphical representations,
• references to external data,
• application-specific attribute data, and
• light-weight parsers.
Unlike many other file formats for graphs, GraphML does not use a custom syntax. Instead, it is based on XML and
hence ideally suited as a common denominator for all kinds of services generating, archiving, or processing graphs.”
http://graphml.graphdrawing.org/
9.7.1 Format
GraphML is an XML format. See http://graphml.graphdrawing.org/specification.html for the specification and
http://graphml.graphdrawing.org/primer/graphml-primer.html for examples.
read_graphml(path[, node_type])
write_graphml(G, path[, encoding, prettyprint])
Read graph in GraphML format from path.
Write G in GraphML XML format to path
9.7.2 read_graphml
read_graphml(path, node_type=<type �str’>)
Read graph in GraphML format from path.
Parameters path : file or string
File or filename to write. Filenames ending in .gz or .bz2 will be compressed.
node_type: Python type (default: str)
Convert node ids to this type
Returns graph: NetworkX graph
If no parallel edges are found a Graph or DiGraph is returned. Otherwise a MultiGraph
or MultiDiGraph is returned.
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Notes
This implementation does not support mixed graphs (directed and unidirected edges together), hypergraphs,
nested graphs, or ports.
For multigraphs the GraphML edge “id” will be used as the edge key. If not specified then they “key” attribute
will be used. If there is no “key” attribute a default NetworkX multigraph edge key will be provided.
Files with the yEd “yfiles” extension will can be read but the graphics information is discarded.
yEd compressed files (“file.graphmlz” extension) can be read by renaming the file to “file.graphml.gz”.
9.7.3 write_graphml
write_graphml(G, path, encoding=’utf-8’, prettyprint=True)
Write G in GraphML XML format to path
Parameters G : graph
A networkx graph
path : file or string
File or filename to write. Filenames ending in .gz or .bz2 will be compressed.
encoding : string (optional)
Encoding for text data.
prettyprint : bool (optional)
If True use line breaks and indenting in output XML.
Notes
This implementation does not support mixed graphs (directed and unidirected edges together) hyperedges,
nested graphs, or ports.
Examples
>>> G=nx.path_graph(4)
>>> nx.write_graphml(G, "test.graphml")
9.8 JSON
Generate and parse JSON serializable data for NetworkX graphs.
These formats are suitable for use with the d3.js examples http://d3js.org/
The three formats that you can generate with NetworkX are:
• node-link like in the d3.js example http://bl.ocks.org/mbostock/4062045
• tree like in the d3.js example http://bl.ocks.org/mbostock/4063550
• adjacency like in the d3.js example http://bost.ocks.org/mike/miserables/
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node_link_data(G[, attrs])
node_link_graph(data[, directed, ...])
adjacency_data(G[, attrs])
adjacency_graph(data[, directed, ...])
tree_data(G, root[, attrs])
tree_graph(data[, attrs])
Return data in node-link format that is suitable for JSON serialization and use in Javascri
Return graph from node-link data format.
Return data in adjacency format that is suitable for JSON serialization and use in Javascr
Return graph from adjacency data format.
Return data in tree format that is suitable for JSON serialization and use in Javascript doc
Return graph from tree data format.
9.8.1 node_link_data
node_link_data(G, attrs={�source’: �source’, �target’: �target’, �key’: �key’, �id’: �id’})
Return data in node-link format that is suitable for JSON serialization and use in Javascript documents.
Parameters G : NetworkX graph
attrs : dict
A dictionary that contains four keys �id’, �source’, �target’ and �key’. The corresponding values provide the attribute names for storing NetworkX-internal graph data. The
values should be unique. Default value: dict(id=’id’, source=’source’,
target=’target’, key=’key’).
If some user-defined graph data use these attribute names as data keys, they may be
silently dropped.
Returns data : dict
A dictionary with node-link formatted data.
Raises NetworkXError
If values in attrs are not unique.
See also:
node_link_graph, adjacency_data, tree_data
Notes
Graph, node, and link attributes are stored in this format but keys for attributes must be strings if you want to
serialize with JSON.
The default value of attrs will be changed in a future release of NetworkX.
Examples
>>> from networkx.readwrite import json_graph
>>> G = nx.Graph([(1,2)])
>>> data = json_graph.node_link_data(G)
To serialize with json
>>> import json
>>> s = json.dumps(data)
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9.8.2 node_link_graph
node_link_graph(data, directed=False, multigraph=True, attrs={�source’: �source’, �target’: �target’,
�key’: �key’, �id’: �id’})
Return graph from node-link data format.
Parameters data : dict
node-link formatted graph data
directed : bool
If True, and direction not specified in data, return a directed graph.
multigraph : bool
If True, and multigraph not specified in data, return a multigraph.
attrs : dict
A dictionary that contains four keys �id’, �source’, �target’ and �key’. The corresponding values provide the attribute names for storing NetworkX-internal graph data.
Default value: dict(id=’id’, source=’source’, target=’target’,
key=’key’).
Returns G : NetworkX graph
A NetworkX graph object
See also:
node_link_data, adjacency_data, tree_data
Notes
The default value of attrs will be changed in a future release of NetworkX.
Examples
>>>
>>>
>>>
>>>
from networkx.readwrite import json_graph
G = nx.Graph([(1,2)])
data = json_graph.node_link_data(G)
H = json_graph.node_link_graph(data)
9.8.3 adjacency_data
adjacency_data(G, attrs={�id’: �id’, �key’: �key’})
Return data in adjacency format that is suitable for JSON serialization and use in Javascript documents.
Parameters G : NetworkX graph
attrs : dict
A dictionary that contains two keys �id’ and �key’. The corresponding values provide
the attribute names for storing NetworkX-internal graph data. The values should be
unique. Default value: dict(id=’id’, key=’key’).
If some user-defined graph data use these attribute names as data keys, they may be
silently dropped.
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Returns data : dict
A dictionary with adjacency formatted data.
Raises NetworkXError
If values in attrs are not unique.
See also:
adjacency_graph, node_link_data, tree_data
Notes
Graph, node, and link attributes will be written when using this format but attribute keys must be strings if you
want to serialize the resulting data with JSON.
The default value of attrs will be changed in a future release of NetworkX.
Examples
>>> from networkx.readwrite import json_graph
>>> G = nx.Graph([(1,2)])
>>> data = json_graph.adjacency_data(G)
To serialize with json
>>> import json
>>> s = json.dumps(data)
9.8.4 adjacency_graph
adjacency_graph(data, directed=False, multigraph=True, attrs={�id’: �id’, �key’: �key’})
Return graph from adjacency data format.
Parameters data : dict
Adjacency list formatted graph data
Returns G : NetworkX graph
A NetworkX graph object
directed : bool
If True, and direction not specified in data, return a directed graph.
multigraph : bool
If True, and multigraph not specified in data, return a multigraph.
attrs : dict
A dictionary that contains two keys �id’ and �key’. The corresponding values provide
the attribute names for storing NetworkX-internal graph data. The values should be
unique. Default value: dict(id=’id’, key=’key’).
See also:
adjacency_graph, node_link_data, tree_data
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Notes
The default value of attrs will be changed in a future release of NetworkX.
Examples
>>>
>>>
>>>
>>>
from networkx.readwrite import json_graph
G = nx.Graph([(1,2)])
data = json_graph.adjacency_data(G)
H = json_graph.adjacency_graph(data)
9.8.5 tree_data
tree_data(G, root, attrs={�children’: �children’, �id’: �id’})
Return data in tree format that is suitable for JSON serialization and use in Javascript documents.
Parameters G : NetworkX graph
G must be an oriented tree
root : node
The root of the tree
attrs : dict
A dictionary that contains two keys �id’ and �children’. The corresponding values provide the attribute names for storing NetworkX-internal graph data. The values should
be unique. Default value: dict(id=’id’, children=’children’).
If some user-defined graph data use these attribute names as data keys, they may be
silently dropped.
Returns data : dict
A dictionary with node-link formatted data.
Raises NetworkXError
If values in attrs are not unique.
See also:
tree_graph, node_link_data, node_link_data
Notes
Node attributes are stored in this format but keys for attributes must be strings if you want to serialize with
JSON.
Graph and edge attributes are not stored.
The default value of attrs will be changed in a future release of NetworkX.
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Examples
>>> from networkx.readwrite import json_graph
>>> G = nx.DiGraph([(1,2)])
>>> data = json_graph.tree_data(G,root=1)
To serialize with json
>>> import json
>>> s = json.dumps(data)
9.8.6 tree_graph
tree_graph(data, attrs={�children’: �children’, �id’: �id’})
Return graph from tree data format.
Parameters data : dict
Tree formatted graph data
Returns G : NetworkX DiGraph
attrs : dict
A dictionary that contains two keys �id’ and �children’. The corresponding values provide the attribute names for storing NetworkX-internal graph data. The values should
be unique. Default value: dict(id=’id’, children=’children’).
See also:
tree_graph, node_link_data, adjacency_data
Notes
The default value of attrs will be changed in a future release of NetworkX.
Examples
>>>
>>>
>>>
>>>
from networkx.readwrite import json_graph
G = nx.DiGraph([(1,2)])
data = json_graph.tree_data(G,root=1)
H = json_graph.tree_graph(data)
9.9 LEDA
Read graphs in LEDA format.
LEDA is a C++ class library for efficient data types and algorithms.
9.9.1 Format
See http://www.algorithmic-solutions.info/leda_guide/graphs/leda_native_graph_fileformat.html
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read_leda(path[, encoding])
parse_leda(lines)
Read graph in LEDA format from path.
Read graph in LEDA format from string or iterable.
9.9.2 read_leda
read_leda(path, encoding=’UTF-8’)
Read graph in LEDA format from path.
Parameters path : file or string
File or filename to read. Filenames ending in .gz or .bz2 will be uncompressed.
Returns G : NetworkX graph
References
[R366]
Examples
G=nx.read_leda(�file.leda’)
9.9.3 parse_leda
parse_leda(lines)
Read graph in LEDA format from string or iterable.
Parameters lines : string or iterable
Data in LEDA format.
Returns G : NetworkX graph
References
[R365]
Examples
G=nx.parse_leda(string)
9.10 YAML
Read and write NetworkX graphs in YAML format.
“YAML is a data serialization format designed for human readability and interaction with scripting languages.” See
http://www.yaml.org for documentation.
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9.10.1 Format
http://pyyaml.org/wiki/PyYAML
read_yaml(path)
write_yaml(G, path[, encoding])
Read graph in YAML format from path.
Write graph G in YAML format to path.
9.10.2 read_yaml
read_yaml(path)
Read graph in YAML format from path.
YAML is a data serialization format designed for human readability and interaction with scripting languages
[R369].
Parameters path : file or string
File or filename to read. Filenames ending in .gz or .bz2 will be uncompressed.
Returns G : NetworkX graph
References
[R369]
Examples
>>> G=nx.path_graph(4)
>>> nx.write_yaml(G,'test.yaml')
>>> G=nx.read_yaml('test.yaml')
9.10.3 write_yaml
write_yaml(G, path, encoding=’UTF-8’, **kwds)
Write graph G in YAML format to path.
YAML is a data serialization format designed for human readability and interaction with scripting languages
[R370].
Parameters G : graph
A NetworkX graph
path : file or string
File or filename to write. Filenames ending in .gz or .bz2 will be compressed.
encoding: string, optional
Specify which encoding to use when writing file.
References
[R370]
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Examples
>>> G=nx.path_graph(4)
>>> nx.write_yaml(G,'test.yaml')
9.11 SparseGraph6
9.11.1 Graph6
Graph6
Read and write graphs in graph6 format.
Format
“graph6 and sparse6 are formats for storing undirected graphs in a compact manner, using only printable ASCII
characters. Files in these formats have text type and contain one line per graph.”
See http://cs.anu.edu.au/~bdm/data/formats.txt for details.
parse_graph6(string)
read_graph6(path)
generate_graph6(G[, nodes, header])
write_graph6(G, path[, nodes, header])
Read a simple undirected graph in graph6 format from string.
Read simple undirected graphs in graph6 format from path.
Generate graph6 format string from a simple undirected graph.
Write a simple undirected graph to path in graph6 format.
parse_graph6
parse_graph6(string)
Read a simple undirected graph in graph6 format from string.
Parameters string : string
Data in graph6 format
Returns G : Graph
Raises NetworkXError
If the string is unable to be parsed in graph6 format
See also:
generate_graph6, read_graph6, write_graph6
References
Graph6 specification: http://cs.anu.edu.au/~bdm/data/formats.txt for details.
Examples
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>>> G = nx.parse_graph6('A_')
>>> sorted(G.edges())
[(0, 1)]
read_graph6
read_graph6(path)
Read simple undirected graphs in graph6 format from path.
Parameters path : file or string
File or filename to write.
Returns G : Graph or list of Graphs
If the file contains multiple lines then a list of graphs is returned
Raises NetworkXError
If the string is unable to be parsed in graph6 format
See also:
generate_graph6, parse_graph6, write_graph6
References
Graph6 specification: http://cs.anu.edu.au/~bdm/data/formats.txt for details.
Examples
>>> nx.write_graph6(nx.Graph([(0,1)]), 'test.g6')
>>> G = nx.read_graph6('test.g6')
>>> sorted(G.edges())
[(0, 1)]
generate_graph6
generate_graph6(G, nodes=None, header=True)
Generate graph6 format string from a simple undirected graph.
Parameters G : Graph (undirected)
nodes: list or iterable
Nodes are labeled 0...n-1 in the order provided. If None the ordering given by G.nodes()
is used.
header: bool
If True add �>>graph6<<’ string to head of data
Returns s : string
String in graph6 format
Raises NetworkXError
If the graph is directed or has parallel edges
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See also:
read_graph6, parse_graph6, write_graph6
Notes
The format does not support edge or node labels, parallel edges or self loops. If self loops are present they are
silently ignored.
References
Graph6 specification: http://cs.anu.edu.au/~bdm/data/formats.txt for details.
Examples
>>> G = nx.Graph([(0, 1)])
>>> nx.generate_graph6(G)
'>>graph6<<A_'
write_graph6
write_graph6(G, path, nodes=None, header=True)
Write a simple undirected graph to path in graph6 format.
Parameters G : Graph (undirected)
path : file or string
File or filename to write.
nodes: list or iterable
Nodes are labeled 0...n-1 in the order provided. If None the ordering given by G.nodes()
is used.
header: bool
If True add �>>graph6<<’ string to head of data
Raises NetworkXError
If the graph is directed or has parallel edges
See also:
generate_graph6, parse_graph6, read_graph6
Notes
The format does not support edge or node labels, parallel edges or self loops. If self loops are present they are
silently ignored.
References
Graph6 specification: http://cs.anu.edu.au/~bdm/data/formats.txt for details.
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Examples
>>> G = nx.Graph([(0, 1)])
>>> nx.write_graph6(G, 'test.g6')
9.11.2 Sparse6
Sparse6
Read and write graphs in sparse6 format.
Format
“graph6 and sparse6 are formats for storing undirected graphs in a compact manner, using only printable ASCII
characters. Files in these formats have text type and contain one line per graph.”
See http://cs.anu.edu.au/~bdm/data/formats.txt for details.
parse_sparse6(string)
read_sparse6(path)
generate_sparse6(G[, nodes, header])
write_sparse6(G, path[, nodes, header])
Read an undirected graph in sparse6 format from string.
Read an undirected graph in sparse6 format from path.
Generate sparse6 format string from an undirected graph.
Write graph G to given path in sparse6 format.
parse_sparse6
parse_sparse6(string)
Read an undirected graph in sparse6 format from string.
Parameters string : string
Data in sparse6 format
Returns G : Graph
Raises NetworkXError
If the string is unable to be parsed in sparse6 format
See also:
generate_sparse6, read_sparse6, write_sparse6
References
Sparse6 specification: http://cs.anu.edu.au/~bdm/data/formats.txt
Examples
>>> G = nx.parse_sparse6(':A_')
>>> sorted(G.edges())
[(0, 1), (0, 1), (0, 1)]
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read_sparse6
read_sparse6(path)
Read an undirected graph in sparse6 format from path.
Parameters path : file or string
File or filename to write.
Returns G : Graph/Multigraph or list of Graphs/MultiGraphs
If the file contains multple lines then a list of graphs is returned
Raises NetworkXError
If the string is unable to be parsed in sparse6 format
See also:
generate_sparse6, read_sparse6, parse_sparse6
References
Sparse6 specification: http://cs.anu.edu.au/~bdm/data/formats.txt
Examples
>>> nx.write_sparse6(nx.Graph([(0,1),(0,1),(0,1)]), 'test.s6')
>>> G = nx.read_sparse6('test.s6')
>>> sorted(G.edges())
[(0, 1)]
generate_sparse6
generate_sparse6(G, nodes=None, header=True)
Generate sparse6 format string from an undirected graph.
Parameters G : Graph (undirected)
nodes: list or iterable
Nodes are labeled 0...n-1 in the order provided. If None the ordering given by G.nodes()
is used.
header: bool
If True add �>>sparse6<<’ string to head of data
Returns s : string
String in sparse6 format
Raises NetworkXError
If the graph is directed
See also:
read_sparse6, parse_sparse6, write_sparse6
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Notes
The format does not support edge or node labels.
http://cs.anu.edu.au/~bdm/data/formats.txt for details.
References ———- Sparse6 specification:
Examples
>>> G = nx.MultiGraph([(0, 1), (0, 1), (0, 1)])
>>> nx.generate_sparse6(G)
'>>sparse6<<:A_'
write_sparse6
write_sparse6(G, path, nodes=None, header=True)
Write graph G to given path in sparse6 format. Parameters ———- G : Graph (undirected)
path [file or string] File or filename to write
nodes: list or iterable Nodes are labeled 0...n-1 in the order provided. If None the ordering given by G.nodes()
is used.
header: bool If True add �>>sparse6<<’ string to head of data
Raises NetworkXError
If the graph is directed
See also:
read_sparse6, parse_sparse6, generate_sparse6
Notes
The format does not support edge or node labels.
References
Sparse6 specification: http://cs.anu.edu.au/~bdm/data/formats.txt for details.
Examples
>>> G = nx.Graph([(0, 1), (0, 1), (0, 1)])
>>> nx.write_sparse6(G, 'test.s6')
9.12 Pajek
Read graphs in Pajek format.
This implementation handles directed and undirected graphs including those with self loops and parallel edges.
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9.12.1 Format
See http://vlado.fmf.uni-lj.si/pub/networks/pajek/doc/draweps.htm for format information.
read_pajek(path[, encoding])
write_pajek(G, path[, encoding])
parse_pajek(lines)
Read graph in Pajek format from path.
Write graph in Pajek format to path.
Parse Pajek format graph from string or iterable.
9.12.2 read_pajek
read_pajek(path, encoding=’UTF-8’)
Read graph in Pajek format from path.
Parameters path : file or string
File or filename to write. Filenames ending in .gz or .bz2 will be uncompressed.
Returns G : NetworkX MultiGraph or MultiDiGraph.
References
See http://vlado.fmf.uni-lj.si/pub/networks/pajek/doc/draweps.htm for format information.
Examples
>>> G=nx.path_graph(4)
>>> nx.write_pajek(G, "test.net")
>>> G=nx.read_pajek("test.net")
To create a Graph instead of a MultiGraph use
>>> G1=nx.Graph(G)
9.12.3 write_pajek
write_pajek(G, path, encoding=’UTF-8’)
Write graph in Pajek format to path.
Parameters G : graph
A Networkx graph
path : file or string
File or filename to write. Filenames ending in .gz or .bz2 will be compressed.
References
See http://vlado.fmf.uni-lj.si/pub/networks/pajek/doc/draweps.htm for format information.
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Examples
>>> G=nx.path_graph(4)
>>> nx.write_pajek(G, "test.net")
9.12.4 parse_pajek
parse_pajek(lines)
Parse Pajek format graph from string or iterable.
Parameters lines : string or iterable
Data in Pajek format.
Returns G : NetworkX graph
See also:
read_pajek
9.13 GIS Shapefile
Generates a networkx.DiGraph from point and line shapefiles.
“The Esri Shapefile or simply a shapefile is a popular geospatial vector data format for geographic information systems
software. It is developed and regulated by Esri as a (mostly) open specification for data interoperability among Esri
and other software products.” See http://en.wikipedia.org/wiki/Shapefile for additional information.
read_shp(path)
write_shp(G, outdir)
Generates a networkx.DiGraph from shapefiles.
Writes a networkx.DiGraph to two shapefiles, edges and nodes.
9.13.1 read_shp
read_shp(path)
Generates a networkx.DiGraph from shapefiles. Point geometries are translated into nodes, lines into edges.
Coordinate tuples are used as keys. Attributes are preserved, line geometries are simplified into start and end
coordinates. Accepts a single shapefile or directory of many shapefiles.
“The Esri Shapefile or simply a shapefile is a popular geospatial vector data format for geographic information
systems software [R367].”
Parameters path : file or string
File, directory, or filename to read.
Returns G : NetworkX graph
References
[R367]
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Examples
>>> G=nx.read_shp('test.shp')
9.13.2 write_shp
write_shp(G, outdir)
Writes a networkx.DiGraph to two shapefiles, edges and nodes. Nodes and edges are expected to have a Well
Known Binary (Wkb) or Well Known Text (Wkt) key in order to generate geometries. Also acceptable are nodes
with a numeric tuple key (x,y).
“The Esri Shapefile or simply a shapefile is a popular geospatial vector data format for geographic information
systems software [R368].”
Parameters outdir : directory path
Output directory for the two shapefiles.
Returns None
References
[R368]
Examples
nx.write_shp(digraph, �/shapefiles’) # doctest +SKIP
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CHAPTER
TEN
DRAWING
10.1 Matplotlib
Draw networks with matplotlib.
10.1.1 See Also
matplotlib: http://matplotlib.org/
pygraphviz: http://pygraphviz.github.io/
draw(G[, pos, ax, hold])
draw_networkx(G[, pos, with_labels])
draw_networkx_nodes(G, pos[, nodelist, ...])
draw_networkx_edges(G, pos[, edgelist, ...])
draw_networkx_labels(G, pos[, labels, ...])
draw_networkx_edge_labels(G, pos[, ...])
draw_circular(G, **kwargs)
draw_random(G, **kwargs)
draw_spectral(G, **kwargs)
draw_spring(G, **kwargs)
draw_shell(G, **kwargs)
draw_graphviz(G[, prog])
Draw the graph G with Matplotlib.
Draw the graph G using Matplotlib.
Draw the nodes of the graph G.
Draw the edges of the graph G.
Draw node labels on the graph G.
Draw edge labels.
Draw the graph G with a circular layout.
Draw the graph G with a random layout.
Draw the graph G with a spectral layout.
Draw the graph G with a spring layout.
Draw networkx graph with shell layout.
Draw networkx graph with graphviz layout.
10.1.2 draw
draw(G, pos=None, ax=None, hold=None, **kwds)
Draw the graph G with Matplotlib.
Draw the graph as a simple representation with no node labels or edge labels and using the full Matplotlib figure
area and no axis labels by default. See draw_networkx() for more full-featured drawing that allows title, axis
labels etc.
Parameters G : graph
A networkx graph
pos : dictionary, optional
A dictionary with nodes as keys and positions as values. If not specified a spring layout
positioning will be computed. See networkx.layout for functions that compute node
positions.
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ax : Matplotlib Axes object, optional
Draw the graph in specified Matplotlib axes.
hold : bool, optional
Set the Matplotlib hold state. If True subsequent draw commands will be added to the
current axes.
**kwds : optional keywords
See networkx.draw_networkx() for a description of optional keywords.
See also:
draw_networkx, draw_networkx_nodes, draw_networkx_edges, draw_networkx_labels,
draw_networkx_edge_labels
Notes
This function has the same name as pylab.draw and pyplot.draw so beware when using
>>> from networkx import *
since you might overwrite the pylab.draw function.
With pyplot use
>>>
>>>
>>>
>>>
>>>
import matplotlib.pyplot as plt
import networkx as nx
G=nx.dodecahedral_graph()
nx.draw(G) # networkx draw()
plt.draw() # pyplot draw()
Also see the NetworkX drawing examples at http://networkx.github.io/documentation/latest/gallery.html
Examples
>>> G=nx.dodecahedral_graph()
>>> nx.draw(G)
>>> nx.draw(G,pos=nx.spring_layout(G)) # use spring layout
10.1.3 draw_networkx
draw_networkx(G, pos=None, with_labels=True, **kwds)
Draw the graph G using Matplotlib.
Draw the graph with Matplotlib with options for node positions, labeling, titles, and many other drawing features. See draw() for simple drawing without labels or axes.
Parameters G : graph
A networkx graph
pos : dictionary, optional
A dictionary with nodes as keys and positions as values. If not specified a spring layout
positioning will be computed. See networkx.layout for functions that compute node
positions.
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with_labels : bool, optional (default=True)
Set to True to draw labels on the nodes.
ax : Matplotlib Axes object, optional
Draw the graph in the specified Matplotlib axes.
nodelist : list, optional (default G.nodes())
Draw only specified nodes
edgelist : list, optional (default=G.edges())
Draw only specified edges
node_size : scalar or array, optional (default=300)
Size of nodes. If an array is specified it must be the same length as nodelist.
node_color : color string, or array of floats, (default=’r’)
Node color. Can be a single color format string, or a sequence of colors with the same
length as nodelist. If numeric values are specified they will be mapped to colors using
the cmap and vmin,vmax parameters. See matplotlib.scatter for more details.
node_shape : string, optional (default=’o’)
The shape of the node.
�so^>v<dph8’.
Specification is as matplotlib.scatter marker, one of
alpha : float, optional (default=1.0)
The node transparency
cmap : Matplotlib colormap, optional (default=None)
Colormap for mapping intensities of nodes
vmin,vmax : float, optional (default=None)
Minimum and maximum for node colormap scaling
linewidths : [None | scalar | sequence]
Line width of symbol border (default =1.0)
width : float, optional (default=1.0)
Line width of edges
edge_color : color string, or array of floats (default=’r’)
Edge color. Can be a single color format string, or a sequence of colors with the same
length as edgelist. If numeric values are specified they will be mapped to colors using
the edge_cmap and edge_vmin,edge_vmax parameters.
edge_cmap : Matplotlib colormap, optional (default=None)
Colormap for mapping intensities of edges
edge_vmin,edge_vmax : floats, optional (default=None)
Minimum and maximum for edge colormap scaling
style : string, optional (default=’solid’)
Edge line style (solid|dashed|dotted,dashdot)
labels : dictionary, optional (default=None)
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Node labels in a dictionary keyed by node of text labels
font_size : int, optional (default=12)
Font size for text labels
font_color : string, optional (default=’k’ black)
Font color string
font_weight : string, optional (default=’normal’)
Font weight
font_family : string, optional (default=’sans-serif’)
Font family
label : string, optional
Label for graph legend
See also:
draw,
draw_networkx_nodes,
draw_networkx_edge_labels
draw_networkx_edges,
draw_networkx_labels,
Examples
>>> G=nx.dodecahedral_graph()
>>> nx.draw(G)
>>> nx.draw(G,pos=nx.spring_layout(G)) # use spring layout
>>> import matplotlib.pyplot as plt
>>> limits=plt.axis('off') # turn of axis
Also see the NetworkX drawing examples at http://networkx.github.io/documentation/latest/gallery.html
10.1.4 draw_networkx_nodes
draw_networkx_nodes(G, pos, nodelist=None, node_size=300, node_color=’r’, node_shape=’o’, alpha=1.0, cmap=None, vmin=None, vmax=None, ax=None, linewidths=None, label=None, **kwds)
Draw the nodes of the graph G.
This draws only the nodes of the graph G.
Parameters G : graph
A networkx graph
pos : dictionary
A dictionary with nodes as keys and positions as values. Positions should be sequences
of length 2.
ax : Matplotlib Axes object, optional
Draw the graph in the specified Matplotlib axes.
nodelist : list, optional
Draw only specified nodes (default G.nodes())
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node_size : scalar or array
Size of nodes (default=300). If an array is specified it must be the same length as
nodelist.
node_color : color string, or array of floats
Node color. Can be a single color format string (default=’r’), or a sequence of colors
with the same length as nodelist. If numeric values are specified they will be mapped
to colors using the cmap and vmin,vmax parameters. See matplotlib.scatter for more
details.
node_shape : string
The shape of the node. Specification is as matplotlib.scatter marker, one of
�so^>v<dph8’ (default=’o’).
alpha : float
The node transparency (default=1.0)
cmap : Matplotlib colormap
Colormap for mapping intensities of nodes (default=None)
vmin,vmax : floats
Minimum and maximum for node colormap scaling (default=None)
linewidths : [None | scalar | sequence]
Line width of symbol border (default =1.0)
label : [None| string]
Label for legend
Returns matplotlib.collections.PathCollection
рќ‘ѓ рќ‘Ћрќ‘Ўв„Ћрќђ¶рќ‘њрќ‘™рќ‘™рќ‘’рќ‘ђрќ‘Ўрќ‘–рќ‘њрќ‘› of the nodes.
See also:
draw,
draw_networkx,
draw_networkx_edge_labels
draw_networkx_edges,
draw_networkx_labels,
Examples
>>> G=nx.dodecahedral_graph()
>>> nodes=nx.draw_networkx_nodes(G,pos=nx.spring_layout(G))
Also see the NetworkX drawing examples at http://networkx.github.io/documentation/latest/gallery.html
10.1.5 draw_networkx_edges
draw_networkx_edges(G, pos, edgelist=None, width=1.0, edge_color=’k’, style=’solid’, alpha=None,
edge_cmap=None, edge_vmin=None, edge_vmax=None, ax=None, arrows=True, label=None, **kwds)
Draw the edges of the graph G.
This draws only the edges of the graph G.
Parameters G : graph
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A networkx graph
pos : dictionary
A dictionary with nodes as keys and positions as values. Positions should be sequences
of length 2.
edgelist : collection of edge tuples
Draw only specified edges(default=G.edges())
width : float
Line width of edges (default =1.0)
edge_color : color string, or array of floats
Edge color. Can be a single color format string (default=’r’), or a sequence of colors
with the same length as edgelist. If numeric values are specified they will be mapped to
colors using the edge_cmap and edge_vmin,edge_vmax parameters.
style : string
Edge line style (default=’solid’) (solid|dashed|dotted,dashdot)
alpha : float
The edge transparency (default=1.0)
edge_ cmap : Matplotlib colormap
Colormap for mapping intensities of edges (default=None)
edge_vmin,edge_vmax : floats
Minimum and maximum for edge colormap scaling (default=None)
ax : Matplotlib Axes object, optional
Draw the graph in the specified Matplotlib axes.
arrows : bool, optional (default=True)
For directed graphs, if True draw arrowheads.
label : [None| string]
Label for legend
Returns matplotlib.collection.LineCollection
рќђїрќ‘–рќ‘›рќ‘’рќђ¶рќ‘њрќ‘™рќ‘™рќ‘’рќ‘ђрќ‘Ўрќ‘–рќ‘њрќ‘› of the edges
See also:
draw,
draw_networkx,
draw_networkx_edge_labels
draw_networkx_nodes,
draw_networkx_labels,
Notes
For directed graphs, “arrows” (actually just thicker stubs) are drawn at the head end. Arrows can be turned off
with keyword arrows=False. Yes, it is ugly but drawing proper arrows with Matplotlib this way is tricky.
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Examples
>>> G=nx.dodecahedral_graph()
>>> edges=nx.draw_networkx_edges(G,pos=nx.spring_layout(G))
Also see the NetworkX drawing examples at http://networkx.github.io/documentation/latest/gallery.html
10.1.6 draw_networkx_labels
draw_networkx_labels(G, pos, labels=None, font_size=12, font_color=’k’, font_family=’sans-serif’,
font_weight=’normal’, alpha=1.0, ax=None, **kwds)
Draw node labels on the graph G.
Parameters G : graph
A networkx graph
pos : dictionary
A dictionary with nodes as keys and positions as values. Positions should be sequences
of length 2.
labels : dictionary, optional (default=None)
Node labels in a dictionary keyed by node of text labels
font_size : int
Font size for text labels (default=12)
font_color : string
Font color string (default=’k’ black)
font_family : string
Font family (default=’sans-serif’)
font_weight : string
Font weight (default=’normal’)
alpha : float
The text transparency (default=1.0)
ax : Matplotlib Axes object, optional
Draw the graph in the specified Matplotlib axes.
Returns dict
рќ‘‘рќ‘–рќ‘ђрќ‘Ў of labels keyed on the nodes
See also:
draw,
draw_networkx,
draw_networkx_edge_labels
draw_networkx_nodes,
draw_networkx_edges,
Examples
>>> G=nx.dodecahedral_graph()
>>> labels=nx.draw_networkx_labels(G,pos=nx.spring_layout(G))
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10.1.7 draw_networkx_edge_labels
draw_networkx_edge_labels(G, pos,
edge_labels=None,
label_pos=0.5,
font_size=10,
font_color=’k’,
font_family=’sans-serif’,
font_weight=’normal’,
alpha=1.0, bbox=None, ax=None, rotate=True, **kwds)
Draw edge labels.
Parameters G : graph
A networkx graph
pos : dictionary
A dictionary with nodes as keys and positions as values. Positions should be sequences
of length 2.
ax : Matplotlib Axes object, optional
Draw the graph in the specified Matplotlib axes.
alpha : float
The text transparency (default=1.0)
edge_labels : dictionary
Edge labels in a dictionary keyed by edge two-tuple of text labels (default=None). Only
labels for the keys in the dictionary are drawn.
label_pos : float
Position of edge label along edge (0=head, 0.5=center, 1=tail)
font_size : int
Font size for text labels (default=12)
font_color : string
Font color string (default=’k’ black)
font_weight : string
Font weight (default=’normal’)
font_family : string
Font family (default=’sans-serif’)
bbox : Matplotlib bbox
Specify text box shape and colors.
clip_on : bool
Turn on clipping at axis boundaries (default=True)
Returns dict
рќ‘‘рќ‘–рќ‘ђрќ‘Ў of labels keyed on the edges
See also:
draw,
draw_networkx,
draw_networkx_labels
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Examples
>>> G=nx.dodecahedral_graph()
>>> edge_labels=nx.draw_networkx_edge_labels(G,pos=nx.spring_layout(G))
Also see the NetworkX drawing examples at http://networkx.github.io/documentation/latest/gallery.html
10.1.8 draw_circular
draw_circular(G, **kwargs)
Draw the graph G with a circular layout.
Parameters G : graph
A networkx graph
**kwargs : optional keywords
See networkx.draw_networkx() for a description of optional keywords, with the exception of the pos parameter which is not used by this function.
10.1.9 draw_random
draw_random(G, **kwargs)
Draw the graph G with a random layout.
Parameters G : graph
A networkx graph
**kwargs : optional keywords
See networkx.draw_networkx() for a description of optional keywords, with the exception of the pos parameter which is not used by this function.
10.1.10 draw_spectral
draw_spectral(G, **kwargs)
Draw the graph G with a spectral layout.
Parameters G : graph
A networkx graph
**kwargs : optional keywords
See networkx.draw_networkx() for a description of optional keywords, with the exception of the pos parameter which is not used by this function.
10.1.11 draw_spring
draw_spring(G, **kwargs)
Draw the graph G with a spring layout.
Parameters G : graph
A networkx graph
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**kwargs : optional keywords
See networkx.draw_networkx() for a description of optional keywords, with the exception of the pos parameter which is not used by this function.
10.1.12 draw_shell
draw_shell(G, **kwargs)
Draw networkx graph with shell layout.
Parameters G : graph
A networkx graph
**kwargs : optional keywords
See networkx.draw_networkx() for a description of optional keywords, with the exception of the pos parameter which is not used by this function.
10.1.13 draw_graphviz
draw_graphviz(G, prog=’neato’, **kwargs)
Draw networkx graph with graphviz layout.
Parameters G : graph
A networkx graph
prog : string, optional
Name of Graphviz layout program
**kwargs : optional keywords
See networkx.draw_networkx() for a description of optional keywords.
10.2 Graphviz AGraph (dot)
Interface to pygraphviz AGraph class.
10.2.1 Examples
>>> G=nx.complete_graph(5)
>>> A=nx.to_agraph(G)
>>> H=nx.from_agraph(A)
10.2.2 See Also
Pygraphviz: http://pygraphviz.github.io/
from_agraph(A[, create_using])
to_agraph(N)
write_dot(G, path)
468
Return a NetworkX Graph or DiGraph from a PyGraphviz graph.
Return a pygraphviz graph from a NetworkX graph N.
Write NetworkX graph G to Graphviz dot format on path.
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Table 10.2 – continued from previous page
read_dot(path)
Return a NetworkX graph from a dot file on path.
graphviz_layout(G[, prog, root, args])
Create node positions for G using Graphviz.
pygraphviz_layout(G[, prog, root, args]) Create node positions for G using Graphviz.
10.2.3 from_agraph
from_agraph(A, create_using=None)
Return a NetworkX Graph or DiGraph from a PyGraphviz graph.
Parameters A : PyGraphviz AGraph
A graph created with PyGraphviz
create_using : NetworkX graph class instance
The output is created using the given graph class instance
Notes
The Graph G will have a dictionary G.graph_attr containing the default graphviz attributes for graphs, nodes
and edges.
Default node attributes will be in the dictionary G.node_attr which is keyed by node.
Edge attributes will be returned as edge data in G. With edge_attr=False the edge data will be the Graphviz edge
weight attribute or the value 1 if no edge weight attribute is found.
Examples
>>>
>>>
>>>
>>>
K5=nx.complete_graph(5)
A=nx.to_agraph(K5)
G=nx.from_agraph(A)
G=nx.from_agraph(A)
10.2.4 to_agraph
to_agraph(N)
Return a pygraphviz graph from a NetworkX graph N.
Parameters N : NetworkX graph
A graph created with NetworkX
Notes
If N has an dict N.graph_attr an attempt will be made first to copy properties attached to the graph (see
from_agraph) and then updated with the calling arguments if any.
10.2. Graphviz AGraph (dot)
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Examples
>>> K5=nx.complete_graph(5)
>>> A=nx.to_agraph(K5)
10.2.5 write_dot
write_dot(G, path)
Write NetworkX graph G to Graphviz dot format on path.
Parameters G : graph
A networkx graph
path : filename
Filename or file handle to write
10.2.6 read_dot
read_dot(path)
Return a NetworkX graph from a dot file on path.
Parameters path : file or string
File name or file handle to read.
10.2.7 graphviz_layout
graphviz_layout(G, prog=’neato’, root=None, args=’�)
Create node positions for G using Graphviz.
Parameters G : NetworkX graph
A graph created with NetworkX
prog : string
Name of Graphviz layout program
root : string, optional
Root node for twopi layout
args : string, optional
Extra arguments to Graphviz layout program
Returns : dictionary
Dictionary of x,y, positions keyed by node.
Notes
This is a wrapper for pygraphviz_layout.
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Examples
>>> G=nx.petersen_graph()
>>> pos=nx.graphviz_layout(G)
>>> pos=nx.graphviz_layout(G,prog='dot')
10.2.8 pygraphviz_layout
pygraphviz_layout(G, prog=’neato’, root=None, args=’�)
Create node positions for G using Graphviz.
Parameters G : NetworkX graph
A graph created with NetworkX
prog : string
Name of Graphviz layout program
root : string, optional
Root node for twopi layout
args : string, optional
Extra arguments to Graphviz layout program
Returns : dictionary
Dictionary of x,y, positions keyed by node.
Examples
>>> G=nx.petersen_graph()
>>> pos=nx.graphviz_layout(G)
>>> pos=nx.graphviz_layout(G,prog='dot')
10.3 Graphviz with pydot
Import and export NetworkX graphs in Graphviz dot format using pydot.
Either this module or nx_pygraphviz can be used to interface with graphviz.
10.3.1 See Also
Pydot: http://code.google.com/p/pydot/ Graphviz: http://www.research.att.com/sw/tools/graphviz/ DOT Language:
http://www.graphviz.org/doc/info/lang.html
from_pydot(P)
to_pydot(N[, strict])
write_dot(G, path)
read_dot(path)
graphviz_layout(G[, prog, root])
pydot_layout(G[, prog, root])
10.3. Graphviz with pydot
Return a NetworkX graph from a Pydot graph.
Return a pydot graph from a NetworkX graph N.
Write NetworkX graph G to Graphviz dot format on path.
Return a NetworkX MultiGraph or MultiDiGraph from a dot file on path.
Create node positions using Pydot and Graphviz.
Create node positions using Pydot and Graphviz.
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10.3.2 from_pydot
from_pydot(P)
Return a NetworkX graph from a Pydot graph.
Parameters P : Pydot graph
A graph created with Pydot
Returns G : NetworkX multigraph
A MultiGraph or MultiDiGraph.
Examples
>>>
>>>
>>>
>>>
K5=nx.complete_graph(5)
A=nx.to_pydot(K5)
G=nx.from_pydot(A) # return MultiGraph
G=nx.Graph(nx.from_pydot(A)) # make a Graph instead of MultiGraph
10.3.3 to_pydot
to_pydot(N, strict=True)
Return a pydot graph from a NetworkX graph N.
Parameters N : NetworkX graph
A graph created with NetworkX
Examples
>>> K5=nx.complete_graph(5)
>>> P=nx.to_pydot(K5)
10.3.4 write_dot
write_dot(G, path)
Write NetworkX graph G to Graphviz dot format on path.
Path can be a string or a file handle.
10.3.5 read_dot
read_dot(path)
Return a NetworkX MultiGraph or MultiDiGraph from a dot file on path.
Parameters path : filename or file handle
Returns G : NetworkX multigraph
A MultiGraph or MultiDiGraph.
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Notes
Use G=nx.Graph(nx.read_dot(path)) to return a Graph instead of a MultiGraph.
10.3.6 graphviz_layout
graphviz_layout(G, prog=’neato’, root=None, **kwds)
Create node positions using Pydot and Graphviz.
Returns a dictionary of positions keyed by node.
Notes
This is a wrapper for pydot_layout.
Examples
>>> G=nx.complete_graph(4)
>>> pos=nx.graphviz_layout(G)
>>> pos=nx.graphviz_layout(G,prog='dot')
10.3.7 pydot_layout
pydot_layout(G, prog=’neato’, root=None, **kwds)
Create node positions using Pydot and Graphviz.
Returns a dictionary of positions keyed by node.
Examples
>>> G=nx.complete_graph(4)
>>> pos=nx.pydot_layout(G)
>>> pos=nx.pydot_layout(G,prog='dot')
10.4 Graph Layout
Node positioning algorithms for graph drawing.
circular_layout(G[, dim, scale])
random_layout(G[, dim])
shell_layout(G[, nlist, dim, scale])
spring_layout(G[, dim, k, pos, fixed, ...])
spectral_layout(G[, dim, weight, scale])
10.4. Graph Layout
Position nodes on a circle.
Position nodes uniformly at random in the unit square.
Position nodes in concentric circles.
Position nodes using Fruchterman-Reingold force-directed algorithm.
Position nodes using the eigenvectors of the graph Laplacian.
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10.4.1 circular_layout
circular_layout(G, dim=2, scale=1)
Position nodes on a circle.
Parameters G : NetworkX graph
dim : int
Dimension of layout, currently only dim=2 is supported
scale : float
Scale factor for positions
Returns dict :
A dictionary of positions keyed by node
Notes
This algorithm currently only works in two dimensions and does not try to minimize edge crossings.
Examples
>>> G=nx.path_graph(4)
>>> pos=nx.circular_layout(G)
10.4.2 random_layout
random_layout(G, dim=2)
Position nodes uniformly at random in the unit square.
For every node, a position is generated by choosing each of dim coordinates uniformly at random on the interval
[0.0, 1.0).
NumPy (http://scipy.org) is required for this function.
Parameters G : NetworkX graph
A position will be assigned to every node in G.
dim : int
Dimension of layout.
Returns dict :
A dictionary of positions keyed by node
Examples
>>> G = nx.lollipop_graph(4, 3)
>>> pos = nx.random_layout(G)
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10.4.3 shell_layout
shell_layout(G, nlist=None, dim=2, scale=1)
Position nodes in concentric circles.
Parameters G : NetworkX graph
nlist : list of lists
List of node lists for each shell.
dim : int
Dimension of layout, currently only dim=2 is supported
scale : float
Scale factor for positions
Returns dict :
A dictionary of positions keyed by node
Notes
This algorithm currently only works in two dimensions and does not try to minimize edge crossings.
Examples
>>> G=nx.path_graph(4)
>>> shells=[[0],[1,2,3]]
>>> pos=nx.shell_layout(G,shells)
10.4.4 spring_layout
spring_layout(G, dim=2, k=None, pos=None, fixed=None, iterations=50, weight=’weight’, scale=1.0)
Position nodes using Fruchterman-Reingold force-directed algorithm.
Parameters G : NetworkX graph
dim : int
Dimension of layout
k : float (default=None)
Optimal distance between nodes. If None the distance is set to 1/sqrt(n) where n is the
number of nodes. Increase this value to move nodes farther apart.
pos : dict or None optional (default=None)
Initial positions for nodes as a dictionary with node as keys and values as a list or tuple.
If None, then nuse random initial positions.
fixed : list or None optional (default=None)
Nodes to keep fixed at initial position.
iterations : int optional (default=50)
Number of iterations of spring-force relaxation
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weight : string or None optional (default=’weight’)
The edge attribute that holds the numerical value used for the edge weight. If None,
then all edge weights are 1.
scale : float (default=1.0)
Scale factor for positions. The nodes are positioned in a box of size [0,scale] x [0,scale].
Returns dict :
A dictionary of positions keyed by node
Examples
>>> G=nx.path_graph(4)
>>> pos=nx.spring_layout(G)
# The same using longer function name >>> pos=nx.fruchterman_reingold_layout(G)
10.4.5 spectral_layout
spectral_layout(G, dim=2, weight=’weight’, scale=1)
Position nodes using the eigenvectors of the graph Laplacian.
Parameters G : NetworkX graph
dim : int
Dimension of layout
weight : string or None optional (default=’weight’)
The edge attribute that holds the numerical value used for the edge weight. If None,
then all edge weights are 1.
scale : float
Scale factor for positions
Returns dict :
A dictionary of positions keyed by node
Notes
Directed graphs will be considered as undirected graphs when positioning the nodes.
For larger graphs (>500 nodes) this will use the SciPy sparse eigenvalue solver (ARPACK).
Examples
>>> G=nx.path_graph(4)
>>> pos=nx.spectral_layout(G)
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EXCEPTIONS
Base exceptions and errors for NetworkX.
class NetworkXException
Base class for exceptions in NetworkX.
class NetworkXError
Exception for a serious error in NetworkX
class NetworkXPointlessConcept
Harary, F. and Read, R. “Is the Null Graph a Pointless Concept?” In Graphs and Combinatorics Conference,
George Washington University. New York: Springer-Verlag, 1973.
class NetworkXAlgorithmError
Exception for unexpected termination of algorithms.
class NetworkXUnfeasible
Exception raised by algorithms trying to solve a problem instance that has no feasible solution.
class NetworkXNoPath
Exception for algorithms that should return a path when running on graphs where such a path does not exist.
class NetworkXUnbounded
Exception raised by algorithms trying to solve a maximization or a minimization problem instance that is unbounded.
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UTILITIES
12.1 Helper Functions
Miscellaneous Helpers for NetworkX.
These are not imported into the base networkx namespace but can be accessed, for example, as
>>> import networkx
>>> networkx.utils.is_string_like('spam')
True
is_string_like(obj)
flatten(obj[, result])
iterable(obj)
is_list_of_ints(intlist)
make_str(x)
cumulative_sum(numbers)
generate_unique_node()
default_opener(filename)
Check if obj is string.
Return flattened version of (possibly nested) iterable object.
Return True if obj is iterable with a well-defined len().
Return True if list is a list of ints.
Return the string representation of t.
Yield cumulative sum of numbers.
Generate a unique node label.
Opens 𝑓 𝑖𝑙𝑒𝑛𝑎𝑚𝑒 using system’s default program.
12.1.1 is_string_like
is_string_like(obj)
Check if obj is string.
12.1.2 flatten
flatten(obj, result=None)
Return flattened version of (possibly nested) iterable object.
12.1.3 iterable
iterable(obj)
Return True if obj is iterable with a well-defined len().
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12.1.4 is_list_of_ints
is_list_of_ints(intlist)
Return True if list is a list of ints.
12.1.5 make_str
make_str(x)
Return the string representation of t.
12.1.6 cumulative_sum
cumulative_sum(numbers)
Yield cumulative sum of numbers.
>>> import networkx.utils as utils
>>> list(utils.cumulative_sum([1,2,3,4]))
[1, 3, 6, 10]
12.1.7 generate_unique_node
generate_unique_node()
Generate a unique node label.
12.1.8 default_opener
default_opener(filename)
Opens 𝑓 𝑖𝑙𝑒𝑛𝑎𝑚𝑒 using system’s default program.
Parameters filename : str
The path of the file to be opened.
12.2 Data Structures and Algorithms
Union-find data structure.
UnionFind.union(*objects)
Find the sets containing the objects and merge them all.
12.2.1 union
UnionFind.union(*objects)
Find the sets containing the objects and merge them all.
12.3 Random Sequence Generators
Utilities for generating random numbers, random sequences, and random selections.
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create_degree_sequence(n[, sfunction, max_tries])
pareto_sequence(n[, exponent])
powerlaw_sequence(n[, exponent])
uniform_sequence(n)
cumulative_distribution(distribution)
discrete_sequence(n[, distribution, ...])
zipf_sequence(n[, alpha, xmin])
zipf_rv(alpha[, xmin, seed])
random_weighted_sample(mapping, k)
weighted_choice(mapping)
Return sample sequence of length n from a Pareto distribution.
Return sample sequence of length n from a power law distribution.
Return sample sequence of length n from a uniform distribution.
Return normalized cumulative distribution from discrete distribution.
Return sample sequence of length n from a given discrete distribution or
Return a sample sequence of length n from a Zipf distribution with expo
Return a random value chosen from the Zipf distribution.
Return k items without replacement from a weighted sample.
Return a single element from a weighted sample.
12.3.1 create_degree_sequence
create_degree_sequence(n, sfunction=None, max_tries=50, **kwds)
12.3.2 pareto_sequence
pareto_sequence(n, exponent=1.0)
Return sample sequence of length n from a Pareto distribution.
12.3.3 powerlaw_sequence
powerlaw_sequence(n, exponent=2.0)
Return sample sequence of length n from a power law distribution.
12.3.4 uniform_sequence
uniform_sequence(n)
Return sample sequence of length n from a uniform distribution.
12.3.5 cumulative_distribution
cumulative_distribution(distribution)
Return normalized cumulative distribution from discrete distribution.
12.3.6 discrete_sequence
discrete_sequence(n, distribution=None, cdistribution=None)
Return sample sequence of length n from a given discrete distribution or discrete cumulative distribution.
One of the following must be specified.
distribution = histogram of values, will be normalized
cdistribution = normalized discrete cumulative distribution
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12.3.7 zipf_sequence
zipf_sequence(n, alpha=2.0, xmin=1)
Return a sample sequence of length n from a Zipf distribution with exponent parameter alpha and minimum
value xmin.
See also:
zipf_rv
12.3.8 zipf_rv
zipf_rv(alpha, xmin=1, seed=None)
Return a random value chosen from the Zipf distribution.
The return value is an integer drawn from the probability distribution ::math:
p(x)=\frac{x^{-\alpha}}{\zeta(\alpha,x_{min})},
where рќњЃ(рќ›ј, рќ‘Ґрќ‘љрќ‘–рќ‘› ) is the Hurwitz zeta function.
Parameters alpha : float
Exponent value of the distribution
xmin : int
Minimum value
seed : int
Seed value for random number generator
Returns x : int
Random value from Zipf distribution
Raises ValueError:
If xmin < 1 or If alpha <= 1
Notes
The rejection algorithm generates random values for a the power-law distribution in uniformly bounded expected
time dependent on parameters. See [1] for details on its operation.
References
..[1] Luc Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, New York, 1986.
Examples
>>> nx.zipf_rv(alpha=2, xmin=3, seed=42)
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12.3.9 random_weighted_sample
random_weighted_sample(mapping, k)
Return k items without replacement from a weighted sample.
The input is a dictionary of items with weights as values.
12.3.10 weighted_choice
weighted_choice(mapping)
Return a single element from a weighted sample.
The input is a dictionary of items with weights as values.
12.4 Decorators
open_file(path_arg[, mode])
Decorator to ensure clean opening and closing of files.
12.4.1 open_file
open_file(path_arg, mode=’r’)
Decorator to ensure clean opening and closing of files.
Parameters path_arg : int
Location of the path argument in args. Even if the argument is a named positional
argument (with a default value), you must specify its index as a positional argument.
mode : str
String for opening mode.
Returns _open_file : function
Function which cleanly executes the io.
Examples
Decorate functions like this:
@open_file(0,'r')
def read_function(pathname):
pass
@open_file(1,'w')
def write_function(G,pathname):
pass
@open_file(1,'w')
def write_function(G, pathname='graph.dot')
pass
@open_file('path', 'w+')
def another_function(arg, **kwargs):
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path = kwargs['path']
pass
12.5 Cuthill-Mckee Ordering
Cuthill-McKee ordering of graph nodes to produce sparse matrices
cuthill_mckee_ordering(G[, heuristic])
reverse_cuthill_mckee_ordering(G[, heuristic])
Generate an ordering (permutation) of the graph nodes to make a sparse
Generate an ordering (permutation) of the graph nodes to make a sparse
12.5.1 cuthill_mckee_ordering
cuthill_mckee_ordering(G, heuristic=None)
Generate an ordering (permutation) of the graph nodes to make a sparse matrix.
Uses the Cuthill-McKee heuristic (based on breadth-first search) [R371].
Parameters G : graph
A NetworkX graph
heuristic : function, optional
Function to choose starting node for RCM algorithm. If None a node from a psuedoperipheral pair is used. A user-defined function can be supplied that takes a graph object
and returns a single node.
Returns nodes : generator
Generator of nodes in Cuthill-McKee ordering.
See also:
reverse_cuthill_mckee_ordering
Notes
The optimal solution the the bandwidth reduction is NP-complete [R372].
References
[R371], [R372]
Examples
>>>
>>>
>>>
>>>
from networkx.utils import cuthill_mckee_ordering
G = nx.path_graph(4)
rcm = list(cuthill_mckee_ordering(G))
A = nx.adjacency_matrix(G, nodelist=rcm)
Smallest degree node as heuristic function:
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>>> def smallest_degree(G):
...
node, deg = min(G.degree().items(), key=lambda x: x[1])
...
return node
>>> rcm = list(cuthill_mckee_ordering(G, heuristic=smallest_degree))
12.5.2 reverse_cuthill_mckee_ordering
reverse_cuthill_mckee_ordering(G, heuristic=None)
Generate an ordering (permutation) of the graph nodes to make a sparse matrix.
Uses the reverse Cuthill-McKee heuristic (based on breadth-first search) [R373].
Parameters G : graph
A NetworkX graph
heuristic : function, optional
Function to choose starting node for RCM algorithm. If None a node from a psuedoperipheral pair is used. A user-defined function can be supplied that takes a graph object
and returns a single node.
Returns nodes : generator
Generator of nodes in reverse Cuthill-McKee ordering.
See also:
cuthill_mckee_ordering
Notes
The optimal solution the the bandwidth reduction is NP-complete [R374].
References
[R373], [R374]
Examples
>>>
>>>
>>>
>>>
from networkx.utils import reverse_cuthill_mckee_ordering
G = nx.path_graph(4)
rcm = list(reverse_cuthill_mckee_ordering(G))
A = nx.adjacency_matrix(G, nodelist=rcm)
Smallest degree node as heuristic function:
>>> def smallest_degree(G):
...
node, deg = min(G.degree().items(), key=lambda x: x[1])
...
return node
>>> rcm = list(reverse_cuthill_mckee_ordering(G, heuristic=smallest_degree))
12.5. Cuthill-Mckee Ordering
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reversed(*args, **kwds)
A context manager for temporarily reversing a directed graph in place.
12.6.1 reversed
reversed(*args, **kwds)
A context manager for temporarily reversing a directed graph in place.
This is a no-op for undirected graphs.
Parameters G : graph
A NetworkX graph.
12.6. Context Managers
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CHAPTER
THIRTEEN
LICENSE
NetworkX is distributed with the BSD license.
Copyright (C) 2004-2012, NetworkX Developers
Aric Hagberg <[email protected]>
Dan Schult <[email protected]>
Pieter Swart <[email protected]>
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above
copyright notice, this list of conditions and the following
disclaimer in the documentation and/or other materials provided
with the distribution.
* Neither the name of the NetworkX Developers nor the names of its
contributors may be used to endorse or promote products derived
from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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Chapter 13. License
CHAPTER
FOURTEEN
CITING
To cite NetworkX please use the following publication:
Aric A. Hagberg, Daniel A. Schult and Pieter J. Swart, “Exploring network structure, dynamics, and function using
NetworkX”, in Proceedings of the 7th Python in Science Conference (SciPy2008), Gäel Varoquaux, Travis Vaught,
and Jarrod Millman (Eds), (Pasadena, CA USA), pp. 11–15, Aug 2008
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Chapter 14. Citing
CHAPTER
FIFTEEN
CREDITS
NetworkX was originally written by Aric Hagberg, Dan Schult, and Pieter Swart, and has been developed with the
help of many others.
Thanks to Guido van Rossum for the idea of using Python for implementing a graph data structure
http://www.python.org/doc/essays/graphs.html
Thanks to David Eppstein for the idea of representing a graph G so that “for n in G” loops over the nodes in G and
G[n] are node n’s neighbors.
Thanks to everyone who has improved NetworkX by contributing code, bug reports (and fixes), documentation, and
input on design, featues, and the future of NetworkX.
Thanks especially to the following contributors:
• Katy Bold contributed the Karate Club graph.
• Hernan Rozenfeld added dorogovtsev_goltsev_mendes_graph and did stress testing.
• Brendt Wohlberg added examples from the Stanford GraphBase.
• Jim Bagrow reported bugs in the search methods.
• Holly Johnsen helped fix the path based centrality measures.
• Arnar Flatberg fixed the graph laplacian routines.
• Chris Myers suggested using None as a default datatype, suggested improvements for the IO routines, added
grid generator index tuple labeling and associated routines, and reported bugs.
• Joel Miller tested and improved the connected components methods fixed bugs and typos in the graph generators,
and contributed the random clustered graph generator.
• Keith Briggs sorted out naming issues for random graphs and wrote dense_gnm_random_graph.
• Ignacio Rozada provided the Krapivsky-Redner graph generator.
• Phillipp Pagel helped fix eccentricity etc. for disconnected graphs.
• Sverre Sundsdal contributed bidirectional shortest path and Dijkstra routines, s-metric computation and graph
generation
• Ross M. Richardson contributed the expected degree graph generator and helped test the pygraphviz interface.
• Christopher Ellison implemented the VF2 isomorphism algorithm and is a core developer.
• Eben Kenah contributed the strongly connected components and DFS functions.
• Sasha Gutfriend contributed edge betweenness algorithms.
• Udi Weinsberg helped develop intersection and difference operators.
• Matteo Dell’Amico wrote the random regular graph generator.
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• Andrew Conway contributed ego_graph, eigenvector centrality, line graph and much more.
• Raf Guns wrote the GraphML writer.
• Salim Fadhley and Matteo Dell’Amico contributed the A* algorithm.
• Fabrice Desclaux contributed the Matplotlib edge labeling code.
• Arpad Horvath fixed the barabasi_albert_graph() generator.
• Minh Van Nguyen contributed the connected_watts_strogatz_graph() and documentation for the Graph and
MultiGraph classes.
• Willem Ligtenberg contributed the directed scale free graph generator.
• Loïc Séguin-C. contributed the Ford-Fulkerson max flow and min cut algorithms, and ported all of NetworkX
to Python3. He is a NetworkX core developer.
• Paul McGuire improved the performance of the GML data parser.
• Jesus Cerquides contributed the chordal graph algorithms.
• Ben Edwards contributed tree generating functions, the rich club coefficient algorithm, the graph product functions, and a whole lot of other useful nuts and bolts.
• Jon Olav Vik contributed cycle finding algorithms.
• Hugh Brown improved the words.py example from the n^2 algorithm.
• Ben Reilly contributed the shapefile reader and writer.
• Leo Lopes contributed the maximal independent set algorithm.
• Jordi Torrents contributed the bipartite clustering, bipartite node redundancy, square clustering, bipartite projection articulation point, and flow-based connectivity algorithms.
• Dheeraj M R contributed the distance-regular testing algorithm
• Franck Kalala contributed the subgraph_centrality and communicability algorithms
• Simon Knight improved the GraphML functions to handle yEd/yfiles data, and to handle types correctly.
• Conrad Lee contributed the k-clique community finding algorithm.
• Sérgio Nery Simões wrote the function for finding all simple paths, and all shortest paths.
• Robert King contributed union, disjoint union, compose, and intersection operators that work on lists of graphs.
• Nick Mancuso wrote the approximation algorithms for dominating set, edge dominating set, independent set,
max clique, and min-weighted vertex cover.
• Brian Cloteaux contributed the linear-time graphicality tests and Havel-Hakimi graph generators
• Alejandro Weinstein contributed the directed Laplacian code
• Dustin Smith wrote the dictionary to numpy array function
• Mathieu Larose sped up the topological sort code
• Vincent Gauthier contributed the Katz centrality algorithm
• Sérgio Nery Simões developed the code for finding all simple paths
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SIXTEEN
GLOSSARY
dictionary A Python dictionary maps keys to values. Also known as “hashes”, or “associative arrays”. See
http://docs.python.org/tutorial/datastructures.html#dictionaries
ebunch An iteratable container of edge tuples like a list, iterator, or file.
edge Edges are either two-tuples of nodes (u,v) or three tuples of nodes with an edge attribute dictionary (u,v,dict).
edge attribute Edges can have arbitrary Python objects assigned as attributes by using keyword/value pairs when
adding an edge assigning to the G.edge[u][v] attribute dictionary for the specified edge u-v.
hashable An object is hashable if it has a hash value which never changes during its lifetime (it needs a __hash__()
method), and can be compared to other objects (it needs an __eq__() or __cmp__() method). Hashable objects
which compare equal must have the same hash value.
Hashability makes an object usable as a dictionary key and a set member, because these data structures use the
hash value internally.
All of Python’s immutable built-in objects are hashable, while no mutable containers (such as lists or dictionaries) are. Objects which are instances of user-defined classes are hashable by default; they all compare unequal,
and their hash value is their id().
Definition from http://docs.python.org/glossary.html
nbunch An nbunch is any iterable container of nodes that is not itself a node in the graph. It can be an iterable or an
iterator, e.g. a list, set, graph, file, etc..
node A node can be any hashable Python object except None.
node attribute Nodes can have arbitrary Python objects assigned as attributes by using keyword/value pairs when
adding a node or assigning to the G.node[n] attribute dictionary for the specified node n.
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BIBLIOGRAPHY
[R158] Boppana, R., & Halldórsson, M. M. (1992). Approximating maximum independent sets by excluding subgraphs. BIT Numerical Mathematics, 32(2), 180–196. Springer. doi:10.1007/BF01994876
[R157] Boppana, R., & Halldórsson, M. M. (1992). Approximating maximum independent sets by excluding subgraphs. BIT Numerical Mathematics, 32(2), 180–196. Springer.
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[R161] Boppana, R., & Halldórsson, M. M. (1992). Approximating maximum independent sets by excluding subgraphs. BIT Numerical Mathematics, 32(2), 180–196. Springer.
[R162] Vazirani, Vijay Approximation Algorithms (2001)
[R163] Bar-Yehuda, R., & Even, S. (1985). A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics, 25, 27–46 http://www.cs.technion.ac.il/~reuven/PDF/vc_lr.pdf
[R167] M. E. J. Newman, Mixing patterns in networks, Physical Review E, 67 026126, 2003
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[R164] M. E. J. Newman, Mixing patterns in networks, Physical Review E, 67 026126, 2003
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[R170] Foster, J.G., Foster, D.V., Grassberger, P. & Paczuski, M. Edge direction and the structure of networks, PNAS
107, 10815-20 (2010).
[R166] A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespignani, “The architecture of complex weighted
networks”. PNAS 101 (11): 3747–3752 (2004).
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Bibliography
PYTHON MODULE INDEX
a
203
networkx.algorithms.components.biconnected,
networkx.algorithms.approximation, 127
205
networkx.algorithms.approximation.clique,
networkx.algorithms.components.connected,
127
196
networkx.algorithms.approximation.clustering_coefficient,
networkx.algorithms.components.semiconnected,
128
210
networkx.algorithms.approximation.dominating_set,
networkx.algorithms.components.strongly_connected,
128
199
networkx.algorithms.approximation.independent_set,
networkx.algorithms.components.weakly_connected,
130
202
networkx.algorithms.approximation.matching,
networkx.algorithms.connectivity,
210
130
networkx.algorithms.connectivity.connectivity,
networkx.algorithms.approximation.ramsey,
211
131
networkx.algorithms.connectivity.cuts,
networkx.algorithms.approximation.vertex_cover,
219
131
networkx.algorithms.connectivity.stoerwagner,
networkx.algorithms.assortativity, 132
225
networkx.algorithms.bipartite, 141
networkx.algorithms.connectivity.utils,
networkx.algorithms.bipartite.basic, 143
227
networkx.algorithms.bipartite.centrality,
networkx.algorithms.core,
227
158
networkx.algorithms.cycles,
231
networkx.algorithms.bipartite.cluster,
networkx.algorithms.dag,
234
153
networkx.algorithms.bipartite.projection,networkx.algorithms.distance_measures,
237
147
networkx.algorithms.distance_regular,
networkx.algorithms.bipartite.redundancy,
239
157
networkx.algorithms.bipartite.spectral, networkx.algorithms.dominance, 240
networkx.algorithms.dominating, 242
152
networkx.algorithms.euler, 243
networkx.algorithms.block, 161
networkx.algorithms.flow, 244
networkx.algorithms.boundary, 162
networkx.algorithms.graphical, 267
networkx.algorithms.centrality, 163
networkx.algorithms.chordal.chordal_alg, networkx.algorithms.hierarchy, 270
networkx.algorithms.isolate, 271
185
networkx.algorithms.isomorphism, 272
networkx.algorithms.clique, 188
networkx.algorithms.isomorphism.isomorphvf2,
networkx.algorithms.cluster, 191
275
networkx.algorithms.coloring, 194
networkx.algorithms.link_analysis.hits_alg,
networkx.algorithms.community, 195
290
networkx.algorithms.community.kclique,
networkx.algorithms.link_analysis.pagerank_alg,
195
285
networkx.algorithms.components, 196
networkx.algorithms.link_prediction, 293
networkx.algorithms.components.attracting,
507
NetworkX Reference, Release 2.0.dev20141229000009
networkx.algorithms.matching, 299
networkx.generators.geometric, 381
networkx.algorithms.mis, 300
networkx.generators.hybrid, 385
networkx.algorithms.mst, 301
networkx.generators.intersection, 392
networkx.algorithms.operators.all, 306
networkx.generators.line, 389
networkx.algorithms.operators.binary,
networkx.generators.random_clustered,
303
377
networkx.algorithms.operators.product,
networkx.generators.random_graphs, 362
308
networkx.generators.small, 358
networkx.algorithms.operators.unary, 303 networkx.generators.social, 394
networkx.algorithms.richclub, 310
networkx.generators.stochastic, 392
networkx.algorithms.shortest_paths.astar,
l
328
networkx.algorithms.shortest_paths.dense,networkx.linalg.algebraicconnectivity,
326
400
networkx.algorithms.shortest_paths.generic,
networkx.linalg.attrmatrix, 403
311
networkx.linalg.graphmatrix, 395
networkx.algorithms.shortest_paths.unweighted,
networkx.linalg.laplacianmatrix, 397
315
networkx.linalg.spectrum, 399
networkx.algorithms.shortest_paths.weighted,
r
318
networkx.algorithms.simple_paths, 329
networkx.readwrite.adjlist, 419
networkx.algorithms.swap, 330
networkx.readwrite.edgelist, 426
networkx.algorithms.traversal.breadth_first_search,
networkx.readwrite.gexf, 433
336
networkx.readwrite.gml, 435
networkx.algorithms.traversal.depth_first_search,
networkx.readwrite.gpickle, 439
332
networkx.readwrite.graph6, 450
networkx.algorithms.traversal.edgedfs,
networkx.readwrite.graphml, 441
338
networkx.readwrite.json_graph, 442
networkx.algorithms.tree.recognition,
networkx.readwrite.leda, 447
339
networkx.readwrite.multiline_adjlist,
networkx.algorithms.vitality, 341
423
networkx.readwrite.nx_shp, 457
c
networkx.readwrite.nx_yaml, 448
networkx.classes.function, 343
networkx.readwrite.pajek, 455
networkx.convert, 409
networkx.readwrite.sparse6, 453
networkx.convert_matrix, 412
d
networkx.drawing.layout, 473
networkx.drawing.nx_agraph, 468
networkx.drawing.nx_pydot, 471
networkx.drawing.nx_pylab, 459
e
u
networkx.utils, 479
networkx.utils.contextmanagers, 486
networkx.utils.decorators, 483
networkx.utils.misc, 479
networkx.utils.random_sequence, 480
networkx.utils.rcm, 484
networkx.utils.union_find, 480
networkx.exception, 477
g
networkx.generators.atlas, 353
networkx.generators.bipartite, 385
networkx.generators.classic, 353
networkx.generators.degree_seq, 371
networkx.generators.directed, 378
networkx.generators.ego, 391
networkx.generators.expanders, 358
508
Python Module Index
INDEX
Symbols
__contains__() (DiGraph method), 56
__contains__() (Graph method), 27
__contains__() (MultiDiGraph method), 116
__contains__() (MultiGraph method), 86
__getitem__() (DiGraph method), 52
__getitem__() (Graph method), 25
__getitem__() (MultiDiGraph method), 113
__getitem__() (MultiGraph method), 84
__init__() (DiGraph method), 38
__init__() (DiGraphMatcher method), 279
__init__() (Graph method), 12
__init__() (GraphMatcher method), 277
__init__() (MultiDiGraph method), 97
__init__() (MultiGraph method), 69
__iter__() (DiGraph method), 48
__iter__() (Graph method), 21
__iter__() (MultiDiGraph method), 108
__iter__() (MultiGraph method), 80
__len__() (DiGraph method), 57
__len__() (Graph method), 29
__len__() (MultiDiGraph method), 118
__len__() (MultiGraph method), 88
A
adamic_adar_index()
(in
module
workx.algorithms.link_prediction), 295
add_cycle() (DiGraph method), 45
add_cycle() (Graph method), 19
add_cycle() (MultiDiGraph method), 105
add_cycle() (MultiGraph method), 77
add_edge() (DiGraph method), 41
add_edge() (Graph method), 15
add_edge() (MultiDiGraph method), 101
add_edge() (MultiGraph method), 72
add_edges_from() (DiGraph method), 42
add_edges_from() (Graph method), 16
add_edges_from() (MultiDiGraph method), 102
add_edges_from() (MultiGraph method), 73
add_node() (DiGraph method), 39
add_node() (Graph method), 12
add_node() (MultiDiGraph method), 98
add_node() (MultiGraph method), 69
add_nodes_from() (DiGraph method), 39
add_nodes_from() (Graph method), 13
add_nodes_from() (MultiDiGraph method), 99
add_nodes_from() (MultiGraph method), 70
add_path() (DiGraph method), 45
add_path() (Graph method), 19
add_path() (MultiDiGraph method), 105
add_path() (MultiGraph method), 76
add_star() (DiGraph method), 44
add_star() (Graph method), 18
add_star() (MultiDiGraph method), 105
add_star() (MultiGraph method), 76
add_weighted_edges_from() (DiGraph method), 43
add_weighted_edges_from() (Graph method), 17
add_weighted_edges_from() (MultiDiGraph method),
102
add_weighted_edges_from() (MultiGraph method), 74
adjacency_data()
(in
module
networkx.readwrite.json_graph), 444
adjacency_graph()
(in
module
networkx.readwrite.json_graph), 445
adjacency_iter() (DiGraph method), 54
adjacency_iter() (Graph method), 26
adjacency_iter() (MultiDiGraph method), 114
adjacency_iter() (MultiGraph method), 85
netadjacency_list() (DiGraph method), 53
adjacency_list() (Graph method), 25
adjacency_list() (MultiDiGraph method), 114
adjacency_list() (MultiGraph method), 84
adjacency_matrix()
(in
module
networkx.linalg.graphmatrix), 395
adjacency_spectrum()
(in
module
networkx.linalg.spectrum), 400
algebraic_connectivity()
(in
module
networkx.linalg.algebraicconnectivity), 400
all_neighbors() (in module networkx.classes.function),
346
all_pairs_dijkstra_path()
(in
module
networkx.algorithms.shortest_paths.weighted),
321
all_pairs_dijkstra_path_length() (in module net-
509
NetworkX Reference, Release 2.0.dev20141229000009
workx.algorithms.shortest_paths.weighted),
workx.algorithms.assortativity), 136
322
average_node_connectivity()
(in
module
netall_pairs_node_connectivity()
(in
module
networkx.algorithms.connectivity.connectivity),
workx.algorithms.connectivity.connectivity),
211
211
average_shortest_path_length()
(in
module
netall_pairs_shortest_path()
(in
module
networkx.algorithms.shortest_paths.generic),
workx.algorithms.shortest_paths.unweighted),
314
317
all_pairs_shortest_path_length() (in module net- B
workx.algorithms.shortest_paths.unweighted), balanced_tree() (in module networkx.generators.classic),
317
354
all_shortest_paths()
(in
module
net- barabasi_albert_graph()
(in
module
networkx.algorithms.shortest_paths.generic),
workx.generators.random_graphs), 368
313
barbell_graph() (in module networkx.generators.classic),
all_simple_paths()
(in
module
net354
workx.algorithms.simple_paths), 329
bellman_ford()
(in
module
netancestors() (in module networkx.algorithms.dag), 234
workx.algorithms.shortest_paths.weighted),
approximate_current_flow_betweenness_centrality() (in
325
module networkx.algorithms.centrality), 171
betweenness_centrality()
(in
module
netarticulation_points()
(in
module
networkx.algorithms.bipartite.centrality), 160
workx.algorithms.components.biconnected),
betweenness_centrality()
(in
module
net209
workx.algorithms.centrality), 166
astar_path()
(in
module
net- bfs_edges()
(in
module
networkx.algorithms.shortest_paths.astar), 328
workx.algorithms.traversal.breadth_first_search),
astar_path_length()
(in
module
net336
workx.algorithms.shortest_paths.astar), 329
bfs_predecessors()
(in
module
netattr_matrix() (in module networkx.linalg.attrmatrix), 403
workx.algorithms.traversal.breadth_first_search),
attr_sparse_matrix()
(in
module
net337
workx.linalg.attrmatrix), 405
bfs_successors()
(in
module
netattracting_component_subgraphs() (in module networkx.algorithms.traversal.breadth_first_search),
workx.algorithms.components.attracting),
338
204
bfs_tree()
(in
module
netattracting_components()
(in
module
networkx.algorithms.traversal.breadth_first_search),
workx.algorithms.components.attracting),
336
204
biadjacency_matrix()
(in
module
netattribute_assortativity_coefficient() (in module networkx.algorithms.bipartite.basic), 146
workx.algorithms.assortativity), 133
biconnected_component_edges() (in module netattribute_mixing_dict()
(in
module
networkx.algorithms.components.biconnected),
workx.algorithms.assortativity), 141
207
attribute_mixing_matrix()
(in
module
net- biconnected_component_subgraphs() (in module networkx.algorithms.assortativity), 139
workx.algorithms.components.biconnected),
authority_matrix()
(in
module
net208
workx.algorithms.link_analysis.hits_alg),
biconnected_components()
(in
module
net293
workx.algorithms.components.biconnected),
average_clustering()
(in
module
net206
workx.algorithms.approximation.clustering_coefficient),
bidirectional_dijkstra()
(in
module
net128
workx.algorithms.shortest_paths.weighted),
average_clustering()
(in
module
net323
workx.algorithms.bipartite.cluster), 154
binomial_graph()
(in
module
netaverage_clustering()
(in
module
networkx.generators.random_graphs), 365
workx.algorithms.cluster), 193
bipartite_alternating_havel_hakimi_graph() (in module
average_degree_connectivity()
(in
module
netnetworkx.generators.bipartite), 387
workx.algorithms.assortativity), 137
bipartite_configuration_model() (in module netaverage_neighbor_degree()
(in
module
networkx.generators.bipartite), 385
510
Index
NetworkX Reference, Release 2.0.dev20141229000009
bipartite_gnmk_random_graph() (in module networkx.generators.bipartite), 389
bipartite_havel_hakimi_graph()
(in
module
networkx.generators.bipartite), 386
bipartite_preferential_attachment_graph() (in module
networkx.generators.bipartite), 388
bipartite_random_graph()
(in
module
networkx.generators.bipartite), 388
bipartite_reverse_havel_hakimi_graph() (in module networkx.generators.bipartite), 387
blockmodel() (in module networkx.algorithms.block),
161
build_auxiliary_edge_connectivity() (in module networkx.algorithms.connectivity.utils), 227
build_auxiliary_node_connectivity() (in module networkx.algorithms.connectivity.utils), 227
build_residual_network()
(in
module
networkx.algorithms.flow), 258
bull_graph() (in module networkx.generators.small), 359
C
candidate_pairs_iter() (DiGraphMatcher method), 280
candidate_pairs_iter() (GraphMatcher method), 278
capacity_scaling() (in module networkx.algorithms.flow),
265
cartesian_product()
(in
module
networkx.algorithms.operators.product), 308
categorical_edge_match()
(in
module
networkx.algorithms.isomorphism), 281
categorical_multiedge_match()
(in
module
networkx.algorithms.isomorphism), 281
categorical_node_match()
(in
module
networkx.algorithms.isomorphism), 280
center()
(in
module
networkx.algorithms.distance_measures), 237
chordal_graph_cliques()
(in
module
networkx.algorithms.chordal.chordal_alg), 186
chordal_graph_treewidth()
(in
module
networkx.algorithms.chordal.chordal_alg), 186
chvatal_graph() (in module networkx.generators.small),
359
circular_ladder_graph()
(in
module
networkx.generators.classic), 355
circular_layout() (in module networkx.drawing.layout),
474
clear() (DiGraph method), 46
clear() (Graph method), 20
clear() (MultiDiGraph method), 106
clear() (MultiGraph method), 77
clique_removal()
(in
module
networkx.algorithms.approximation.clique),
128
cliques_containing_node()
(in
module
networkx.algorithms.clique), 190
Index
closeness_centrality()
(in
module
networkx.algorithms.bipartite.centrality), 158
closeness_centrality()
(in
module
networkx.algorithms.centrality), 165
closeness_vitality()
(in
module
networkx.algorithms.vitality), 341
clustering()
(in
module
networkx.algorithms.bipartite.cluster), 153
clustering() (in module networkx.algorithms.cluster), 192
cn_soundarajan_hopcroft()
(in
module
networkx.algorithms.link_prediction), 296
collaboration_weighted_projected_graph() (in module
networkx.algorithms.bipartite.projection), 149
color() (in module networkx.algorithms.bipartite.basic),
144
common_neighbors()
(in
module
networkx.classes.function), 347
communicability()
(in
module
networkx.algorithms.centrality), 178
communicability_betweenness_centrality() (in module
networkx.algorithms.centrality), 181
communicability_centrality()
(in
module
networkx.algorithms.centrality), 180
communicability_centrality_exp() (in module networkx.algorithms.centrality), 181
communicability_exp()
(in
module
networkx.algorithms.centrality), 179
complement()
(in
module
networkx.algorithms.operators.unary), 303
complete_bipartite_graph()
(in
module
networkx.generators.classic), 355
complete_graph()
(in
module
networkx.generators.classic), 355
compose()
(in
module
networkx.algorithms.operators.binary), 304
compose_all()
(in
module
networkx.algorithms.operators.all), 306
condensation()
(in
module
networkx.algorithms.components.strongly_connected),
202
configuration_model()
(in
module
networkx.generators.degree_seq), 371
connected_component_subgraphs() (in module networkx.algorithms.components.connected),
198
connected_components()
(in
module
networkx.algorithms.components.connected),
197
connected_double_edge_swap() (in module networkx.algorithms.swap), 331
connected_watts_strogatz_graph() (in module networkx.generators.random_graphs), 367
copy() (DiGraph method), 63
copy() (Graph method), 33
511
NetworkX Reference, Release 2.0.dev20141229000009
copy() (MultiDiGraph method), 124
copy() (MultiGraph method), 92
core_number() (in module networkx.algorithms.core),
228
cost_of_flow() (in module networkx.algorithms.flow),
263
could_be_isomorphic()
(in
module
networkx.algorithms.isomorphism), 274
create_degree_sequence()
(in
module
networkx.utils.random_sequence), 481
create_empty_copy()
(in
module
networkx.classes.function), 344
cubical_graph() (in module networkx.generators.small),
360
cumulative_distribution()
(in
module
networkx.utils.random_sequence), 481
cumulative_sum() (in module networkx.utils.misc), 480
current_flow_betweenness_centrality() (in module networkx.algorithms.centrality), 169
current_flow_closeness_centrality() (in module networkx.algorithms.centrality), 168
cuthill_mckee_ordering()
(in
module
networkx.utils.rcm), 484
cycle_basis() (in module networkx.algorithms.cycles),
231
cycle_graph() (in module networkx.generators.classic),
355
D
davis_southern_women_graph() (in module networkx.generators.social), 394
default_opener() (in module networkx.utils.misc), 480
degree() (DiGraph method), 57
degree() (Graph method), 29
degree() (in module networkx.classes.function), 343
degree() (MultiDiGraph method), 118
degree() (MultiGraph method), 88
degree_assortativity_coefficient() (in module networkx.algorithms.assortativity), 132
degree_centrality()
(in
module
networkx.algorithms.bipartite.centrality), 159
degree_centrality()
(in
module
networkx.algorithms.centrality), 163
degree_histogram()
(in
module
networkx.classes.function), 343
degree_iter() (DiGraph method), 58
degree_iter() (Graph method), 30
degree_iter() (MultiDiGraph method), 118
degree_iter() (MultiGraph method), 89
degree_mixing_dict()
(in
module
networkx.algorithms.assortativity), 140
degree_mixing_matrix()
(in
module
networkx.algorithms.assortativity), 140
512
degree_pearson_correlation_coefficient() (in module networkx.algorithms.assortativity), 135
degree_sequence_tree()
(in
module
networkx.generators.degree_seq), 376
degrees()
(in
module
networkx.algorithms.bipartite.basic), 145
dense_gnm_random_graph()
(in
module
networkx.generators.random_graphs), 364
density()
(in
module
networkx.algorithms.bipartite.basic), 145
density() (in module networkx.classes.function), 343
desargues_graph()
(in
module
networkx.generators.small), 360
descendants() (in module networkx.algorithms.dag), 234
dfs_edges()
(in
module
networkx.algorithms.traversal.depth_first_search),
332
dfs_labeled_edges()
(in
module
networkx.algorithms.traversal.depth_first_search),
335
dfs_postorder_nodes()
(in
module
networkx.algorithms.traversal.depth_first_search),
334
dfs_predecessors()
(in
module
networkx.algorithms.traversal.depth_first_search),
333
dfs_preorder_nodes()
(in
module
networkx.algorithms.traversal.depth_first_search),
334
dfs_successors()
(in
module
networkx.algorithms.traversal.depth_first_search),
333
dfs_tree()
(in
module
networkx.algorithms.traversal.depth_first_search),
332
diameter()
(in
module
networkx.algorithms.distance_measures), 237
diamond_graph() (in module networkx.generators.small),
360
dictionary, 495
difference()
(in
module
networkx.algorithms.operators.binary), 306
DiGraph() (in module networkx), 35
dijkstra_path()
(in
module
networkx.algorithms.shortest_paths.weighted),
319
dijkstra_path_length()
(in
module
networkx.algorithms.shortest_paths.weighted),
319
dijkstra_predecessor_and_distance() (in module networkx.algorithms.shortest_paths.weighted),
324
directed_configuration_model()
(in
module
networkx.generators.degree_seq), 373
Index
NetworkX Reference, Release 2.0.dev20141229000009
directed_havel_hakimi_graph()
(in
module
networkx.generators.degree_seq), 375
directed_laplacian_matrix()
(in
module
networkx.linalg.laplacianmatrix), 398
discrete_sequence()
(in
module
networkx.utils.random_sequence), 481
disjoint_union()
(in
module
networkx.algorithms.operators.binary), 305
disjoint_union_all()
(in
module
networkx.algorithms.operators.all), 307
dispersion() (in module networkx.algorithms.centrality),
184
dodecahedral_graph()
(in
module
networkx.generators.small), 360
dominance_frontiers()
(in
module
networkx.algorithms.dominance), 241
dominating_set()
(in
module
networkx.algorithms.dominating), 242
dorogovtsev_goltsev_mendes_graph() (in module networkx.generators.classic), 355
double_edge_swap()
(in
module
networkx.algorithms.swap), 330
draw() (in module networkx.drawing.nx_pylab), 459
draw_circular() (in module networkx.drawing.nx_pylab),
467
draw_graphviz()
(in
module
networkx.drawing.nx_pylab), 468
draw_networkx()
(in
module
networkx.drawing.nx_pylab), 460
draw_networkx_edge_labels()
(in
module
networkx.drawing.nx_pylab), 466
draw_networkx_edges()
(in
module
networkx.drawing.nx_pylab), 463
draw_networkx_labels()
(in
module
networkx.drawing.nx_pylab), 465
draw_networkx_nodes()
(in
module
networkx.drawing.nx_pylab), 462
draw_random() (in module networkx.drawing.nx_pylab),
467
draw_shell() (in module networkx.drawing.nx_pylab),
468
draw_spectral() (in module networkx.drawing.nx_pylab),
467
draw_spring() (in module networkx.drawing.nx_pylab),
467
edge_boundary()
(in
module
networkx.algorithms.boundary), 162
edge_connectivity()
(in
module
networkx.algorithms.connectivity.connectivity),
212
edge_current_flow_betweenness_centrality() (in module
networkx.algorithms.centrality), 170
edge_dfs()
(in
module
networkx.algorithms.traversal.edgedfs), 338
edge_load() (in module networkx.algorithms.centrality),
184
edges() (DiGraph method), 48
edges() (Graph method), 22
edges() (in module networkx.classes.function), 347
edges() (MultiDiGraph method), 108
edges() (MultiGraph method), 80
edges_iter() (DiGraph method), 49
edges_iter() (Graph method), 22
edges_iter() (in module networkx.classes.function), 348
edges_iter() (MultiDiGraph method), 109
edges_iter() (MultiGraph method), 81
edmonds_karp() (in module networkx.algorithms.flow),
252
ego_graph() (in module networkx.generators.ego), 391
eigenvector_centrality()
(in
module
networkx.algorithms.centrality), 173
eigenvector_centrality_numpy() (in
module
networkx.algorithms.centrality), 174
empty_graph() (in module networkx.generators.classic),
356
enumerate_all_cliques()
(in
module
networkx.algorithms.clique), 188
erdos_renyi_graph()
(in
module
networkx.generators.random_graphs), 365
estrada_index()
(in
module
networkx.algorithms.centrality), 182
eulerian_circuit() (in module networkx.algorithms.euler),
244
expected_degree_graph()
(in
module
networkx.generators.degree_seq), 374
F
fast_could_be_isomorphic()
(in
module
networkx.algorithms.isomorphism), 274
fast_gnp_random_graph()
(in
module
networkx.generators.random_graphs), 362
E
faster_could_be_isomorphic()
(in
module
netebunch, 495
workx.algorithms.isomorphism), 275
eccentricity()
(in
module
net- fiedler_vector()
(in
module
networkx.algorithms.distance_measures), 238
workx.linalg.algebraicconnectivity), 401
edge, 495
find_cliques() (in module networkx.algorithms.clique),
edge attribute, 495
189
edge_betweenness_centrality()
(in
module
net- find_cycle() (in module networkx.algorithms.cycles), 233
workx.algorithms.centrality), 167
Index
513
NetworkX Reference, Release 2.0.dev20141229000009
find_induced_nodes()
(in
module
networkx.algorithms.chordal.chordal_alg), 187
flatten() (in module networkx.utils.misc), 479
florentine_families_graph()
(in
module
networkx.generators.social), 394
flow_hierarchy()
(in
module
networkx.algorithms.hierarchy), 271
floyd_warshall()
(in
module
networkx.algorithms.shortest_paths.dense),
326
floyd_warshall_numpy()
(in
module
networkx.algorithms.shortest_paths.dense),
327
floyd_warshall_predecessor_and_distance() (in module
networkx.algorithms.shortest_paths.dense),
327
freeze() (in module networkx.classes.function), 350
from_agraph() (in module networkx.drawing.nx_agraph),
469
from_dict_of_dicts() (in module networkx.convert), 410
from_dict_of_lists() (in module networkx.convert), 411
from_edgelist() (in module networkx.convert), 412
from_numpy_matrix()
(in
module
networkx.convert_matrix), 415
from_pydot() (in module networkx.drawing.nx_pydot),
472
from_scipy_sparse_matrix()
(in
module
networkx.convert_matrix), 417
frucht_graph() (in module networkx.generators.small),
360
G
general_random_intersection_graph() (in module networkx.generators.intersection), 393
generate_adjlist() (in module networkx.readwrite.adjlist),
422
generate_edgelist()
(in
module
networkx.readwrite.edgelist), 431
generate_gml() (in module networkx.readwrite.gml), 438
generate_graph6()
(in
module
networkx.readwrite.graph6), 451
generate_multiline_adjlist()
(in
module
networkx.readwrite.multiline_adjlist), 426
generate_sparse6()
(in
module
networkx.readwrite.sparse6), 454
generate_unique_node() (in module networkx.utils.misc),
480
generic_edge_match()
(in
module
networkx.algorithms.isomorphism), 284
generic_multiedge_match()
(in
module
networkx.algorithms.isomorphism), 285
generic_node_match()
(in
module
networkx.algorithms.isomorphism), 284
514
generic_weighted_projected_graph() (in module networkx.algorithms.bipartite.projection), 151
geographical_threshold_graph() (in module networkx.generators.geometric), 382
get_edge_attributes()
(in
module
networkx.classes.function), 349
get_edge_data() (DiGraph method), 51
get_edge_data() (Graph method), 23
get_edge_data() (MultiDiGraph method), 112
get_edge_data() (MultiGraph method), 82
get_node_attributes()
(in
module
networkx.classes.function), 349
global_parameters()
(in
module
networkx.algorithms.distance_regular), 240
gn_graph() (in module networkx.generators.directed),
379
gnc_graph() (in module networkx.generators.directed),
380
gnm_random_graph()
(in
module
networkx.generators.random_graphs), 364
gnp_random_graph()
(in
module
networkx.generators.random_graphs), 363
gnr_graph() (in module networkx.generators.directed),
379
google_matrix()
(in
module
networkx.algorithms.link_analysis.pagerank_alg),
289
Graph() (in module networkx), 9
graph_atlas_g() (in module networkx.generators.atlas),
353
graph_clique_number()
(in
module
networkx.algorithms.clique), 190
graph_number_of_cliques()
(in
module
networkx.algorithms.clique), 190
graphviz_layout()
(in
module
networkx.drawing.nx_agraph), 470
graphviz_layout()
(in
module
networkx.drawing.nx_pydot), 473
greedy_color()
(in
module
networkx.algorithms.coloring), 194
grid_2d_graph() (in module networkx.generators.classic),
356
grid_graph() (in module networkx.generators.classic),
356
H
has_edge() (DiGraph method), 56
has_edge() (Graph method), 28
has_edge() (MultiDiGraph method), 116
has_edge() (MultiGraph method), 87
has_node() (DiGraph method), 55
has_node() (Graph method), 27
has_node() (MultiDiGraph method), 116
has_node() (MultiGraph method), 86
Index
NetworkX Reference, Release 2.0.dev20141229000009
has_path()
(in
module
networkx.algorithms.shortest_paths.generic),
315
hashable, 495
havel_hakimi_graph()
(in
module
networkx.generators.degree_seq), 375
heawood_graph() (in module networkx.generators.small),
360
hits()
(in
module
networkx.algorithms.link_analysis.hits_alg),
290
hits_numpy()
(in
module
networkx.algorithms.link_analysis.hits_alg),
291
hits_scipy()
(in
module
networkx.algorithms.link_analysis.hits_alg),
292
house_graph() (in module networkx.generators.small),
360
house_x_graph() (in module networkx.generators.small),
360
hub_matrix()
(in
module
networkx.algorithms.link_analysis.hits_alg),
293
hypercube_graph()
(in
module
networkx.generators.classic), 356
I
icosahedral_graph()
(in
module
networkx.generators.small), 360
immediate_dominators()
(in
module
networkx.algorithms.dominance), 241
in_degree() (DiGraph method), 58
in_degree() (MultiDiGraph method), 119
in_degree_centrality()
(in
module
networkx.algorithms.centrality), 164
in_degree_iter() (DiGraph method), 59
in_degree_iter() (MultiDiGraph method), 120
in_edges() (DiGraph method), 51
in_edges() (MultiDiGraph method), 111
in_edges_iter() (DiGraph method), 51
in_edges_iter() (MultiDiGraph method), 111
incidence_matrix()
(in
module
networkx.linalg.graphmatrix), 396
info() (in module networkx.classes.function), 344
initialize() (DiGraphMatcher method), 279
initialize() (GraphMatcher method), 277
intersection()
(in
module
networkx.algorithms.operators.binary), 305
intersection_all()
(in
module
networkx.algorithms.operators.all), 308
intersection_array()
(in
module
networkx.algorithms.distance_regular), 239
is_aperiodic() (in module networkx.algorithms.dag), 236
Index
is_attracting_component()
(in
module
networkx.algorithms.components.attracting),
203
is_biconnected()
(in
module
networkx.algorithms.components.biconnected),
205
is_bipartite()
(in
module
networkx.algorithms.bipartite.basic), 143
is_bipartite_node_set()
(in
module
networkx.algorithms.bipartite.basic), 143
is_chordal()
(in
module
networkx.algorithms.chordal.chordal_alg), 185
is_connected()
(in
module
networkx.algorithms.components.connected),
196
is_digraphical()
(in
module
networkx.algorithms.graphical), 268
is_directed() (in module networkx.classes.function), 344
is_directed_acyclic_graph()
(in
module
networkx.algorithms.dag), 236
is_distance_regular()
(in
module
networkx.algorithms.distance_regular), 239
is_dominating_set()
(in
module
networkx.algorithms.dominating), 243
is_eulerian() (in module networkx.algorithms.euler), 243
is_forest()
(in
module
networkx.algorithms.tree.recognition), 341
is_frozen() (in module networkx.classes.function), 351
is_graphical()
(in
module
networkx.algorithms.graphical), 267
is_isolate() (in module networkx.algorithms.isolate), 271
is_isomorphic() (DiGraphMatcher method), 279
is_isomorphic() (GraphMatcher method), 278
is_isomorphic()
(in
module
networkx.algorithms.isomorphism), 272
is_kl_connected()
(in
module
networkx.generators.hybrid), 385
is_list_of_ints() (in module networkx.utils.misc), 480
is_multigraphical()
(in
module
networkx.algorithms.graphical), 268
is_pseudographical()
(in
module
networkx.algorithms.graphical), 269
is_semiconnected()
(in
module
networkx.algorithms.components.semiconnected),
210
is_string_like() (in module networkx.utils.misc), 479
is_strongly_connected()
(in
module
networkx.algorithms.components.strongly_connected),
199
is_tree()
(in
module
networkx.algorithms.tree.recognition), 340
is_valid_degree_sequence_erdos_gallai() (in module networkx.algorithms.graphical), 270
is_valid_degree_sequence_havel_hakimi() (in module
515
NetworkX Reference, Release 2.0.dev20141229000009
networkx.algorithms.graphical), 269
literal_stringizer() (in module networkx.readwrite.gml),
is_weakly_connected()
(in
module
net439
workx.algorithms.components.weakly_connected),load_centrality()
(in
module
net202
workx.algorithms.centrality), 183
isolates() (in module networkx.algorithms.isolate), 272
local_edge_connectivity()
(in
module
netisomorphisms_iter() (DiGraphMatcher method), 279
workx.algorithms.connectivity.connectivity),
isomorphisms_iter() (GraphMatcher method), 278
213
iterable() (in module networkx.utils.misc), 479
local_node_connectivity()
(in
module
networkx.algorithms.connectivity.connectivity),
J
215
jaccard_coefficient()
(in
module
net- lollipop_graph() (in module networkx.generators.classic),
357
workx.algorithms.link_prediction), 294
K
M
(in
module
netk_clique_communities()
(in
module
net- make_clique_bipartite()
workx.algorithms.clique),
190
workx.algorithms.community.kclique), 196
make_max_clique_graph()
(in
module
netk_core() (in module networkx.algorithms.core), 228
workx.algorithms.clique),
189
k_corona() (in module networkx.algorithms.core), 231
make_small_graph()
(in
module
netk_crust() (in module networkx.algorithms.core), 230
workx.generators.small), 358
k_nearest_neighbors()
(in
module
netmake_str() (in module networkx.utils.misc), 480
workx.algorithms.assortativity), 138
k_random_intersection_graph()
(in
module
net- match() (DiGraphMatcher method), 280
match() (GraphMatcher method), 278
workx.generators.intersection), 393
max_clique()
(in
module
netk_shell() (in module networkx.algorithms.core), 229
workx.algorithms.approximation.clique),
karate_club_graph()
(in
module
net127
workx.generators.social), 394
(in
module
netkatz_centrality()
(in
module
net- max_flow_min_cost()
workx.algorithms.flow), 264
workx.algorithms.centrality), 175
(in
module
netkatz_centrality_numpy()
(in
module
net- max_weight_matching()
workx.algorithms.matching), 300
workx.algorithms.centrality), 176
(in
module
netkl_connected_subgraph()
(in
module
net- maximal_independent_set()
workx.algorithms.mis), 301
workx.generators.hybrid), 385
maximal_matching()
(in
module
netkosaraju_strongly_connected_components()
workx.algorithms.matching), 299
(in
module
netmaximum_flow() (in module networkx.algorithms.flow),
workx.algorithms.components.strongly_connected),
245
201
(in
module
netkrackhardt_kite_graph()
(in
module
net- maximum_flow_value()
workx.algorithms.flow), 247
workx.generators.small), 361
maximum_independent_set()
(in
module
networkx.algorithms.approximation.independent_set),
L
130
ladder_graph() (in module networkx.generators.classic),
min_cost_flow()
(in module networkx.algorithms.flow),
357
262
laplacian_matrix()
(in
module
netmin_cost_flow_cost()
(in
module
networkx.linalg.laplacianmatrix), 397
workx.algorithms.flow),
261
laplacian_spectrum()
(in
module
netmin_edge_dominating_set()
(in
module
networkx.linalg.spectrum), 399
workx.algorithms.approximation.dominating_set),
latapy_clustering()
(in
module
net129
workx.algorithms.bipartite.cluster), 155
min_maximal_matching()
(in
module
netLCF_graph() (in module networkx.generators.small), 359
workx.algorithms.approximation.matching),
lexicographic_product()
(in
module
net131
workx.algorithms.operators.product), 309
min_weighted_dominating_set()
(in module netline_graph() (in module networkx.generators.line), 390
workx.algorithms.approximation.dominating_set),
literal_destringizer()
(in
module
net129
workx.readwrite.gml), 438
516
Index
NetworkX Reference, Release 2.0.dev20141229000009
min_weighted_vertex_cover()
(in
module
networkx.algorithms.approximation.vertex_cover),
132
minimum_cut() (in module networkx.algorithms.flow),
248
minimum_cut_value()
(in
module
networkx.algorithms.flow), 250
minimum_edge_cut()
(in
module
networkx.algorithms.connectivity.cuts), 219
minimum_node_cut()
(in
module
networkx.algorithms.connectivity.cuts), 220
minimum_spanning_edges()
(in
module
networkx.algorithms.mst), 302
minimum_spanning_tree()
(in
module
networkx.algorithms.mst), 301
minimum_st_edge_cut()
(in
module
networkx.algorithms.connectivity.cuts), 222
minimum_st_node_cut()
(in
module
networkx.algorithms.connectivity.cuts), 224
moebius_kantor_graph()
(in
module
networkx.generators.small), 361
MultiDiGraph() (in module networkx), 95
MultiGraph() (in module networkx), 66
N
navigable_small_world_graph()
(in
module
networkx.generators.geometric), 384
nbunch, 495
nbunch_iter() (DiGraph method), 54
nbunch_iter() (Graph method), 26
nbunch_iter() (MultiDiGraph method), 115
nbunch_iter() (MultiGraph method), 85
negative_edge_cycle()
(in
module
networkx.algorithms.shortest_paths.weighted),
326
neighbors() (DiGraph method), 52
neighbors() (Graph method), 24
neighbors() (MultiDiGraph method), 113
neighbors() (MultiGraph method), 83
neighbors_iter() (DiGraph method), 52
neighbors_iter() (Graph method), 24
neighbors_iter() (MultiDiGraph method), 113
neighbors_iter() (MultiGraph method), 83
network_simplex()
(in
module
networkx.algorithms.flow), 258
networkx.algorithms.approximation (module), 127
networkx.algorithms.approximation.clique (module), 127
networkx.algorithms.approximation.clustering_coefficient
(module), 128
networkx.algorithms.approximation.dominating_set
(module), 128
networkx.algorithms.approximation.independent_set
(module), 130
Index
networkx.algorithms.approximation.matching (module),
130
networkx.algorithms.approximation.ramsey
(module),
131
networkx.algorithms.approximation.vertex_cover (module), 131
networkx.algorithms.assortativity (module), 132
networkx.algorithms.bipartite (module), 141
networkx.algorithms.bipartite.basic (module), 143
networkx.algorithms.bipartite.centrality (module), 158
networkx.algorithms.bipartite.cluster (module), 153
networkx.algorithms.bipartite.projection (module), 147
networkx.algorithms.bipartite.redundancy (module), 157
networkx.algorithms.bipartite.spectral (module), 152
networkx.algorithms.block (module), 161
networkx.algorithms.boundary (module), 162
networkx.algorithms.centrality (module), 163
networkx.algorithms.chordal.chordal_alg (module), 185
networkx.algorithms.clique (module), 188
networkx.algorithms.cluster (module), 191
networkx.algorithms.coloring (module), 194
networkx.algorithms.community (module), 195
networkx.algorithms.community.kclique (module), 195
networkx.algorithms.components (module), 196
networkx.algorithms.components.attracting
(module),
203
networkx.algorithms.components.biconnected (module),
205
networkx.algorithms.components.connected (module),
196
networkx.algorithms.components.semiconnected (module), 210
networkx.algorithms.components.strongly_connected
(module), 199
networkx.algorithms.components.weakly_connected
(module), 202
networkx.algorithms.connectivity (module), 210
networkx.algorithms.connectivity.connectivity (module),
211
networkx.algorithms.connectivity.cuts (module), 219
networkx.algorithms.connectivity.stoerwagner (module),
225
networkx.algorithms.connectivity.utils (module), 227
networkx.algorithms.core (module), 227
networkx.algorithms.cycles (module), 231
networkx.algorithms.dag (module), 234
networkx.algorithms.distance_measures (module), 237
networkx.algorithms.distance_regular (module), 239
networkx.algorithms.dominance (module), 240
networkx.algorithms.dominating (module), 242
networkx.algorithms.euler (module), 243
networkx.algorithms.flow (module), 244
networkx.algorithms.graphical (module), 267
networkx.algorithms.hierarchy (module), 270
517
NetworkX Reference, Release 2.0.dev20141229000009
networkx.algorithms.isolate (module), 271
networkx.algorithms.isomorphism (module), 272
networkx.algorithms.isomorphism.isomorphvf2 (module), 275
networkx.algorithms.link_analysis.hits_alg
(module),
290
networkx.algorithms.link_analysis.pagerank_alg (module), 285
networkx.algorithms.link_prediction (module), 293
networkx.algorithms.matching (module), 299
networkx.algorithms.mis (module), 300
networkx.algorithms.mst (module), 301
networkx.algorithms.operators.all (module), 306
networkx.algorithms.operators.binary (module), 303
networkx.algorithms.operators.product (module), 308
networkx.algorithms.operators.unary (module), 303
networkx.algorithms.richclub (module), 310
networkx.algorithms.shortest_paths.astar (module), 328
networkx.algorithms.shortest_paths.dense (module), 326
networkx.algorithms.shortest_paths.generic
(module),
311
networkx.algorithms.shortest_paths.unweighted (module), 315
networkx.algorithms.shortest_paths.weighted (module),
318
networkx.algorithms.simple_paths (module), 329
networkx.algorithms.swap (module), 330
networkx.algorithms.traversal.breadth_first_search (module), 336
networkx.algorithms.traversal.depth_first_search (module), 332
networkx.algorithms.traversal.edgedfs (module), 338
networkx.algorithms.tree.recognition (module), 339
networkx.algorithms.vitality (module), 341
networkx.classes.function (module), 343
networkx.convert (module), 409
networkx.convert_matrix (module), 412
networkx.drawing.layout (module), 473
networkx.drawing.nx_agraph (module), 468
networkx.drawing.nx_pydot (module), 471
networkx.drawing.nx_pylab (module), 459
networkx.exception (module), 477
networkx.generators.atlas (module), 353
networkx.generators.bipartite (module), 385
networkx.generators.classic (module), 353
networkx.generators.degree_seq (module), 371
networkx.generators.directed (module), 378
networkx.generators.ego (module), 391
networkx.generators.expanders (module), 358
networkx.generators.geometric (module), 381
networkx.generators.hybrid (module), 385
networkx.generators.intersection (module), 392
networkx.generators.line (module), 389
networkx.generators.random_clustered (module), 377
518
networkx.generators.random_graphs (module), 362
networkx.generators.small (module), 358
networkx.generators.social (module), 394
networkx.generators.stochastic (module), 392
networkx.linalg.algebraicconnectivity (module), 400
networkx.linalg.attrmatrix (module), 403
networkx.linalg.graphmatrix (module), 395
networkx.linalg.laplacianmatrix (module), 397
networkx.linalg.spectrum (module), 399
networkx.readwrite.adjlist (module), 419
networkx.readwrite.edgelist (module), 426
networkx.readwrite.gexf (module), 433
networkx.readwrite.gml (module), 435
networkx.readwrite.gpickle (module), 439
networkx.readwrite.graph6 (module), 450
networkx.readwrite.graphml (module), 441
networkx.readwrite.json_graph (module), 442
networkx.readwrite.leda (module), 447
networkx.readwrite.multiline_adjlist (module), 423
networkx.readwrite.nx_shp (module), 457
networkx.readwrite.nx_yaml (module), 448
networkx.readwrite.pajek (module), 455
networkx.readwrite.sparse6 (module), 453
networkx.utils (module), 479
networkx.utils.contextmanagers (module), 486
networkx.utils.decorators (module), 483
networkx.utils.misc (module), 479
networkx.utils.random_sequence (module), 480
networkx.utils.rcm (module), 484
networkx.utils.union_find (module), 480
NetworkXAlgorithmError (class in networkx), 477
NetworkXError (class in networkx), 477
NetworkXException (class in networkx), 477
NetworkXNoPath (class in networkx), 477
NetworkXPointlessConcept (class in networkx), 477
NetworkXUnbounded (class in networkx), 477
NetworkXUnfeasible (class in networkx), 477
newman_watts_strogatz_graph() (in module networkx.generators.random_graphs), 366
node, 495
node attribute, 495
node_boundary()
(in
module
networkx.algorithms.boundary), 163
node_clique_number()
(in
module
networkx.algorithms.clique), 190
node_connected_component()
(in
module
networkx.algorithms.components.connected),
198
node_connectivity()
(in
module
networkx.algorithms.connectivity.connectivity),
218
node_link_data()
(in
module
networkx.readwrite.json_graph), 443
Index
NetworkX Reference, Release 2.0.dev20141229000009
node_link_graph()
(in
module
net- number_weakly_connected_components()
workx.readwrite.json_graph), 444
(in
module
netnode_redundancy()
(in
module
networkx.algorithms.components.weakly_connected),
workx.algorithms.bipartite.redundancy),
203
157
numeric_assortativity_coefficient() (in module netnodes() (DiGraph method), 47
workx.algorithms.assortativity), 134
nodes() (Graph method), 20
numerical_edge_match()
(in
module
netnodes() (in module networkx.classes.function), 346
workx.algorithms.isomorphism), 282
nodes() (MultiDiGraph method), 107
numerical_multiedge_match()
(in
module
netnodes() (MultiGraph method), 79
workx.algorithms.isomorphism), 283
nodes_iter() (DiGraph method), 47
numerical_node_match()
(in
module
netnodes_iter() (Graph method), 21
workx.algorithms.isomorphism), 282
nodes_iter() (in module networkx.classes.function), 346
O
nodes_iter() (MultiDiGraph method), 107
nodes_iter() (MultiGraph method), 79
octahedral_graph()
(in
module
netnodes_with_selfloops() (DiGraph method), 62
workx.generators.small), 361
nodes_with_selfloops() (Graph method), 31
open_file() (in module networkx.utils.decorators), 483
nodes_with_selfloops() (MultiDiGraph method), 122
order() (DiGraph method), 56
nodes_with_selfloops() (MultiGraph method), 91
order() (Graph method), 28
non_edges() (in module networkx.classes.function), 348 order() (MultiDiGraph method), 117
non_neighbors() (in module networkx.classes.function), order() (MultiGraph method), 87
346
out_degree() (DiGraph method), 60
normalized_laplacian_matrix()
(in
module
net- out_degree() (MultiDiGraph method), 120
workx.linalg.laplacianmatrix), 397
out_degree_centrality()
(in
module
netnull_graph() (in module networkx.generators.classic),
workx.algorithms.centrality), 164
357
out_degree_iter() (DiGraph method), 60
number_attracting_components() (in module net- out_degree_iter() (MultiDiGraph method), 121
workx.algorithms.components.attracting),
out_edges() (DiGraph method), 49
204
out_edges() (MultiDiGraph method), 110
number_connected_components() (in module net- out_edges_iter() (DiGraph method), 50
workx.algorithms.components.connected),
out_edges_iter() (MultiDiGraph method), 110
197
overlap_weighted_projected_graph() (in module netnumber_of_cliques()
(in
module
networkx.algorithms.bipartite.projection), 150
workx.algorithms.clique), 190
number_of_edges() (DiGraph method), 61
P
number_of_edges() (Graph method), 31
pagerank()
(in
module
netnumber_of_edges()
(in
module
networkx.algorithms.link_analysis.pagerank_alg),
workx.classes.function), 347
286
number_of_edges() (MultiDiGraph method), 122
pagerank_numpy()
(in
module
netnumber_of_edges() (MultiGraph method), 90
workx.algorithms.link_analysis.pagerank_alg),
number_of_nodes() (DiGraph method), 57
287
number_of_nodes() (Graph method), 29
pagerank_scipy()
(in
module
netnumber_of_nodes()
(in
module
networkx.algorithms.link_analysis.pagerank_alg),
workx.classes.function), 346
288
number_of_nodes() (MultiDiGraph method), 117
pappus_graph() (in module networkx.generators.small),
number_of_nodes() (MultiGraph method), 88
361
number_of_selfloops() (DiGraph method), 63
pareto_sequence()
(in
module
netnumber_of_selfloops() (Graph method), 32
workx.utils.random_sequence), 481
number_of_selfloops() (MultiDiGraph method), 123
parse_adjlist() (in module networkx.readwrite.adjlist),
number_of_selfloops() (MultiGraph method), 92
421
number_strongly_connected_components()
parse_edgelist() (in module networkx.readwrite.edgelist),
(in
module
net432
workx.algorithms.components.strongly_connected),
parse_gml() (in module networkx.readwrite.gml), 437
199
Index
519
NetworkX Reference, Release 2.0.dev20141229000009
parse_graph6() (in module networkx.readwrite.graph6),
450
parse_leda() (in module networkx.readwrite.leda), 448
parse_multiline_adjlist()
(in
module
networkx.readwrite.multiline_adjlist), 425
parse_pajek() (in module networkx.readwrite.pajek), 457
parse_sparse6() (in module networkx.readwrite.sparse6),
453
path_graph() (in module networkx.generators.classic),
357
periphery()
(in
module
networkx.algorithms.distance_measures), 238
petersen_graph() (in module networkx.generators.small),
361
powerlaw_cluster_graph()
(in
module
networkx.generators.random_graphs), 369
powerlaw_sequence()
(in
module
networkx.utils.random_sequence), 481
predecessor()
(in
module
networkx.algorithms.shortest_paths.unweighted),
318
predecessors() (DiGraph method), 53
predecessors() (MultiDiGraph method), 114
predecessors_iter() (DiGraph method), 53
predecessors_iter() (MultiDiGraph method), 114
preferential_attachment()
(in
module
networkx.algorithms.link_prediction), 295
preflow_push() (in module networkx.algorithms.flow),
256
projected_graph()
(in
module
networkx.algorithms.bipartite.projection), 147
pydot_layout() (in module networkx.drawing.nx_pydot),
473
pygraphviz_layout()
(in
module
networkx.drawing.nx_agraph), 471
R
ra_index_soundarajan_hopcroft() (in module networkx.algorithms.link_prediction), 297
radius()
(in
module
networkx.algorithms.distance_measures), 238
ramsey_R2()
(in
module
networkx.algorithms.approximation.ramsey),
131
random_clustered_graph()
(in
module
networkx.generators.random_clustered), 377
random_degree_sequence_graph() (in module networkx.generators.degree_seq), 376
random_geometric_graph()
(in
module
networkx.generators.geometric), 381
random_layout() (in module networkx.drawing.layout),
474
random_lobster()
(in
module
networkx.generators.random_graphs), 369
520
random_powerlaw_tree()
(in
module
networkx.generators.random_graphs), 370
random_powerlaw_tree_sequence() (in module networkx.generators.random_graphs), 371
random_regular_graph()
(in
module
networkx.generators.random_graphs), 368
random_shell_graph()
(in
module
networkx.generators.random_graphs), 370
random_weighted_sample()
(in
module
networkx.utils.random_sequence), 483
read_adjlist() (in module networkx.readwrite.adjlist), 419
read_dot() (in module networkx.drawing.nx_agraph), 470
read_dot() (in module networkx.drawing.nx_pydot), 472
read_edgelist() (in module networkx.readwrite.edgelist),
427
read_gexf() (in module networkx.readwrite.gexf), 433
read_gml() (in module networkx.readwrite.gml), 436
read_gpickle() (in module networkx.readwrite.gpickle),
439
read_graph6() (in module networkx.readwrite.graph6),
451
read_graphml() (in module networkx.readwrite.graphml),
441
read_leda() (in module networkx.readwrite.leda), 448
read_multiline_adjlist()
(in
module
networkx.readwrite.multiline_adjlist), 423
read_pajek() (in module networkx.readwrite.pajek), 456
read_shp() (in module networkx.readwrite.nx_shp), 457
read_sparse6() (in module networkx.readwrite.sparse6),
454
read_weighted_edgelist()
(in
module
networkx.readwrite.edgelist), 429
read_yaml() (in module networkx.readwrite.nx_yaml),
449
relabel_gexf_graph()
(in
module
networkx.readwrite.gexf), 435
remove_edge() (DiGraph method), 43
remove_edge() (Graph method), 17
remove_edge() (MultiDiGraph method), 103
remove_edge() (MultiGraph method), 74
remove_edges_from() (DiGraph method), 44
remove_edges_from() (Graph method), 18
remove_edges_from() (MultiDiGraph method), 104
remove_edges_from() (MultiGraph method), 75
remove_node() (DiGraph method), 40
remove_node() (Graph method), 14
remove_node() (MultiDiGraph method), 100
remove_node() (MultiGraph method), 71
remove_nodes_from() (DiGraph method), 41
remove_nodes_from() (Graph method), 14
remove_nodes_from() (MultiDiGraph method), 100
remove_nodes_from() (MultiGraph method), 71
resource_allocation_index()
(in
module
networkx.algorithms.link_prediction), 293
Index
NetworkX Reference, Release 2.0.dev20141229000009
reverse() (DiGraph method), 66
reverse()
(in
module
networkx.algorithms.operators.unary), 303
reverse() (MultiDiGraph method), 126
reverse_cuthill_mckee_ordering() (in module networkx.utils.rcm), 485
reversed() (in module networkx.utils.contextmanagers),
487
rich_club_coefficient()
(in
module
networkx.algorithms.richclub), 310
robins_alexander_clustering()
(in
module
networkx.algorithms.bipartite.cluster), 157
S
scale_free_graph()
(in
module
networkx.generators.directed), 380
sedgewick_maze_graph()
(in
module
networkx.generators.small), 361
selfloop_edges() (DiGraph method), 62
selfloop_edges() (Graph method), 32
selfloop_edges() (MultiDiGraph method), 123
selfloop_edges() (MultiGraph method), 91
semantic_feasibility() (DiGraphMatcher method), 280
semantic_feasibility() (GraphMatcher method), 278
set_edge_attributes()
(in
module
networkx.classes.function), 349
set_node_attributes()
(in
module
networkx.classes.function), 348
sets() (in module networkx.algorithms.bipartite.basic),
144
shell_layout() (in module networkx.drawing.layout), 475
shortest_augmenting_path()
(in
module
networkx.algorithms.flow), 254
shortest_path()
(in
module
networkx.algorithms.shortest_paths.generic),
312
shortest_path_length()
(in
module
networkx.algorithms.shortest_paths.generic),
313
simple_cycles() (in module networkx.algorithms.cycles),
232
single_source_dijkstra()
(in
module
networkx.algorithms.shortest_paths.weighted),
322
single_source_dijkstra_path()
(in
module
networkx.algorithms.shortest_paths.weighted),
320
single_source_dijkstra_path_length() (in module networkx.algorithms.shortest_paths.weighted),
321
single_source_shortest_path()
(in
module
networkx.algorithms.shortest_paths.unweighted),
316
Index
single_source_shortest_path_length() (in module networkx.algorithms.shortest_paths.unweighted),
316
size() (DiGraph method), 61
size() (Graph method), 30
size() (MultiDiGraph method), 121
size() (MultiGraph method), 89
spectral_bipartivity()
(in
module
networkx.algorithms.bipartite.spectral), 152
spectral_layout() (in module networkx.drawing.layout),
476
spectral_ordering()
(in
module
networkx.linalg.algebraicconnectivity), 402
spring_layout() (in module networkx.drawing.layout),
475
square_clustering()
(in
module
networkx.algorithms.cluster), 194
star_graph() (in module networkx.generators.classic), 357
stochastic_graph()
(in
module
networkx.generators.stochastic), 392
stoer_wagner()
(in
module
networkx.algorithms.connectivity.stoerwagner),
225
strong_product()
(in
module
networkx.algorithms.operators.product), 309
strongly_connected_component_subgraphs()
(in
module
networkx.algorithms.components.strongly_connected),
200
strongly_connected_components() (in module networkx.algorithms.components.strongly_connected),
200
strongly_connected_components_recursive()
(in
module
networkx.algorithms.components.strongly_connected),
201
subgraph() (DiGraph method), 65
subgraph() (Graph method), 35
subgraph() (MultiDiGraph method), 126
subgraph() (MultiGraph method), 94
subgraph_is_isomorphic() (DiGraphMatcher method),
279
subgraph_is_isomorphic() (GraphMatcher method), 278
subgraph_isomorphisms_iter()
(DiGraphMatcher
method), 280
subgraph_isomorphisms_iter() (GraphMatcher method),
278
successors() (DiGraph method), 53
successors() (MultiDiGraph method), 113
successors_iter() (DiGraph method), 53
successors_iter() (MultiDiGraph method), 113
symmetric_difference()
(in
module
networkx.algorithms.operators.binary), 306
syntactic_feasibility() (DiGraphMatcher method), 280
521
NetworkX Reference, Release 2.0.dev20141229000009
syntactic_feasibility() (GraphMatcher method), 278
T
tensor_product()
(in
module
networkx.algorithms.operators.product), 310
tetrahedral_graph()
(in
module
networkx.generators.small), 361
to_agraph() (in module networkx.drawing.nx_agraph),
469
to_dict_of_dicts() (in module networkx.convert), 410
to_dict_of_lists() (in module networkx.convert), 411
to_directed() (DiGraph method), 64
to_directed() (Graph method), 34
to_directed() (MultiDiGraph method), 125
to_directed() (MultiGraph method), 93
to_edgelist() (in module networkx.convert), 411
to_networkx_graph() (in module networkx.convert), 409
to_numpy_matrix()
(in
module
networkx.convert_matrix), 413
to_numpy_recarray()
(in
module
networkx.convert_matrix), 414
to_pydot() (in module networkx.drawing.nx_pydot), 472
to_scipy_sparse_matrix()
(in
module
networkx.convert_matrix), 416
to_undirected() (DiGraph method), 64
to_undirected() (Graph method), 33
to_undirected() (MultiDiGraph method), 124
to_undirected() (MultiGraph method), 93
topological_sort() (in module networkx.algorithms.dag),
235
topological_sort_recursive()
(in
module
networkx.algorithms.dag), 235
transitivity() (in module networkx.algorithms.cluster),
191
tree_data() (in module networkx.readwrite.json_graph),
446
tree_graph() (in module networkx.readwrite.json_graph),
447
triangles() (in module networkx.algorithms.cluster), 191
trivial_graph() (in module networkx.generators.classic),
357
truncated_cube_graph()
(in
module
networkx.generators.small), 361
truncated_tetrahedron_graph()
(in
module
networkx.generators.small), 362
tutte_graph() (in module networkx.generators.small), 362
union() (UnionFind method), 480
union_all()
(in
module
workx.algorithms.operators.all), 307
net-
W
watts_strogatz_graph()
(in
module
networkx.generators.random_graphs), 366
waxman_graph()
(in
module
networkx.generators.geometric), 383
weakly_connected_component_subgraphs()
(in
module
networkx.algorithms.components.weakly_connected),
203
weakly_connected_components() (in module networkx.algorithms.components.weakly_connected),
203
weighted_choice()
(in
module
networkx.utils.random_sequence), 483
weighted_projected_graph()
(in
module
networkx.algorithms.bipartite.projection), 148
wheel_graph() (in module networkx.generators.classic),
358
within_inter_cluster()
(in
module
networkx.algorithms.link_prediction), 298
write_adjlist() (in module networkx.readwrite.adjlist),
420
write_dot() (in module networkx.drawing.nx_agraph),
470
write_dot() (in module networkx.drawing.nx_pydot), 472
write_edgelist() (in module networkx.readwrite.edgelist),
428
write_gexf() (in module networkx.readwrite.gexf), 434
write_gml() (in module networkx.readwrite.gml), 436
write_gpickle() (in module networkx.readwrite.gpickle),
440
write_graph6() (in module networkx.readwrite.graph6),
452
write_graphml()
(in
module
networkx.readwrite.graphml), 442
write_multiline_adjlist()
(in
module
networkx.readwrite.multiline_adjlist), 424
write_pajek() (in module networkx.readwrite.pajek), 456
write_shp() (in module networkx.readwrite.nx_shp), 458
write_sparse6() (in module networkx.readwrite.sparse6),
455
write_weighted_edgelist()
(in
module
networkx.readwrite.edgelist), 430
U
write_yaml() (in module networkx.readwrite.nx_yaml),
449
uniform_random_intersection_graph() (in module networkx.generators.intersection), 392
uniform_sequence()
(in
module
net- Z
zipf_rv() (in module networkx.utils.random_sequence),
workx.utils.random_sequence), 481
482
union()
(in
module
networkx.algorithms.operators.binary), 304
522
Index
NetworkX Reference, Release 2.0.dev20141229000009
zipf_sequence()
(in
module
workx.utils.random_sequence), 482
Index
net-
523
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