Carnes et al.

Carnes et al.
Maximizing Influence in a Competitive Social Network:
A Follower’s Perspective
[Extended Abstract]
∗
†
‡
Tim Carnes , Chandrashekhar Nagarajan , Stefan M. Wild , and Anke van Zuylen
†
School of Operations Research and Industrial Engineering
Cornell University
Ithaca, NY 14853
{tcarnes, chandra, stefan, anke}@orie.cornell.edu
ABSTRACT
1.
We consider the problem faced by a company that wants to
use viral marketing to introduce a new product into a market where a competing product is already being introduced.
We assume that consumers will use only one of the two products and will influence their friends in their decision of which
product to use. We propose two models for the spread of influence of competing technologies through a social network
and consider the influence maximization problem from the
follower’s perspective. In particular we assume the follower
has a fixed budget available that can be used to target a
subset of consumers and show that, although it is NP-hard
to select the most influential subset to target, it is possible
to give an efficient algorithm that is within 63% of optimal.
Our computational experiments show that by using knowledge of the social network and the set of consumers targeted
by the competitor, the follower may in fact capture a majority of the market by targeting a relatively small set of the
right consumers.
The spread of a new idea or product is often studied by
modeling a social network as a graph where the nodes represent individuals, and edges represent interactions between
individuals. These interactions could include the recommendation of a particular product and such recommendation
networks and their effects on consumer purchasing have recently been analyzed in [15] and [16]. Further, there has
been recent statistical support that such network linkage
can directly affect product adoption [9]. Based on these empirical studies, we can formulate assumptions on how people
affect the people they interact with. We can then use these
graphs to answer questions such as: “If customers influence
each other in their decisions to buy products, which customers should be targeted to maximize the expected profit
of a new product?” and “How large of a consumer base needs
to be targeted for a new technology, product, or idea to capture a significant share of the market?”
Motivated by the declining effectiveness of traditional mass
marketing techniques [15], many recent papers have studied
these and similar types of questions. The algorithmic problem of designing viral marketing strategies, marketing techniques which exploit pre-existing social networks to reach
consumers, was studied by Richardson and Domingos [21],
and Kempe, Kleinberg and Tardos [12, 13]. Their research
builds on a “word-of-mouth” approach examined in a marketing context by Goldenberg et al. in [8]. In the aforementioned works, the producer of a new product is assumed to
have the ability to “influence” a particular set of consumers
within the social network – either through targeted advertising, providing free samples, or adding monetary incentive
– to adopt the new product. If these people influence some
of their friends to also try the product, and these friends in
turn recommend it to others, and so forth, the producer
can create a cascade of recommendations. The question
then becomes how to choose an initial subset of so-called
early adopters to maximize the number of people that will
eventually be reached, and hence be likely to purchase the
product. The size of the subsets allowed is assumed to be
limited due to marketing budget constraints. Kempe et al.
develop general models for the spreading of influence, show
that finding the most influential set of nodes is NP-hard, and
give an approximation algorithm for finding a set of nodes
that approximately maximizes the expected influence.
The models developed by Kempe et al. assume that there
Categories and Subject Descriptors
F.2 [Analysis of Algorithms & Problem Complexity]:
Nonnumerical Algorithms and Problems; G.2.1 [Discrete
Mathematics]: Combinatorics—combinatorial algorithms
General Terms
Algorithms, Performance, Theory
Keywords
Approximation Algorithms, Social Networks, Viral Marketing, Network Analysis, Targeted Marketing
∗Research supported by NSF grants CCR-0635121 & DMI-0500263.
†Research supported by NSF grant CCF-0514628.
‡
Research supported by a DOE Computational Science Graduate Fellowship under grant number DE-FG02-97ER25308 and by NSF grant
CCF-0305583.
Permission to make digital or hard copies of all or part of this work for
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permission and/or a fee.
ICEC’07, August 19–22, 2007, Minneapolis, Minnesota, USA.
Copyright 2007 ACM 978-1-59593-700-1/07/0008 ...$5.00.
INTRODUCTION
is only one company introducing a product. However, producers of consumer technologies often must introduce a new
product into a market where a competitor will offer a comparable product. The introduction of Nintendo’s Wii, to
compete with Sony’s Playstation 3, and Blu-ray discs, competing with Toshiba’s HD DVD, are recent canonical examples of such behavior. When adoption of the technology is
not free, it is unlikely that a typical consumer will use both
products. Furthermore, even if a competing product is superior, consumers are often reluctant to switch technologies if
they must bear a cost of transition which may outweigh any
direct benefits of the technology [6]. The question whether
in this setting a competing product can survive and will be
adopted by a significant fraction of the market, or if it will
eventually disappear, has been studied in numerous works,
including [10], [17], and [23].
It is not always the case that the product with the largest
number of early adopters can translate this initial edge into
market dominance. A classical example where such tipping
did occur is the demise of the BETA format due to the VHS
format’s initial popularity. However, Katz and Shapiro note
that consumer heterogeneity coupled with distinct features
of rival products tends to limit tipping in markets where consumers care more about a product’s features than its overall
prevalence [11]. Hence it is an interesting question to consider how a company with a smaller marketing budget may
effectively infiltrate a market in which a stronger competing
company is also present.
Historically, competition between two products has largely
been addressed from an economic modeling perspective and
focused on areas such as market equilibrium. For example, in [2] and [3], primarily network-independent properties
are employed to model the propagation of two technologies
through a market. Tomochi et al. [23] offer a more gametheoretic approach which relies on the network for spatial
coordination games. However, they do not address the problem of taking advantage of the social network and viral marketing when introducing a new technology into a market.
In this paper, we study the algorithmic problem of how
to introduce a new product into an environment where a
competing product is also being introduced. We focus on
the case when a company can keep itself hidden from a
competitor until the moment of introduction. We assume
that the company has a fixed budget for targeting consumers
and knows who its competitor’s early adopters are – either
through extensive market research or industrial espionage.
We first develop two models for the spread of adoption of
the two products through the network. We show that finding the most influential set of a given size for the company
to target – the set that maximizes the expected number of
people that will adopt the new product – is NP-hard under
the proposed models in this setting. From a game theoretic
point-of-view, this can also be viewed as calculating the company’s best response to a competitor’s move in a Stackelberg
game [7].
Following Kempe et al. we show that using well known
results on submodular functions [18], we can give a (1 − 1e −
ε)-approximation algorithm for finding the most influential
set of nodes. Additionally, using a result of Sviridenko [22],
we generalize the allowed subsets to be limited based upon
cost rather than simply size, hence allowing different costs to
be associated with targeting different subsets of customers.
We will empirically show that a company can obtain a larger
market share than its unsuspecting competitor even if the
competitor has a much larger marketing budget. Further,
we show that knowing who the competitor’s early adopters
are, hence being able to apply our algorithm, will allow the
company to capture a given percentage of the market using
a much smaller marketing budget.
In the sequel we use the words technology and product
synonymously. We discuss useful results from related work
in Section 2. Building on these results, we describe the models we developed for the spread of two competing technologies in Section 3 and the results derived for these models in
Section 4. In Section 5 we give the results of some numerical
simulations of the behavior of these models, and we present
conclusions and further research directions in Section 6.
2.
BACKGROUND
We begin by recalling some existing results central to the
present work.
Submodular function maximization: Given a ground
set V , a function f : 2V → R is said to be submodular if
f (S ∪ {v}) − f (S) ≥ f (T ∪ {v}) − f (T ) for all v ∈ V and
sets S ⊆ T ⊆ V . We further say that f is monotone if
f (S ∪ {v}) ≥ f (S) for all v ∈ V and subsets S ⊆ V .
For a non-negative, submodular, and monotone function
f , and the optimization problem
max {f (I) : |I| = k, I ⊆ V } ,
(1)
the greedy Hill Climbing Algorithm repeatedly adds the element from V that gives the greatest improvement, by solving
max {f (I ∪ {v}) : v ∈ V − I}
(2)
until |I| = k. In [18] Nemhauser et al. show that hill climbing
yields a 1 − 1e -approximation: if I is the set found by the
Hill Climbing
Algorithm, and I ∗ maximizes (1), then f (I) ≥
∗
1
1 − e f (I ). This result has been extended [12] to show
that for any ε > 0, there is a γ > 0 such that when using a
(1 + γ)-approximation of f (·) in (2), we obtain a (1 − 1e − ε)approximation.
Sviridenko recently generalized the result from Nemhauser
et al. to include problems of the form (1) with an additional
knapsack-type constraint [22]. In particular, for a set of nonnegative weights {ci : i ∈ V } and a budget B ≥ 0, we now
consider the problem:
(
)
X
max f (I) :
ci ≤ B, I ⊆ V ,
(3)
i∈I
where f is again a non-negative, submodular, and monotone set function. An extension of hill climbing, iteratively
adding to I elements v ∈ V − I which maximize
(
)
X
f (I ∪ {v}) − f (I)
max
: cv +
ci ≤ B ,
(4)
cv
i∈I
P
until cv > B − i∈I ci for all v ∈ V − I, is described in [14].
Sviridenko
showed that this version of hill climbing yields a
1 − 1e -approximation to (3).
Influence maximization on a network in the single
technology case: The spread of a single technology through
a network has been approached using different diffusion models (see for example [12, 13, 21, 24]). Here we describe the
independent cascade model introduced by Kempe et al. [12]
which resembles the models we will develop in the case of
competing technologies.
We assume some set of nodes I initially uses the technology. The diffusion process then unfolds in discrete time
steps. When a node u first adopts the technology, it gets a
single chance to make its neighbor v adopt the technology.
It succeeds with probability puv independently of the history
so far. In the next time step, the nodes which just adopted
the technology get a chance to influence their neighbors and
so on. Note that the process is progressive: once a node has
adopted the technology, it will not go back to the state of
not having adopted it.
The quantity of interest is then the influence function
σ(I), signifying the expected number of nodes that eventually adopt the technology given the initial set of adopters
I. In [12] Kempe et al. address how to choose an initial
set I of some fixed size k to maximize σ(I). Kempe et al.
previously showed that solving (1) when f is the influence
function σ is NP-hard but that σ is submodular. Hence if σ
can be approximated (say with numerical simulations) arbitrarily well,
then for any given ε > 0, hill climbing gives a
1 − 1e − ε -approximation algorithm for finding an influential initial k-set I.
Our results: We propose two models for the simultaneous diffusion of two competing technologies on any network
given an initial set of early adopters for each technology. Influence functions σ(IA |IB ) are defined to quantify the success of a technology’s choice of initial adopters. While the
proposed models for diffusion are conceptually simple, we
show that maximizing such influence functions subject to
a budget is computationally intractable. However, in each
case we are able to show that the influence function is nonnegative, submodular, and monotone, and hence hill climbing provides an approximation algorithm. We have also generalized these results to address heterogeneous costs for targeting consumers.
3. MODELING THE DIFFUSION OF TWO
TECHNOLOGIES
We now extend the independent cascade model to the case
of two competing technologies. In particular, we propose two
models for describing how two technologies simultaneously
diffuse over a given network.
Consumers are again modeled as nodes in a network and
links between nodes represent interaction between consumers.
We assume that our network is an undirected1 graph G =
(V, E) with |V | = n nodes and |E| = m edges. Nodes can
take on one of three states – A and B referring to the two
technologies of interest, and C denoting that neither technology is adopted. We can specify two initial sets of nodes
– a set of initial adopters of A, IA ⊆ V , and a set of initial
adopters of B, IB ⊆ V (with the implicit assumption that
IC = V − (IA ∪ IB )). We assume that IA ∩ IB = ∅. We
assume that once a node has chosen a technology, it will
not change to another technology, but that nodes that are
using one of the two technologies can influence their neighbors that are not using either technology in their decisions
to adopt one of the two technologies.
As in the independent cascade model for a single technology, we assume that if u has adopted a particular technology,
then u influences neighbor v with probability puv . Hence1
The results in this paper can be easily extended to the more
general directed case.
forth, we say that an edge is “active” with probability puv .
However, it is now possible that v is influenced by multiple
neighbors that use different technologies. We will propose
two models that govern this “diffusion” of technologies A and
B, starting from the sets of initial adopters, given the set of
active edges Ea of the network. In other words, the models we develop operate on a random subgraph of the social
network G, where each edge is included independently with
probability puv .
Our first model will describe the diffusion of a technology
where the product/technology itself can only be obtained
from an initial adopter, and a consumer who becomes interested in the technology (i.e. the corresponding node is
influenced by a neighbor via an active edge) will pick one of
the closest early adopters at random. In the second model,
the technology availability is not tied to the network, and
the consumer who becomes interested in the technology will
choose one of its neighbors and adopt the same technology
as this neighbor.
Given such a diffusion model, and the assumption that
initially a set of consumers, IB , is already using technology
B, company A would like to choose a set of k consumers,
IA , to target so as to maximize the expected number of
consumers reached eventually.
Let the influence function f (IA |IB ) be the expected number of consumers that will adopt technology A, given that
initially the set IA is using technology A and the set IB is
using technology B. We are hence after a solution of the
influence maximization problem:
max {f (IA |IB ) : IA ⊆ V − IB , |IA | = k} .
(5)
If the cost of targeting consumers varies from consumer to
consumer, a company may instead wish to maximize its revenue subject to some budget B. Given non-negative costs
{ci : i ∈ V }, the more general from of (5) is then:




X
max f (IA |IB ) : IA ⊆ V − IB ,
ci ≤ B .
(6)


i∈IA
For ease of exposition, throughout the sequel we suppress
these costs and will assume that cv = 1 for all v ∈ V .
3.1
A Distance-based model
Our first model is related to competitive facility location
[5] on a network. In this model, the location of a node in the
network is important, as well as the connectivity of a node.
The idea is that a consumer will be more likely to mimic the
behavior of an early adopter if their distance in the social
network is small.
We assume that for each edge (u, v) ∈ E, we are also
given a length duv . If no length is specified we assume that
all edges have length 1. In the following we will assume
all edges have length 1, however the results can easily be
extended for arbitrary non-negative edge lengths. We let
I = IA ∪ IB be the set of all initial adopters.
Let du (I, Ea ) denote the shortest distance from u to I
along the edges in Ea , with the notation du (I, Ea ) = ∞2
if and only if u is not connected to any node of I using
only active edges. If du (I, Ea ) < ∞, let νu (IA , du (I, Ea ))
2
If du (I, Ea ) = ∞, we say that u will adopt neither technology (state C) because it is not connected to any of the
initial adopters by active edges. Henceforth, we assume that
any node u under consideration is connected to some v ∈ I
in G.
v
11
00
00
11
00
11
IA
I
B
Figure 1: Given the set of active edges drawn, the probability that node v adopts technology A is 32 in the distance-based
model, and 12 in the wave propagation model.
and νu (IB , du (I, Ea )) be the number of nodes in IA and IB ,
respectively, at distance du (I, Ea ) from u along edges in Ea .
Given that du (I, Ea ) is the shortest distance from u to I
along the active edges of G, we will say that node u adopts
technology i ∈ {A, B} with probability
νu (Ii , du (I, Ea ))
.
νu (IA , du (I, Ea )) + νu (IB , du (I, Ea ))
(7)
Note that – conditioned on set Ea – node u is thus only influenced by nodes in IA and IB that are at distance du (I, Ea ).
We note that these are well-defined (conditional) probabilities that sum to one, since at least one of νu (IA , du (I, Ea ))
and νu (IB , du (I, Ea )) is strictly positive.
In this model the expected number of nodes which adopt
A will be denoted by
"
#
X
νu (IA , du (I, Ea ))
ρ(IA |IB ) = E
,
νu (IA , du (I, Ea )) + νu (IB , du (I, Ea ))
u∈V
where the expectation is over the set of active edges. We fix
IB and try to determine IA so as to maximize the expected
number of nodes that adopt technology A:
max{ρ(IA |IB ) : IA ⊆ (V − IB ), |IA | = k}.
(8)
3.2 Wave propagation model
Although both of the models we propose here reduce to
the independent cascade model of Kempe et al. [12] if there
is no competition (if we let IB = ∅), our second model for
propagation is closer in spirit to the independent cascade
model. We motivate this model through the example shown
in Figure 1.
In this example, with the edges shown being active, our
previous distance-based model gives node v a probability of
2
of adopting technology A, even though it has only two
3
neighbors, one of which adopts technology A and one of
which adopts technology B. Under the alternative model
presented here a node copies the technology adoption of a
neighboring node randomly chosen from the set of its neighbors that are closest to the initial sets (IA , IB ). In the example, and given the set of active edges shown, this corresponds to giving node v a probability 12 of adopting A and
1
of adopting B.
2
We can think of the propagation as happening in discrete
steps. In step d, all nodes that are at distance at most d − 1
from some node in the initial sets have adopted technology
A or B, and all nodes for which the closest initial node
is farther than d − 1 do not have a technology yet (where
the distance is again with respect to active edges). The
nodes at a distance d from the initial sets now choose one
of their neighbors that are at distance d − 1 independently
at random, and adopt the same technology as this neighbor.
As in the previous section, we assume that the node under
consideration is in the same connected component of at least
one of the nodes u ∈ I.
Formally, let P (v|IA , IB , Ea ) be the probability that node
v adopts technology A when the initial sets for technologies
A and B are IA and IB , respectively, and the set of active
edges is Ea . Let u be a node for which the closest node in
I = IA ∪ IB is at distance d. Let S be the set of neighbors
of u that are at distance d − 1 from I, where all distances
are again with respect to active edges. Then
P
v∈S P (v|IA , IB , Ea )
P (u|IA , IB , Ea ) =
.
(9)
|S|
For initial sets IA , IB , let
"
#
X
π(IA |IB ) = E
P (v|IA , IB , Ea )
(10)
v∈V
denote the expected number of nodes that adopt technology
A. For fixed IB , we seek a solution to:
max{π(IA |IB ) : IA ⊆ (V − IB ), |IA | = k}.
4.
(11)
APPROXIMATION ALGORITHMS FOR
INFLUENCE MAXIMIZATION
For each of the diffusion models proposed in Section 3
we now show that the decision versions of (8) and (11) are
NP-hard but that the corresponding influence functions are
nonnegative, monotone and submodular. It will then follow
from [18] and [22] that we can use a greedy hill-climbing algorithm to get a (1 − 1e )-approximation algorithm for these
problems. In general it will not be possible to exactly solve
the subproblems (2) and (4), as this requires exact evaluation of ρ(·|IB ) and π(·|IB ). However, using sampling we can
get arbitrarily close approximations of the values needed in
(2) and (4). This then allows us to obtain (1 − 1e − ε)approximation algorithms for both models [12].
Theorem 1. For any given IB with |V − IB | ≥ k, the
Hill Climbing Algorithm gives a 1 − 1e − ε -approximation
algorithm for (8).
Proof. Given inputs (IA , IB ), and a set of active edges
Ea , ρ(IA |IB ) can be efficiently evaluated using an algorithm
which relies on a single all-pairs shortest paths computation
and has overall complexity O(|V |3 ). Using sampling, we
can then approximate ρ(IA |IB ) = E [ρ(IA |IB )|Ea ] to within
(1+γ) for any γ > 0 (where the running time depends on γ1 ).
Hence we can implement the greedy hill-climbing algorithm
using (1 + γ)-approximate values for ρ(IA ∪ {v}|IB ) in polynomial time. Monotonicity and submodularity of ρ(·|IB )
will be shown in Lemma 2 and Lemma 3, respectively. The
approximation guarantee is then an immediate consequence
of the results in Section 2.
Theorem 2. For any given IB with |V − IB | ≥ k, the
Hill Climbing Algorithm gives a 1 − 1e − ε -approximation
algorithm for (11).
Proof. Given inputs (IA , IB ) and a set of active edges
Ea , π(IA |IB ) can be efficiently evaluated using an algorithm
which relies on a single all-pairs shortest path computation
x
and has overall complexity O(|V |3 ). We can approximate
π(IA |IB ) = E [π(IA |IB )|Ea ] to within (1 + γ) for any γ > 0,
hence we can implement the greedy hill-climbing algorithm
using (1 + γ)-approximate values for π(IA ∪ {v}|IB ) in polynomial time. Monotonicity and submodularity of π(·|IB )
will be shown Lemma 5 and Lemma 6, respectively. The
approximation guarantee is again an immediate consequence
of the results in Section 2.
d1
d2
C1
d3
d4
s
1
e1
C2
dn
e2
s2
Before proceeding, we note that to show NP-hardness and
the desired properties of the influence functions, it suffices to
consider the case when puv = 1 for all edges (or equivalently,
Ea = E): NP-hardness of a special case clearly implies NPhardness of the more general case, and the expected value of
a function of Ea is nonnegative, monotone, and submodular
if for any Ea the function is nonnegative, monotone, and
submodular. For ease of exposition, we therefore restrict
ourselves to this special case in the remainder of this section.
Theorem 3. The decision version of (8) is NP-hard 3 .
Proof. Given a ground set of elements E = {ei : 1 ≤ i ≤
n} and a collection of sets S = {si : 1 ≤ i ≤ m} such that
each si ⊆ E, the decision version of the set cover problem
asks if there is a collection of k sets covering all elements
in E. Without loss of generality we assume that every element is covered by at least one set and that k < min(m, n).
We reduce the NP-hard set cover decision problem to the
decision version of (8).
Given an instance (S, E, k) of set cover, we construct a
graph H as follows. We add a node si for each set si ∈ S
and a node ej for each element ej ∈ E. We add an edge
(si , ej ) if and only if ej ∈ si . We add an additional node
x and connect it to each ej through another node dj (see
Figure 2). Lastly, for a constant κ > 0 to be specified in the
subsequent lemma, we construct a cluster Cj of κ nodes for
each j = 1, . . . , n and connect each of the nodes in Cj to ej .
The following lemma completes the reduction by specifying
the value of κ.
Lemma 1. Let κ > (k + 1)(m + n) and IB = {x}. There
is a collection of k sets which cover E if and only if there
is a set IA of k nodes in the graph H such that ρ(IA |IB ) ≥
n(κ + 1).
Proof. If there a collection of k sets covering E then
take IA to be the k nodes corresponding to those sets. This
gives ρ(IA |IB ) ≥ n(κ + 1) by the following argument. Each
of the nodes ej is adjacent to one of the nodes in IA since
the corresponding sets form a cover. Each ej and the nodes
in each cluster Cj adopt A with probability one since initial
adopters of A are the closest for these nodes. This results
in at least n(κ + 1) nodes with technology A.
If there is no collection of k sets that cover E then we prove
that no set IA of k nodes can be the initial adopters of A
and still achieve ρ(IA |IB ) ≥ n(κ + 1). Consider the nodes
e1 , . . . , en . There exists a node ej which adopts technology B
3
Kempe et al. showed that this problem is NP-hard for the
directed case when IB = ∅. However the problem is not
NP-hard for the undirected case when IB = ∅.
C3
e4
C4
sm
en
4.1 A Distance-based model
Let the decision version of (8) be to determine if there is
a set IA of size k with ρ(IA |IB ) ≥ M for any M ∈ Q. We
then have the following result.
e3
s3
Cn
Figure 2: Graph H for set cover reduction.
1
with probability at least k+1
since any set IA of size k cannot
be within a distance of 1 from all ei (by the construction of
H and lack of a cover). This implies that ej and all nodes
1
. So
in Cj adopt technology B with probability at least k+1
ρ(IA |IB )
≤
1
(n −
)(κ + 1)
k+1
{z
}
|
+
from e1 ,...,en and C1 ,...,Cn
= n(κ + 1) + (m + n) −
m+n
| {z }
from s1 ,...,sm and d1 ,...,dn
1
(κ + 1) < n(κ + 1).
k+1
Having shown the hardness of (8), we now turn to the
Lemmas required in Theorem 1 and show that the influence
function ρ is both monotone and submodular. We assume
without loss of generality that every edge is active with probability 1, and for ease of notation, we will write du (I) instead
of du (I, Ea ). Furthermore, we will drop the subscript u when
u is clear from the context.
Lemma 2. For any IB , ρ(IA |IB ) is a monotone function
of IA .
Proof. For a fixed u ∈ V and initial set IB , it suffices to
show for any v ∈ V − IB , and any IA ⊆ V , the probability
that u adopts A given the initial set IA is at most the probability that u adopts A when the initial set is IA ∪ {v}, that
is:
ν(IA , d(I))
ν(IA , d(I)) + ν(IB , d(I))
≤
ν(IA ∪ {v}, d(I ∪ {v}))
.
ν(IA ∪ {v}, d(I ∪ {v})) + ν(IB , d(I ∪ {v}))
We note that the shortest distance from u to a node in I
is not smaller than the shortest distance from u to a node
in I ∪ {v}, so d(I) ≥ d(I ∪ {v}).
Now, if d(I ∪ {v}) < d(I), then ν(IB , d(I ∪ {v})) = 0,
so the right hand side is 1, and the inequality clearly holds.
Otherwise, ν(IB , d(I∪{v})) = ν(IB , d(I)), and ν(IA , d(I)) =
ν(IA , d(I ∪ {v})) ≤ ν(IA ∪ {v}, d(I ∪ {v}) and the inequality
c
holds since for real numbers c ≥ a ≥ 0 and b > 0, c+b
≥
a
.
a+b
Lemma 3. For any IB , ρ(IA |IB ) is a submodular function
of IA .
Proof. For a set of initial adopters of A, S, and a node
x ∈ V − (S ∪ IB ), we define the increase in the probability
that node u adopts technology A when adding x to the initial
set S as:
P (u, S, x)
=
ν(S ∪ {x}, d(S ∪ {x} ∪ IB ))
ν(S ∪ {x}, d(S ∪ {x} ∪ IB )) + ν(IB , d(S ∪ {x} ∪ IB ))
ν(S, d(S ∪ IB ))
−
.
ν(S, d(S ∪ IB )) + ν(IB , d(S ∪ IB ))
We need to show that for any node u ∈ V and S ⊆ T ⊆ V ,
P (u, S, x) ≥ P (u, T, x).
Let d1 = d(S), d2 = d(T ), d3 = d(IB ). Since S ⊆ T ,
d1 ≥ d2 . We analyze three cases:
Case 1 (d1 ≥ d2 ≥ d3 ): If d(u, x) > d3 , adding x does
not change the probability of u adopting A. So P (u, S, x) =
P (u, T, x) = 0. If d(u, x) < d3 , then adding x makes u adopt
A with probability 1. It then follows from the monotonicity
of ρ that
P (u, S, x)
=
≥
=
ν(S, d(S ∪ IB ))
ν(S, d(S ∪ IB )) + ν(IB , d(S ∪ IB ))
ν(T, d(T ∪ IB ))
1−
ν(T, d(T ∪ IB )) + ν(IB , d(T ∪ IB ))
P (u, T, x).
1−
If d(u, x) = d3 , then ν(S, d(S∪IB )) = ν(S, d3 ) and ν(T, d(T ∪
IB )) = ν(T, d3 ) and these both increase by 1 if x is added.
Furthermore ν(IB , d(X ∪ IB )) = ν(IB , d3 ) for X ∈ {S, S ∪
{x}, T, T ∪ {x}}. So we need to show that
ν(S, d3 ) + 1
ν(S, d3 )
−
ν(S, d3 ) + 1 + ν(IB , d3 )
ν(S, d3 ) + ν(IB , d3 )
ν(T, d3 )
ν(T, d3 ) + 1
−
.
≥
ν(T, d3 ) + 1 + ν(IB , d3 )
ν(T, d3 ) + ν(IB , d3 )
This equation can be easily checked to be true using the fact
that ν(S, d3 ) ≤ ν(T, d3 ).
Case 2 (d3 > d1 ≥ d2 ): In this case ν(IB , d(X ∪IB )) = 0 for
X ∈ {S, S∪{x}, T, T ∪{x}}. In this case the probability that
u adopts A is 1 for initial sets X ∈ {S, S ∪ {x}, T, T ∪ {x}},
so P (u, S, x) = P (u, T, x) = 0.
Case 3 (d1 ≥ d3 > d2 ): Since d3 > d2 , u will adopt technology A with probability 1 if the initial set is T or T ∪ {x}. So
P (u, T, x) = 0, and P (u, S, x) ≥ 0 holds by Lemma 2.
4.2 Wave propagation model
Since it suffices to show that π(IA |IB ) is monotone and
submodular in the special case that every edge in the graph
is active with probability 1, we will restrict ourselves to this
case and write P (u, IA , IB ) instead of P (u|IA , IB , Ea ) for the
probability that node u adopts technology A when the initial
sets for technology A and B are IA and IB , respectively and
the set of active edges is Ea .
Let the decision version of (11) be to determine if there is
a set IA of size k with π(IA |IB ) ≥ M for any M ∈ Q. We
then have the following result.
Theorem 4. The decision version of (11) is NP-hard.
Proof. We reduce the NP-hard set cover decision problem to the decision version of (11) as in Theorem 3. Given an
instance (S, E, k) of set cover, we construct the same graph
H constructed in the proof of Theorem 3. The following
lemma completes the proof.
Lemma 4. Let κ > (m + 1)(m + n) and IB = {x}. There
is a collection of k sets which cover E if and only if there
is a set IA of k nodes in the graph H such that π(IA |IB ) ≥
n(κ + 1).
Proof. If there a collection of k sets covering E then
take IA to be the k nodes corresponding to those sets. This
gives π(IA |IB ) ≥ n(κ + 1) by the same argument given in
the proof of Lemma 1.
If there is no collection of k sets that cover E then we
prove that no set IA of k nodes can be the initial adopters
of A and still achieve π(IA |IB ) ≥ n(κ + 1). Consider the
nodes e1 , . . . , en . Any set IA of size k cannot be within a
distance of 1 from all ej (by the construction of H and lack of
a cover). So there exists a node ej which adopts technology
1
B with probability at least m+1
because one of its neighbors
dj has P (dj |IA , IB ) = 0 and is at distance 1 from I and at
most m + 1 of its neighbors are at distance 1 from I. So
m
P (ej |IA , IB ) ≤ m+1
. This implies that ej and all the nodes
m
. So
in Cj adopt technology A with probability at most m+1
for any initial set IA of size k,
X
π(IA |IB ) ≤
P (v, IA , IB )
v∈V
≤
(m + n)
| {z }
for v∈{s1 ,...,sm }∪{d1 ,...,dn }
+ (n − 1)(κ + 1)
|
{z
}
for v∈ei ∪Ci ,i6=j
≤
+ (κ + 1)P (ej |IA , IB )
{z
}
|
<
<
(n − 1)(κ + 1) + (m + 1)(m + n)
n(κ + 1).
v∈ej ∪Cj
(m + n) + (n − 1)(κ + 1) + (κ + 1)
m
m+1
We again benefit from the valuable properties of monotonicity and submodularity.
Lemma 5. For any IB , π(IA |IB ) is a monotone function
of IA .
Proof. To prove monotonicity we need to show that P (u|S∪
x, IB ) ≥ P (u|S, IB ) for all x ∈ V − IB . We employ the same
notation as in Section 4.1 and let n(v) = {u : (u, v) ∈ E}
denote the neighbors of node v. Note that d(u, S ∪ x ∪
IB ) ≤ d(u, S ∪ IB ). If d(u, S ∪ x ∪ IB ) < d(u, S ∪ IB ) then
P (u|S ∪ x, IB ) = 1 ≥ P (u|S, IB ) which proves monotonicity.
So the interesting case is when d(u, S ∪x∪IB ) = d(u, S ∪IB ).
We prove P (u|S ∪ x, IB ) ≥ P (u|S, IB ) for this case by induction on the distance d = d(u, S ∪ IB ).
Base case: d = 1. If x is not a neighbor of u then P (u|S ∪
x, IB ) = P (u|S, IB ). If x is a neighbor of u then P (u|S ∪
1+|n(u)∩S|
|n(u)∩S|
x, IB ) = 1+|n(u)∩(S∪I
≥ |n(u)∩(S∪I
= P (u|S, IB ).
B )|
B )|
Induction step: Now we prove monotonicity for nodes u
such that d(u, S ∪ IB ) = d assuming monotonicity for all the
nodes v with d(v, S ∪ IB ) < d. Let S be the set of neighbors
of u which are at a distance d − 1 from S ∪ IB . Let K be
the set of neighbors of u which are at a distance d − 1 from
x but at a distance greater than d − 1 from S ∪ IB . Let
K = |K|. Note that all v ∈ K have P (v|S ∪ x, IB ) = 1. The
probability of u accepting technology A is then:
P
K + v∈S P (v|S ∪ x, IB )
P (u|S ∪ x, IB ) =
K + |S|
P
P
(v|S
∪ x, IB )
v∈S
≥
|S|
P
v∈S P (v|S, IB )
≥
= P (u|S, IB ).
|S|
The second inequality follows from the induction assumption
that monotonicity holds for the nodes v with d(v, S ∪IB ) < d
and the fact that all nodes in S are at a distance d − 1 from
S ∪ IB .
Similarly, let T be the set of neighbors of u for whom the
closest node from T ∪ IB is at distance d − 1, and let L be
the number of neighbors of u that are at distance d − 1 from
x, and at distance greater than d − 1 from T ∪ IB . Then:
P (u|T ∪ x, IB ) − P (u|T, IB )
P
=
We now establish the following three inequalities:
K
K + |S|
P
Lemma 6. For any IB , π(IA |IB ) is a submodular function of IA .
by induction on d = d(u, x). If d = 0, then clearly the inequality holds. Suppose it holds for any v such that d(v, x) =
d − 1.
As in the proof of Lemma 3, we consider different cases
for the distance from u to the closest node in S, T and
IB . Let d1 = d(u, S), d2 = d(u, T ), d3 = d(u, IB ). It is
easy to see that the proof of Lemma 3 also works for our
alternative model, except for the case when d1 ≥ d2 ≥ d3
and d3 = d(u, x).
Let S be the set of neighbors of u for whom the closest
node from S ∪ IB is at distance d − 1 so that:
P
v∈S P (v|S, IB )
.
P (u|S, IB ) =
|S|
Note that each neighbor of u that is at distance d − 1 from
x but is at distance greater than d − 1 from the nodes in
S ∪ IB , adopts A with probability 1. Let K be the number
of such nodes, then:
P
K + v∈S P (v|S ∪ x, IB )
P (u|S ∪ x, IB ) =
.
K + |S|
P
v∈S (1
(12)
∪ x, IB ) − P (v|S, IB ))
K + |S|
P
v∈T (P (v|T ∪ x, IB ) − P (v|T, IB ))
≥
(13)
L + |T |
− P (v|S, IB ))
|S|
P
≥
v∈T
(1 − P (v|T, IB ))
|T |
(14)
Clearly these inequalities imply that
P (u|S ∪ x, IB ) − P (u|S, IB ) ≥ P (u|T ∪ x, IB ) − P (u|T, IB ).
To prove (12), let K and L be the set of neighbors of u that
are at distance d−1 from x, and at distance greater than d−1
from S ∪ IB and T ∪ IB , respectively. (So K = |K|, L = |L|).
Since S ⊆ T , we have K ⊇ L and hence K ≥ L. Now,
T ∪ L is the set of neighbors of u that are at distance d − 1
from T ∪ x ∪ IB , and S ∪ K is the set of neighbors of u that
are at distance d − 1 from S ∪ x ∪ IB , so S ∪ K ⊆ T ∪ L.
Since T ∩ L = S ∩ K = ∅, we get that K + |S| ≤ L + |T |.
Combining this with K ≥ L we obtain (12).
To prove (13), we note that for v ∈ T − S, we must have
P (v|T, IB ) = P (v|T ∪ x, IB ) = 1. Since v 6∈ S, the shortest
distance from v to any node in IB is greater than d − 1, and
since v ∈ T , there must be a node in T that is at distance
d − 1 from v. Hence:
X
[P (v|T ∪ x, IB ) − P (v|T, IB )]
v∈T
=
Therefore the difference in the probability of u adopting A
is:
P (u|S ∪ x, IB ) − P (u|S, IB )
P
P
P (v|S, IB )
K + v∈S P (v|S ∪ x, IB )
− v∈S
=
K + |S|
|S|
P
P
P (v|S, IB )
P (v|S ∪ x, IB ) −
K
v∈S
v∈S
=
+
K + |S|
K + |S|
P
P (v|S, IB )
K
v∈S
−
(K + |S|)|S|
P
v∈S (P (v|S ∪ x, IB ) − P (v|S, IB ))
=
K + |S|
P
K
v∈S (1 − P (v|S, IB ))
+
.
K + |S|
|S|
L
L + |T |
≥
v∈S (P (v|S
Proof. We will show that for two sets S ⊆ T ⊆ V − IB ,
and a node x ∈ V − IB , we have that ∀u ∈ V
P (u|S ∪ x, IB ) − P (u|S, IB ) ≥ P (u|T ∪ x, IB ) − P (u|T, IB )
(P (v|T ∪ x, IB ) − P (v|T, IB ))
L + |T |
P
L
v∈T (1 − P (v|T, IB ))
+
L + |T |
|T |
v∈T
X
[P (v|T ∪ x, IB ) − P (v|T, IB )]
v∈S
≥
X
[P (v|S ∪ x, IB ) − P (v|S, IB )] ,
v∈S
where the inequality follows from induction. We established
above that K + |S| ≤ L + |T |, which completes the proof of
(13).
For (14), we again use the fact that P (v|T, IB ) = 0 for
v ∈ T − S and obtain:
X
X
[1 − P (v|T, IB )] =
[1 − P (v|T, IB )]
v∈T
v∈S
≤
X
[1 − P (v|S, IB )] ,
v∈S
where the inequality follows from monotonicity. The fact
that |T | ≥ |S| gives (14).
800
700
700
600
600
Size of Active Set for A
Size of Active Set for A
800
500
400
300
200
400
300
200
Greedy A
High Degree A
Central A
100
0
500
0
20
40
60
80
100
k = |IA|
Figure 3: Distance-based model: high-degree IB
5. NUMERICAL SIMULATIONS
In this section we analyze the behavior of both models
and the resulting influence sets of each on a real network –
the coauthorship graph based on papers in theoretical highenergy physics. Empirical evidence suggests that coauthorship graphs are representative of typical social networks [19].
By choosing to run our experiments on the data from an actual social network as opposed to generating random graphs,
we are able to obtain results that are more specifically applicable to the motivations for our models.
The specific dataset we employed was the PROXIMITY
HEP-Th database based on data from the arXiv archive
and the Stanford Linear Accelerator Center SPIRES-HEP
database provided for the 2003 KDD Cup competition with
additional preparation performed by the Knowledge Discovery Laboratory, University of Massachusetts Amherst [20].
After minor preprocessing, the network consisted of 8392
distinct authors and 461 separate connected components (of
size at least 2), the largest of which contained 7034 authors.
We compared different choices for companies A and B,
where company B first chooses a certain subset of the nodes,
IB , unaware that company A will also try to enter the market, and company A subsequently targets a subset IA , after
which we look at the spreading of influence from IA and IB
according to the processes described in Section 3.
We ran simulations where the set IB was chosen according to several different heuristics. As discussed in [12] the
heuristics of choosing high-degree nodes and central nodes
are often used in the sociology literature to find influential
sets of nodes. Here the high-degree heuristic chooses nodes
in order of highest degree, while the central node heuristic chooses nodes with low average distance to other nodes.
The average distance is calculated by taking the average
of a node’s distance to all other nodes, where the distance
between unconnected nodes is the number of nodes in the
graph. In addition to these two heuristics, we also ran simulations where the nodes of IB were chosen according to the
greedy Hill Climbing Algorithm for the single technology
case [12].
We used each of these three heuristics to choose an initial
set IB of fixed size |IB | = 100 corresponding to a little more
than 1% of the nodes in the network. For each of these
IB sets, and for both diffusion models from Section 3, we
Greedy A
High Degree A
Central A
100
0
0
20
40
60
80
100
k = |IA|
Figure 4: Wave propagation model: high-degree IB
ran the greedy Hill Climbing Algorithm to determine the
most influential IA set, where |IA | ranged from 1 to 100
nodes. Since the problem of finding the best IA of a fixed
size is NP-complete, we compare the results of the algorithm
against two heuristics for choosing the IA set from V − IB :
high-degree nodes and central nodes.
As in Kempe et al. [12], we begin by giving each edge (u, v)
in the network a probability puv = .1 of being active. We
suppress the details of the simulation procedure employed,
but note here that when puv ∈ (0, 1), many random subgraphs must be generated to both obtain a node with the
largest marginal expected influence and to evaluate the overall influence of all methods.
Figure 3 compares the size of the market which product
A captures for increasing values of k = |IA | in the distancebased model from Section 4.1 if A uses different heuristics,
where the 100 nodes of IB are chosen according to the high
degree heuristic. Figure 4 is similar to Figure 3 but uses
the wave-propagation model from Section 3.2. Due to space
limitations, we only present results for the wave propagation result in the sequel. In all of the experiments which
we conducted, the Hill Climbing Algorithm for company A
outperformed the other heuristics. This can be attributed to
the fact that the Hill Climbing Algorithm takes into account
the effect of both the nodes in IB and the nodes already selected in IA .
In Figure 5, we fix company A’s strategy to the greedy Hill
Climbing Algorithm and compare the strategies of company
B. We see that for large enough |IA |, company B’s market
share is smallest when B used the Hill Climbing Algorithm
for a single technology, even though Kempe et al. [12] experimentally showed that in the absence of competition, this
algorithm performed best. This can be explained by the fact
that the greedy algorithm for a single technology iteratively
adds the node to IB that maximizes the expected number of
additional nodes influenced when there is no competition,
which inherently does not make the solution very robust
when an unexpected competitor also tries to influence nodes.
When |IA | > 5, the high degree heuristic helped company
B maintain the largest possible market share in all of our
experiments.
Figure 6 shows the percentage of the market captured
by A and B, respectively and in total, for increasing sizes
950
0.14
Greedy B
High Degree B
Central B
900
0.12
Percentage of Total Market
Size of Active Set for B
850
800
750
700
650
600
0.1
0.08
0.06
0.04
550
A
B
A+B
0.02
500
450
0
20
40
60
80
0
100
k = |IA|
0
20
40
60
80
100
k = |IA|
Figure 5: B’s share of the market against a greedy
IA (Wave propagation model)
Figure 6: Percentage of the total market captured
(Wave propagation model: high-degree IB )
of IA when using our greedy algorithm for choosing IA in
the wave propagation model. Since the growth in the total
number of adopters of A and B is slower than the growth of
A’s influence, this figure shows that A’s increase in market
share is due to both reaching new consumers and drawing
consumers away from its competitor. For IB chosen using
the high-degree heuristic, we see that B maintains a larger
market share even when |IA | = |IB | = 100. When B chooses
IB with either of the other two methods considered, A is able
to obtain a lead with a considerably smaller initial set.
Figure 7 shows how much larger an initial set each of the
heuristics require relative to the greedy algorithm’s initial
set to attain some specified level of influence. Here we see
that to influence 300-400 consumers, the high-degree nodes
heuristic requires an initial set which is approximately 15%
larger than that required by the greedy algorithm. In particular, this quantifies how much better the greedy algorithm
performs relative to the popular heuristics. This information
could also be used by A to determine the value of knowing precisely what consumers B will target, since the other
heuristics do not require this knowledge.
Figure 8 shows the marginal gain in influence which A
enjoys from targeting an additional consumer versus the
three different strategies for B. Here we observe that simply greedily targeting the most influential consumer yields
an approximate expected return of more than 90 eventual
adopters if B chose greedily and more than 10 eventual
adopters if B used the high-degree heuristic. We note that
given costs for targeting consumers, this figure could help a
company decide at what point the cost of targeting an additional consumer outweighs the marginal return expected
from this action.
network, in which the two technologies propagate in exactly
the same way. These models and our results can be easily
extended to handle additional competitors. Furthermore,
by adding dummy nodes to the network, our model easily
allows for the case when companies can target customers,
but the targeted customer adopts the technology only with
a certain probability.
We believe our results could also be extended to include
more general cases, for example by having different acceptance probabilities for the two technologies, or by allowing
the rate at which influence travels in the graph differ for the
two technologies.
From a game-theoretic perspective, the question we study
is that of finding a best response to the first player’s move
in a Stackelberg game. A natural next step would be to
study the optimal behavior of the first player, given that
she knows that the second player will use our approximate
best response, and ultimately to study the Nash equilibria
of this Stackelberg game. We have shown that A can obtain
a significant portion of a market despite choosing consumers
second. In our experiments, the heuristic that chooses high
degree nodes was the best strategy for the first player among
the three strategies we compared. If B doesn’t choose their
initial set in this way, A can in fact outperform B with
a relatively small budget. It would be interesting to find a
provably good heuristic for the first player. Other interesting
games that could be considered using our models are the
simultaneous version of this game, and the game where the
two players take turns in targeting nodes.
Lastly, using our first model with edge probabilities equal
to 1, these problems can also be seen in the context of competitive facility location [1, 4] on a network, but we are not
aware of any previous results for competitive location games
on a network.
6. CONCLUDING REMARKS
In this paper we studied the spreading of two competing
technologies, A and B, in a social network. We addressed
the question of finding an initial set of nodes to target for
technology A, given that the initial set of nodes adopting
technology B is known. To our knowledge, this work represents the first treatment of such questions. We proposed
two basic models for the spreading of technologies through a
7.
ACKNOWLEDGMENTS
We are grateful to Eric Friedman, Jon Kleinberg, David
Williamson, and an anonymous referee for their helpful comments on an earlier version of this paper. We also thank
David Shmoys, Christine Shoemaker, and David Williamson
for financial support.
Ratio of Inital Set Size to Greedy Initial Set Size Needed
2
90
Greedy A
High Degree A
Central A
1.9
Greedy B
High Degree B
Central B
80
1.8
70
1.7
60
1.6
50
1.5
40
1.4
30
1.3
1.2
20
1.1
10
1
50
100
150
200
250
300
Size of Active Set for A
350
0
400
Figure 7: The benefit of knowing IB , (Wave propagation model: high-degree IB )
20
40
60
80
100
k = |IA|
Figure 8: A’s gain in adding an early adopter (Wave
propagation model: greedy IA )
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