INDIGO: An In-Situ Distributed Gossip

INDIGO: An In-Situ Distributed Gossip Framework
for Sensor Networks
Paritosh Ramanan
Goutham Kamath
Wen-Zhan Song
Department of Computer Science
Georgia State University
{pramanan1,gkamath1}@student.gsu.edu, wsong@gsu.edu
Abstract—With the onset of Cyber-Physical Systems (CPS),
distributed algorithms on Wireless Sensor Networks(WSNs) have
been receiving renewed attention. The distributed consensus
problem is a well studied problem having a myriad of applications
which can be accomplished using asynchronous distributed gossip
algorithms on Wireless Sensor Networks(WSN). However, a
practical realization of gossip algorithms for WSNs is found
lacking in the current state of the art. In this paper, we propose
the design, development and analysis of a novel in-situ distributed
gossip framework called INDIGO. A key aspect of INDIGO is
its ability to execute on a generic system platform as well as
on a hardware oriented testbed platform in a seamless manner
allowing easy portability of existing algorithms. We evaluate the
performance of INDIGO with respect to the distributed consensus
problem as well as the distributed optimization problem. We also
present a data driven analysis of the effect, certain operating
parameters like sleep time and wait time have on the performance
of the framework and empirically attempt to determine a sweet
spot. The results obtained from various experiments on INDIGO
validate its efficacy, reliability and robustness and demonstrate
its utility as a framework for the evaluation and implementation
of asynchronous distributed algorithms.
Index Terms—Asynchronous Distributed Gossip, System Design, Distributed Consensus, Distributed Optimization
I. I NTRODUCTION
Sensor networks are becoming an important part of monitoring activities across various interdisciplinary domains.
They have been successfully applied to solve problems like
seismic activity monitoring and tomography[24], exploratory
geophysics [4], wildfire and wildlife monitoring[5] among
many things. Extracting optimal performance from sensors has
always been a challenge[6] and it has led to a flurry of active
research in recent times. Sensor networks come with their own
set of constraints which cannot be overlooked. For instance,
sensor networks often come with a very limited energy source,
which makes it imperative to use system resources judiciously
as well as keep communication costs at as minimum a level as
possible. It is also quite likely that due to energy constraints the
sensor network might be able to provide only limited amount
of bandwidth for data transfer, which makes communication
a more precious affair.
Therefore, recent state-of-the art research in the area of
sensor networks suggests that the trends appear to be focussing
on striking a balance between power consumption attributed
to communication and system utilization. With sensor nodes
becoming computationally more powerful and less resource
hungry, the bottleneck of communication as a barrier for
efficient utilization of system resources seems to persist. Due
to the rise of increasingly power efficient sensor nodes it now
makes more sense in some cases to delegate computation
based tasks to the nodes themselves than to have them use
up precious resources to depend on a central entity for computation. In recent times, the interleaving of the computational
aspect of sensor networks with that of physical processes such
as sensing has opened up new research avenues like CyberPhysical Systems [23] and in-network computing [12].
One such research problem in which the centralized approach to problem solving is less efficient than an in-network
approach is that of achieving consensus in a sensor network. Sensor nodes are heavily reliant on batteries. Wireless
transmission of sensed data requires bandwidth which consumes considerably more energy than processing data locally
[28],[29]. In some cases, for example in seismic networks,
the sensed data at a particular node does not vary drastically
in a spatial sense with respect to its neighbors. Therefore, in
case of applications like seismic sensing, one would be more
interested in obtaining the global picture with respect to the
data obtained from the network rather than focus on the fine
grained nuance of the data pertaining to each node. This would
typically involve the solving of a global optimization problem
as a function of the sensed data. By adopting a distributed
approach, we would also avoid loss of packets in and around
the sink node owing to congestion. In the light of these
observations, by using a decentralized, in-network approach
we could exploit the spatial co-relation of data among neighbor
nodes, avoid redundant transmission of data to a central entity
by pushing the computation to the end nodes and in turn hope
to save on energy consumption owing to costly transmissions
[27].
The consensus problem in sensor network epitomizes the
above-mentioned ideas. It deals with each node arriving at
a consensus of a measured parameter solely on the basis
of exchange of information with its neighbor nodes. As an
extension of the distributed consensus problem, the distributed
consensus optimization problem involves using consensus to
propagate information to other nodes in the network and then
solving a local optimization problem with constraints local to
each particular node.
It is in this regard that the problem of asynchronous distributed gossip has been proposed for consensus as well as
consensus optimization in sensor networks. The idea is to be
able to solve a computationally intensive problem by mutual
exchange of information among nodes. The very basic case
of distributed gossip is the distributed consensus problem. By
attacking the distributed consensus problem, we can expect to
solve much more computationally intensive problems.
The distributed gossip approach is a very promising one
in the world of Cyber-Physical Systems [22]. In the recent
past, a key implementation of CPS has been in the area of
seismic monitoring [25][24]. As an extension of the above
work, research is being conducted for performing seismic
tomography [3] in a distributed fashion. Seismic tomography
is the process of determining with good accuracy, a profile of
the earth under the surface. It is extremely helpful in the area
of geophysics for disaster planning and preparedness.
Currently, most tomography approaches use a centralized
technique where information is relayed to a sink in order to
solve a global optimization problem. However, with distributed
gossip, one can hope to minimize this cost, make the system
and the network more efficient and expect it to be more
reactive. In this regard distributed gossip techniques have an
edge over existing algorithms.
Although asynchronous distributed gossip protocols have
been well studied in theory, there is very little work done
with respect to characterizing the behavior and performance of
distributed gossip protocols on an actual WSN set up. There is
also not much study done in terms of performance characterization in solving a distributed consensus optimization problem
over a wireless network. In order to address these issues, we
present INDIGO, a novel in-situ distributed gossip framework
aimed at solving distributed consensus optimization problems
using distributed gossip techniques. INDIGO is a practical,
flexible and a highly versatile framework, which can be
seamlessly implemented on a specific hardware platform as
well as on a generic TCP/IP based network. This feature
enables a wide variety of use cases for INDIGO, from testing
and evaluation of novel distributed approaches to actual field
deployment. By incorporating two diverse gossip protocols,
i.e. broadcast and random, in its design, INDIGO also enables
users to conduct a rich set of tests and compare results
of distributed consensus optimization on an actual wireless
network. Our results indicate INDIGO’s strong performance
with respect to real world case studies in the field of seismic
sensing and a strong co-relation to the results as predicted by
theory.
The rest of the paper is organized as follows. Section
II talks about the existing state-of-the-art gossip algorithms
which INDIGO implements. Section III presents an overview
of the random and broadcast gossip protocol as implemented
under INDIGO and presents an empirical analysis of the
performance of the framework on the basis of some newly
introduced parameters like sleep time and wait time. Section
IV talks about how we have implemented the aforementioned
algorithms both on system and testbed platforms. Section
V demonstrates the various results we have obtained using
INDIGO and Section VI concludes the study by highlighting
the various aspects of the study as well as pointing at the
future direction of research in this area.
II. R ELATED W ORK
Distributed Gossip in sensor networks is a well studied
problem. The types of gossip can be broadly categorized into
three types i.e. broadcast, random and geographic [7][8][15].
Geographic gossip uses geographic routing, which is not
preferable in the case of our sensor network as it is hard to
implement on a proprietary hardware stack such as XBee. In
this study we limit ourselves to the domain of only broadcast
and random gossip and describe the various published works,
which have inspired this study. As already mentioned the main
aim of this study is to implement established gossip algorithms
on a system level and help in observing their behavior in
different scenarios.
Random Gossip was first proposed by Boyd et al. [7] based
on the asynchronous time model. Random Gossip chooses
nodes at random from its neighbors to exchange information
and calculate the average. The important thing about random
gossip is that at any time instant there can be only one
exchange taking place between two particular nodes. This
implies that while the process of averaging or gossip is going
on, no other third node can indulge either of the nodes
in gossip. It is only after both the nodes have successfully
performed gossip that they are free to choose other nodes to
perform gossip with at random. The paper also proves that the
algorithm converges to the true average and further goes on to
determine the convergence rate. It also provide upper bounds
with respect to the averaging time of the algorithms. These
conclusively provide sufficient evidence of the robust nature of
the random gossip algorithm. With respect to broadcast gossip
however, the work done by Aysal et al. in [8] prove that the
algorithm converges only in expectation. The paper also goes
on to provide a comparison between different approaches (i.e.
broadcast, random and geographic) in terms of the variance as
well as the mean squared error per node against the number
of radio transmissions with respect to different network sizes.
The work done in [31] provides a very good explanation of the
rate of convergence in a more practical setting by assuming
each link has a fixed delay.
Although both random and broadcast gossip aim to achieve
average consensus among nodes, their style of performing
gossip is radically different. While random gossip chooses
to perform gossip with its immediate neighbors, a node can
only perform gossip with only one other particular node at
any given time. Broadcast gossip on the other hand performs
gossip by broadcasting its values to its neighbors. While
random gossip is suited to any type of network with a static
topology, broadcast gossip is more relevant in case of wireless
sensor networks where the underlying communication pattern
is broadcast driven.
The work done by Dimakis et al. [11] presents a broad
overview of the recent developments in the area of gossip
protocols. It describes the convergence rate of gossip protocols
in relation to the number of transmitted messages as well as
energy consumption and also discuss about gossip characteristics over wireless links. Further, the work done by Denantes et
al. [16] presents an interesting evaluation on a mathematical
basis of certain metrics which may be useful in choosing
an apt algorithm for performing distributed gossip. Instead
of focussing on a time-invariant scenario, these metrics are
evaluated on the basis of time-varying networks culminating
in the provision of an upper bound on the convergence speed.
The work done by Braca et al. [17] investigate an important
and crucial problem of when to begin averaging and when to
end sensing. They propose an alternative novel approach of
running consensus where the sensing and averaging happen in
a simultaneous fashion. The paper [18] provides a very novel
application of gossip protocols. By investigating the problem
of consensus in a multi-agent system, it demonstrates a practical application of gossip protocols towards a Distributed Flight
Array (DFA). DFA is a set of multiple agents, which coordinate amongst themselves to arrive at a consensus and fly
in a variety of combinations While both the works [8] and [7]
present an astute theoretical analysis of their respective gossip
technique, they make a number of assumptions which may not
hold good in case of a real implementation.
The work done by Tsianos et al. [21] presents a practical
approach for asynchronous gossip protocols but they do not
use a bi-directional mechanism and opt for a one-directional
variant instead and their evaluations are performed on an MPI
cluster which has different constraints from an actual WSN.
We now proceed to provide a detailed explanation of the
problem to be solved coupled with an exhaustive overview of
the INDIGO framework design.
III. P ROBLEM F ORMULATION AND F RAMEWORK D ESIGN
A. Decentralized consensus optimization
A seismic tomography problem can be modeled as a linear
least squares problem of the following form.
1
xLS = arg min ||Ax − b||22
2
x
(1)
where x ∈ Rn , A ∈ Rm×n and b ∈ Rm . Equation 1 represents
the global least squares problem that needs to be solved. If
we let F (x) = minx 12 ||Ax − b||22 , Equation 1 transforms
into an optimization problem of minimizing the objective
function F (x) with respect to x. Given the high dimensional
nature of seismic tomography, the global system of equations
represented by Equation 1 is very large and as a result the
process of obtaining a good solution to the optimization
problem is a tedious affair. Further, solving such a problem
over a loosely connected and often unreliable Wireless Sensor
Network where each of the nodes hold part of the global
optimization puzzle is a challenge in its own right.
To solve this issue, we construct a decentralized approach
from the above system of equations, by partitioning A and
b row wise over p nodes of the network to yield A =
{A1 , A2 , . . . , Ap } and b = {b1 , b2 , . . . , bp } respectively. The
system of equations represented by Ai ∈ Rmi ×n and bi ∈ Rmi
form the subsystem at the ith node where each node holds a
part of the input data. This decentralized version in turn leads
to the formation of a relatively smaller, local optimization
problem with the following individual objective function
1
fi (x) = min ||Ai x − bi ||22
x 2
(2)
at the ith node. The work done in [30] proposes one such
decentralized algorithm which aims to solve this consensus
optimization problem. Hence, we obtain a decentralized consensus optimization of the following form.
n
minimize F (x) =
1X
fi (xi )
n i=1
(3)
where xi ,χi andfi are the local estimate of the observed
value and the local objective function on the ith node respectively. The individual optimization is solved using Bayesian
ART(Algebraic Reconstruction Technique)[20]. The Bayesian
ART is an iterative technique used to solve a system of
equations like those in Equation 1 or Equation 2 by driving
the solution towards the minima as pointed to by the gradient
of the objective function.
Therefore, we now only have to minimize each node’s
objective function independent of its peers and with the help of
mutual exchange of information, i.e. each node’s own estimate
of x, among neighbors we can expect a convergence among all
the nodes to a solution of the global optimization problem in
Equation 1. Mutual exchange of information occurs among
neighboring nodes with the help of the gossip algorithms
mentioned in the previous section.
At the very heart of each gossip algorithm is the intention
to obtain global average of a measured parameter. Therefore,
a gossip algorithm attempts to solve the following averaging
problem in an entirely distributed way.
n
x̄ =
1X
xi
n i=1
(4)
where, x1 , x2 . . . xn are the individual observations recorded
by each of the n nodes in the network. For instance, a bunch
of temperature sensing nodes measuring the temperature of a
room may do so by relaying their measured values to a central
sink or exchange information amongst themselves and arrive
at an average which would be the consensus.
With the help of the INDIGO framework we can now
apply the concepts illustrated above to solve our problem of
seismic tomography. First, we construct a local optimization
problem at each node in the network. Nodes then average their
respective local estimate of x with their neighbors using the
gossip protocols mentioned in the previous section. Once the
averaging has taken place, the local optimization problem is
solved by each individual node to obtain its own next estimate
of x.
B. INDIGO Framework Overview and Design
As described in the previous section, gossip protocols can
be broadly categorized into random and broadcast gossip
protocols. In this section we present a novel and practical
framework design that aims to bring forth the true spirit of
the aforementioned protocols. INDIGO has the capability to
be configured to execute either the broadcast or the random
gossip protocol at run time. The idea is to create a flexible
framework design, which can be extended into a platform on
the basis of which various algorithms can be evaluated upon.
Let us consider a graph G(V, E), with V, E being the vertex
set and edge set respectively. Since distributed gossip occurs
among neighbors, we denote the neighborhood of any node
i ∈ V as follows,
Ni = {j|j ∈ V, Wij = 1},
(5)
where W is the adjacency matrix of graph G.
j1
j1
j3
j4
i
i
j4
j3
j2
(a) Broadcast Gossip
j2
(b) Random Gossip
Fig. 1. Illustration of Random and Broadcast gossip with respect to Node i
and its neighborhood jk ∈ Ni , ∀k ∈ {1, |Ni |}
The actual model followed by broadcast and random gossip
algorithms is an asynchronous time model, which models a
rate 1 Poisson clock on each node [7][8]. We introduce the
concepts of exclusivity and stochasticity of the framework as
an approximation to enforce this behavior. Exclusivity implies
that a node when in the process of performing gossip cannot
entertain gossip requests from a third party node, thereby
discarding any other packets until the ongoing gossip exchange
succeeds. An important outcome of exclusivity is that the
node which is soliciting has no way of knowing whether its
destination has received its request or not. In a real setting it
is important to take into account the fact that, packets may
get lost and moreover, even if the packet is received, the
destination might be involved in gossip with some other of
its neighbor and may simply discard this request. If these
situations are not handled properly, the gossip protocol may
never terminate or worse it may lead to contradictory results.
In order to solve this problem a concept of wait time, denoted
by σ is introduced. It denotes the duration of time any node
waits before it deems the gossip exchange to have failed. Wait
time insulates nodes from the phenomenon of waiting forever
to hear from their solicited neighbors and also handles the
aspect of packet loss. With the wait time concept in place, if
the packet has not been received or has been discarded by the
receiver, the sender can resume gossip afresh.
Another important feature that needs to be preserved is
the stochastic nature of the gossip process. There has to
be a degree of randomness associated when a particular
node begins gossip. Failure to maintain this feature would
lead to a deterministic output. Absence of this feature may
also cause deadlock among nodes or cause a heavy rate of
failure of gossip exchanges. To maintain stochastic behavior
a parameter known as maximum sleep time, denoted by ρ has
been introduced which is nothing but an upper bound on the
random interval of time a node sleeps before attempting a
gossip exchange.
We now describe the various terminologies related to both
random and broadcast gossip and proceed to give a detailed
description of the sequence of events in each. Figure 1 provides the pictorial representation of both random and broadcast
gossip protocols.
• xself : The estimate of a node’s measurement where
xself ∈ Cn×1
• σ: The maximum duration of time after which a gossip
exchange is deemed a failure.
• ρ: The upper bound on the random interval of time a
node sleeps before initiating gossip.
• M: The maximum number of gossip updates to be
performed by all nodes.
• Ni : The neighborhood of node i.
• recv(k, xk ): An estimate xk recieved from node k.
• send(k, xself ): A node’s self estimate unicasted to node
k.
• χself = [xj . . . xj+m ], matrix of values recieved from m
nodes to be averaged where χ ∈ Cn×m
• broadcast(xself ): A node’s self estimate broadcasted to
all neighbors.
C. Random Gossip
Based on the above features and using aforementioned terminologies we have Algorithm 1 which describes the Random
Gossip protocol encapsulated as a function. In the beginning
Algorithm 1 Random Gossip Algorithm
function RANDOM-GOSSIP (σ, ρ, M, xself )
while updates < M do
sleep for time t, s.t. 0 ≤ t ≤ ρ
if solicited by j ∈ Nself with value xj then
(x +x
)
xself = j 2 self
send(j, xself )
updates ← updates + 1
else
pick random neighbor j ∈ Nself
send(j, xself ) and start timer for σ
if recv(j, xj )&!timer.expire() then
xself = xj
updates ← updates + 1
end if
end if
end while
return xself
end function
of each batch of gossip each node goes to sleep for a random
interval of time t ≤ ρ. A node wakes up from sleep and
chooses a random peer from its routing table and solicits an
average. It starts a timer for t ≤ σ in order to wait for the
solicited node to respond. If a node is in solicitation mode,
it will discard any other solicitation request by a third party
node. The σ timer expires with the solicited node failing to
respond. In such a case the node again goes to sleep for a
random interval of time t ≤ ρ. The solicited node responds
before timer expires. It updates its current value with the newly
received value and goes to sleep for time t ≤ ρ. A node wakes
up from sleep and finds that there is already a request for
average by one of its peer. In such a case the node performs
the average and sends back the result to the solicitor node.
This process is summarized by Figure 2(b) which summarizes
the sequence of events discussed in Algorithm 1.
D. Broadcast Gossip
Broadcast gossip varies from random gossip in its demand
for exclusivity. Since broadcast gossip exploits the underlying broadcast nature of the network, there is no explicit
requirement for exclusivity. However, in broadcast gossip, a
node still needs to maintain the stochastic nature and for this
purpose the concept of maximum sleep time is maintained.
Also, in broadcast gossip, a node is expected to wait for
receiving values from its neighbors. During this process, there
should be a way to determine when to stop accepting the
values and perform the average. This can be done in two
ways, either wait for a fixed number of neighbors to respond
and then do the average or wait for a fixed amount of time
and do the average with whatever values have been received
until then. Logically, the latter is a better way due to many
reasons. Firstly, this technique does not depend on the node
degree. Secondly, it does not go into an indefinite wait on
not receiving anything from a fixed set of neighbors. Lastly,
it preserves the stochastic and asynchronous nature of the
algorithm. Therefore, we incorporate the concept of wait time
to mark the cut-off time for performing the average. While
the average is being computed any received requests will be
dropped. Based on the above features Algorithm 2 presents the
algorithm for the broadcast gossip protocol encapsulated as a
function. In Broadcast Gossip too each node goes to sleep
for a random interval of time t ≤ ρ. A node that has just
woken up from sleep and broadcasts its value to neighbors. It
then waits for interval of time t ≤ σ. It performs the average
with whatever values have been received in the interim period
and again goes to sleep for random interval of time t ≤ σ.
Figure 2(a) summarizes the sequence of events disscussed in
Algorithm 2. We will now turn our attention to the effects ρ
and σ have on the gossip performance.
E. Sweet Spot Analysis
It is of primary interest to determine whether these parameters have any bearing on the success of a gossip exchange.
Moreover, it is also of importance to find out whether there
exists a Sweet Spot, i.e a range of values of ρ and σ value
which could yield a near optimal probability of success. To
Algorithm 2 Broadcast Gossip Algorithm
function BROADCAST-GOSSIP (σ, ρ, M, xself )
while updates < M do
broadcast(xself )
sleep for time t, s.t. 0 ≤ t ≤ ρ
χ ← null
no of msgs ← 0
start timer for σ
while !timer.expire() do
recv(j, xj ), ∃j ∈ Nself
χ[no of msgs] = xj
no of msgs ← no of msgs + 1
end whilePno of msgs
( i=1
χ[i])+xself
xself =
no of msgs+1
updates ← updates + 1
end while
return xself
end function
accomplish this, numerous experiments were conducted with
0 ≤ ρ ≤ 10 on a 3×3 simulation setup configured for random
gossip. We varied the value of σ with respect to ρ and plotted
the average probability of success of each gossip exchange.
The result is presented in Figure 3 Figure 3 depicts the ps , the
probability of success on the y-axis and the ρ values on the xaxis respectively. The probability of success ps is determined
by the relation,
n
X
Nsi
ps =
(6)
Nti
i=1
where Nsi is the total number of successful gossip attempts
and Nti is the total number of attempts obtained on the ith
node. Each curve in Figure 3 represents a particular relation
between ρ and σ. With σ being the dependent variable and ρ
being the independent variable, we collect values for a variety
of combinations of ρ and σ. From the figure, it can be observed
that there indeed exists a sweet spot for the set of relations ρ =
kσ where 0 ≤ k ≤ 1 while for the relation ρ = 2σ, the value
of ps turns out to be sub optimal.Although this experiment is in
no way exhaustive and further trends may emerge on detailed
analysis with other values of ρ, σ, we can draw a number
of inferences from this figure. Firstly, the trends follow the
intuitive notion that if the maximum time a node can sleep
is less than the maximum time it is ready to wait then the
probability of success increases and vice versa. Secondly, with
further reduction in the ratio ρ : σ, there appears to be a
saturation point and further decrease will not yield greater
improvement. Lastly, for this network setup, the region around
ρ ≥ 6 seems to be a favorable position because in all relations,
there is a noticeable improvement of performance. From this
analysis it becomes quite clear that ρ, σ do have an effect on
the probability of success of gossip exchanges and there does
exist a sweet spot for these values.
The sweet spot value depends quite heavily on the underlying graph characteristics. With the use of linear regression and
RANDOM-GOSSIP( , ⇢, M, xself )
BROADCAST-GOSSIP( , ⇢, M, xself )
NEIGHBOR SOLICITS AVERAGE
•
solicited by j 2 Nself with value xj
•
xself =
broadcast(xself )
•
= null, no of messages = 0
if updates  M
• sleep for time t, s.t 0  t  ⇢
else
• return xself
•
•
•
•
after timer expires
•
xself =
•
increment updates
(
Pno of
start timer for and while timer
active
receive xj , 9j 2 Nself
msgs
[i]) + xself
no of msgs + 1
i=1
1
•
•
0.8
Ps
0.2
5
6
7
8
9
10
� in secs
Fig. 3.
•
•
•
if j sends back average before timer expires
xself = xj
increment updates
a testbed platform as well comprising of BeagleBone Black
coupled with an XBee radios. Since the testbed platform is
an indoor setup, the nodes form a network, which resembles a
complete graph due to close radio proximity. A unique feature
of INDIGO is its platform agnostic way of functioning which
provides a flexible, rich and diverse testing environment. We
draw a comparison between the two before proceeding towards
evaluation with the help of case studies. Figure 4 depicts a
schematic comparing the design of the testbed and the system
platforms.
0.4
4
pick random neighbor j 2 Nself
send xself to j , start timer for
Flow Diagram depicting Broadcast and Random Gossip algorithm
0.6
3
increment updates
(b) Random Gossip
� = 2�
�=�
� = 1/2�
� = 1/3�
2
send xself to j,
•
AVERAGE SOLICITED BY SELF
Fig. 2.
1
•
if updates  M
• sleep for time t, s.t 0≤ t ≤ ⇢
else
• return xself
[no of msgs] = xj
no of msgs + +
(a) Broadcast Gossip
0
(xj + xself )
2
Sweet Spot Analysis
hypothesis testing methods, one could determine the optimum
value based on empirical data pertaining to a given network
graph.
In the following sections, we discuss the implementation
details and a testbed setup description of INDIGO before
proceeding forward to analyze the results in the form of
various case studies.
IV. S YSTEM I MPLEMENTATION AND T ESTBED D ESIGN
In this section, we describe in greater detail, the technical
aspects of two evaluation platforms, i.e. a system platform and
a testbed platform. System platform is intended to provide
a generic evaluation platform using the standard TCP/IP
stack based wireless mesh network. Although for evaluation
purposes, such a robust system platform should be sufficient,
we also require a testbed platform to emulate on-field environments using the very same hardware, which would be used
for deployment. Hence we propose and eventually describe
A. System Design
We utilize a mesh network model for implementing INDIGO. Mesh networks are those in which each node not
only communicates with its peers but also serves as a relay
point by facilitating the transfer of messages between two
different nodes. Since maintaining proper end-to-end connectivity in a mesh network is a costly affair due to low link
reliability, we employ a mechanism known as the Bundle
Layer which is a delay tolerant technique of transmission.
The key objective behind the Bundle Layer is to improve
reliable transmission over wireless media over the TCP/IP
stack. To accomplish this the Bundle Layer breaks down the
notion of end-to-end among the various hops in between which
would significantly reduce retransmission of packets. Under
the Bundle Layer lies the actual transport layer which uses
normal TCP and beneath which runs a distance vector routing
protocol known as BATMAN (Better Approach to Mobile Adhoc Networking)[2]. The advantage of BATMAN lies in the
fact that routing overhead is minimized by maintaining only
the next hop neighbor entry to forward messages to instead
of maintaining the full route to the destination. The Bundle
Layer along with BATMAN ensure reliable transmission of
messages between source and destination.
B. Testbed Design
Our testbed setup comprises of the BeagleBone Black(BBB)
interfaced with the XBee radio. The BBB is an inexpensive
Application Layer
• Distributed Consensus
• Distributed Optimization
Gossip Layer
Broadcast Gossip
Logic
Random Gossip
Logic
receive
neighbor info
Testbed Platform
System Platform
sendBndl(dest,val) recvBndl(src,val)
Bundle Layer
Cache Management Sub-Layer
Queue in Mem
2 nd Queue
persist
in Disk
sendBndl(dest,val) recvBndl(src,val)
Serial Port Layer
Linux Buffer Management
fcntl.h
Serial Port File Descriptor
Synchronized I/O(SYNC),RDRW, BLOCK
load
send2nexthop
recv
Convergence Sub-Layer
Bundle
TCP
Adapter
Serial Port File
ACK
ACKACK
Transport Layer
Unicast
TCP
Network Layer
BATMAN
Fig. 4.
XBee DigiMesh MAC
Layer
MAC Routing
Mesh network formulation
Reliability
Address Table
System design and Testbed design : A comparison
small palm sized computer which runs the Angstrom operating
system which is a flavor of embedded linux. The BBB has a
memory of 512 MB and has a single core CPU with clock rate
of 1GHz. For radio communication we use the XBee PRO S3B
been abstracted by XBee including routing and mesh network
capability. The programmable control allows us to operate
the XBee in a variety of modes, which makes it application
flexible. Among the most important features, we could set
the Power Level(PL) parameter which indicates the amount
of power consumed during transmission. During run time, we
can issue commands encapsulated in a pre-decided frame and
pass it on to the device and expect to get encapsulated replies.
Through programmable control one can even choose from a
variety of sleep patterns already offered by the device. This
greatly simplifies the process of deployment by having a robust
network maintenance framework. Figure 5(a) presents a blow
up of the different components which go into making one
node on our testbed platform, while Figure 5(b) shows how
the various hardware components fit together. For interfacing
the BBB with the XBee it is configured as a peripheral UART
(Universal Asynchronous Receiver Transmitter). Using the
device tree overlay we are able to bring up a serial port for
communication with the XBee. This serial port is memory
mapped to the on board memory of the underlying XBee.
Once this configuration is in place, we can communicate
with the XBee and its peers through this serial port. For
accomplishing this we have developed a host of XBee specific
functions for sending and receiving information. The hallmark
of these functions is that they allow for a flexible operation of
the XBee with varying message types and message lengths.
Figure 6 provides an overview of the XBee message structure
for conducting distributed gossip. Another interesting point to
note is that through a configuration of the serial port through
the POSIX compliant serial port libraries in Linux, we can
man this serial port with the effect of achieving simultaneous
receiving and transmitting of data.
V. C ASE S TUDIES
This section focusses on the application based evaluation of
INDIGO. We focus on two forms of evaluations.
• TYPE 1: Distributed consensus gathering of the form.
n
x̄ =
•
1X
xi
n i=1
(7)
TYPE 2: Distributed consensus optimization of the form.
n
minimize F (x) =
(a) Blowup of the BBB-XBee Node
1X
fi (xi )
n i=1
(8)
subject to xi ∈ χi
(b) Interfacing of BBB with XBee
Fig. 5.
Actual Testbed Node setup involving BBB and XBee
900 MHz version which is mesh network capable. The module
comes with an onboard flash memory of 512 bytes and has
a Freescale MC9S08QE32 micro controller which allows for
programmable control. Various network functionalities have
We start with the simple case of distributed consensus gathering, which is of TYPE 1 in both the system as well as
the testbed platform. Then we move to more complex cases
like distributed event location on the testbed and finally to
distributed tomography computation on a simulation setup
which are problems of TYPE 2. For the system evaluation
platform we employ a network emulator named CORE [1].
CORE creates virtual Network Interface Cards (NICs) for a
specific network on a single host machine allowing emulation
of actual network settings. The advantage of CORE is that
1:start delimiter
4:frame type
5:frame ID
15-16:XBee
Reserved Bytes
18:transmit
options
0x7E MSB LSB 0x10 0x01 64-bit destination address 0xFF 0xFE 0x00 0x00
length of packet from
4th Byte to CheckSum
2:MSB of length
3:LSB of length
6:MSB of destination address
14:LSB of destination address
Data Payload
CHK
DISTRIBUTED EXCHANGE TYPE
17:broadcast
radius
19 :Type
DIST_EX
CHG
Fig. 6.
N:checksum
20-23:
SIZE OF
VECTOR
VECTOR AS A BYTE STREAM
Distributed Gossip XBee Message Structure
traditional Unix like environment can be obtained on each of
the nodes in the network which makes porting code to actual
physical devices from the virtual nodes straightforward. For
the testbed evaluation platform, we use the testbed consisting
of 6 BBBs each connected to an XBee. The BBBs are
connected to an Ethernet switch which is in turn connected to
a host machine. While the distributed gossip occurs amongst
the BBBs using the XBee radio, the Ethernet interface helps
maintain control of the gossip process with a rich set of scripts
via the host machine.
A. Simple Consensual Average
Distributed gossip protocols are evaluated [19] on the basis
of their ability to converge to consensus based on two different
types of initializations of data i.e. slope and spike initialization.
We plot the values on each node in the experiment at each
iteration to track and demonstrate convergence. Since there
are two setups, the system and the testbed, we conduct
experiments relating to each of the initializations on each of
the setups leading to a total of 8 combinations as depicted in
Figure 7 and in Figure 8. All the experiments were performed
on the testbed platform using 6 Beaglebone Blacks and XBees
and on system emulation platform comprising of 9 nodes with
ρ = 3 and σ = 3.
1) Slope Initialization: All nodes in the network are initialized with a scalar value x = k ∗ nodeId, where k is constant
for all the nodes. The resultant set of values form a slope on
a network of nodes. It is expected that on termination of the
gossip protocol, the slope will have given way to a flat surface
tending to average of the initial set. Figure 7 depicts the gossip
trends arising out of slope initialization on the testbed platform
and the system platform. As can be seen from the figure, the
gossip yields very good results, with the protocol converging
to a consensus which falls under a very close margin of the
actual average.
2) Spike Initialization: All but one of the nodes is initialized to a very high scalar value and the rest are set to 0.
With this initialization it is expected that all the nodes will
have the average of the spike value on termination. Figure 8
depicts the gossip trends using a spike initialization on. While
the random gossip scheme performs well and converges to
consensus within a close margin of average, the broadcast
gossip converges to a consensus but isn’t close to the actual
average. This is expected behavior as it has been anticipated in
[8] that broadcast gossip only converges to average consensus
in expectation.
The results obtained in this section demonstrate the robustness of the INDIGO framework in successfully realizing the
gossip algorithms with respect to a real world scenario and is
commensurate with what was expected in theory.
B. Distributed Event Location
Distributed Event Location is a process of localizing a
seismic event. This is done through a process known as
Geigers method [9] wherein a system of equations of the form
represented in Equation 8 is solved. Therefore, distributed
event location falls under TYPE 2. We can solve these system
of equations using any least squares technique like Bayesian
ART [20]. The whole idea behind this experiment is to make
the process of Event Location as mentioned in [9] distributed.
We are primarily interested in the rate of decrease of error of
the location vector as an indication of success in localizing the
seismic event. Therefore, we plot the relative error against the
iterations to demonstrate effective event location. The relative
error η is calculated as
ηi =
||xi − x∗ ||
,
||x∗ ||
(9)
where i is the iteration number and x∗ is the ground truth and
||.|| is the 2 norm of the vector. For performing this experiment
we used the system testbed which comprised of 6 Beaglebone
Blacks communicating with each other using the XBee radio.
Figure 9 represents the experiment involving random and
broadcast gossip while performing distributed event location
for one particular event where the y-axis represents the relative
error. We observe a monotonously decreasing error trend in
Figure 9(a) and Figure 9(b) before reaching an acceptable
error margin in both the random and broadcast gossip case.
Figure 10 shows the number of packets lost while performing
distributed event location among the different nodes in both
cases. From Figure 9 and Figure 10, we can safely assert
that the framework can tolerate packet losses observed in the
network. As a result, each node solves its local system of
equations referred to by Equation 8 by using an initial guess.
Next, it generates the new x value and performs gossip with
40
30
20
20
10
10
0
0
5
10
15
0
20
70
60
90
Node 6
Node 7
Node 8
Node 9
average
Node 1
Node 2
Node 3
Node 4
Node 5
80
70
60
50
40
0
5
10
Iteration Number
15
20
25
40
30
20
20
0
30
50
30
10
Node 6
Node 7
Node 8
Node 9
average
Node 1
Node 2
Node 3
Node 4
Node 5
80
Value
30
90
Node 1
Node 2
Node 3
Node 4
Node 5
Node 6
average
50
Value
40
Value
60
Node 1
Node 2
Node 3
Node 4
Node 5
Node 6
average
50
Value
60
10
0
5
10
Iteration Number
15
0
20
0
5
10
15
Iteration Number
20
25
30
35
40
Iteration Number
(a) Broadcast Gossip Slope Intializa- (b) Random Gossip Slope Intializa- (c) Broadcast Gossip Slope Intializa- (d) Random Gossip Slope Intialization on Testbed
tion on Testbed
tion on System Emulation
tion on System Emulation
60
80
60
40
40
20
0
0
5
10
15
20
140
Node 1
Node 2
Node 3
Node 4
Node 5
Node 6
average
100
Value
80
Value
120
Node 1
Node 2
Node 3
Node 4
Node 5
Node 6
average
100
100
140
Node 6
Node 7
Node 8
Node 9
average
Node 1
Node 2
Node 3
Node 4
Node 5
120
Value
120
Results of Slope Initialization
100
80
60
80
60
40
40
20
20
20
0
0
0
5
Iteration Number
10
15
20
Iteration Number
0
5
10
15
20
Node 6
Node 7
Node 8
Node 9
average
Node 1
Node 2
Node 3
Node 4
Node 5
120
Value
Fig. 7.
0
0
5
10
Iteration Number
15
20
25
30
35
40
Iteration Number
(a) Broadcast Gossip Spike Intializa- (b) Random Gossip Spike Intializa- (c) Broadcast Gossip Spike Intializa- (d) Random Gossip Spike Intialization on Testbed
tion on Testbed
tion on System Emulation
tion on System Emulation
Fig. 8.
0.12
0.1
0.09
0.08
0.1
0.095
0.09
0.085
0.08
0.07
0.06
Node 1
Node 2
Node 3
Node 4
Node 5
Node 6
0.11
0.105
Relative Error
Relative Error
0.115
Node 1
Node 2
Node 3
Node 4
Node 5
Node 6
0.11
Results of Spike Initialization
0.075
0
20
40
60
80
100
120
0.07
0
20
40
Iteration Number
(a) Random Gossip Trends
60
80
100
120
Iteration Number
again generate a new estimate of x and the process continues
till a given tolerance is reached or the maximum number
of iterations are reached. This technique embodies a true
asynchronous gossip approach as the objective function being
solved is directly coupled with exactly one gossip update.
With this result, it becomes apparent that INDIGO can be
fruitfully applied to solve the event location problem in a
distributed way.
(b) Broadcast Gossip Trends
C. Distributed Seismic Tomography
Fig. 9. Results of Distributed Event location using random and broadcast
gossip performed on the Testbed
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minimize ||x||
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subject to Ax = b
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Another application of INDIGO is to perform distributed
seismic tomography [26] which is a TYPE 2 problem and can
be modeled as a distributed consensus optimization problem.
Centralized seismic tomography involves solving an objective
function of the type,
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Fig. 10. Packet Loss of Random and Broadcast Gossip while performing
Distributed Event Location with 100 iterations
some other of its neighbor node. After the completion of this
gossip exchange, it uses the obtained x value as basis to
(10)
where x ∈ Cn , A ∈ Cm×n , b ∈ Cm and i ∈ {0, n}.
In distributed seismic tomography, k th node has its own bk
and Ak and an initial xkinit which it uses to solve a local
optimization problem (LOP). referred to by Equation 10.
However, in the distributed scenario, the k th node performs
a gossip update with its neighbor(s) to obtain a new estimate
of its value xk . This value is in turn used to solve the local
optimization problem and the process repeats till a threshold
is reached. In other words, the distributed gossip and the LOP
are tightly coupled leading to true asynchronous behavior.
To execute this problem on INDIGO, we used a synthetic
data model. Our resolution was 16 × 16, which meant that
our x matrix was of size 256. Our setup was simulated on a
||Axi − b||
ηi =
||b||
(11)
||xi − xgt ||
||xgt ||
(12)
βi =
Relative residual at each iteration helps in determining how
close the system is to the actual observed parameters denoted
by the vector b. The relative error helps determine how close
we are to the actual ground truth. In order to depict the
uniformity in convergence among all nodes, we employ an
error bar technique of plotting the results. The points on the
curve denote the mean relative residual and the mean relative
error in Figure 11 and Figure 12 respectively among the
49 nodes in the network while the vertical bars denote the
standard deviation observed at each iteration. We are able
to assert with certainty that the behavior of all the nodes
is monotonously decreasing, consistent with theory and there
were no rapid deviations in any node at any point of time.
The figures therefore provide a holistic picture in relation to
convergence to the centralized solution without loss or underrepresentation of any facet of the experiment.
Relative Residual Error (η)
1
ηRG
ηBG
0.8
0.6
0.4
Fig. 11.
βRG
β
BG
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
10
Fig. 12.
20
30
40
50
60
Iteration Number
70
80
90
100
Distributed Seismic Tomography relative error(β)
Observing both figures, one can instantly notice a healthy
converging trend with respect to relative residual norm and
the relative error norm. There is a slight jitter in case of
random gossip as against broadcast gossip due to the fact that
random gossip needs to maintain exclusivity with respect to
averaging. The standard deviation of broadcast gossip seems
to reduce much more drastically owing to a higher degree
of mixing among neighbors leading to a higher flow of information through the network. This experiment conclusively
demonstrates a real world working implementation of the
INDIGO framework for solving the decentralized consensus
optimization problem with the help of distributed gossip
protocols.
Lastly we examine the communication cost in terms of the
number of messages sent and received, depicted in Figure
13. The number of messages is plotted as a function of the
grid points on the XY plane representing the nodes. These
messages include the total number of incoming and outgoing
messages handled by the wireless radio. While random gossip
exhibits a relatively uneven surface in Figure , broadcast gossip
has a highly consistent communication cost among nodes as
depicted in 13(b). This fact can be attributed to the relatively
higher stochastic nature of random gossip as compared to
broadcast gossip.
From the above discussions on the various applications
and investigations into the behavior of gossip protocols in
each, it becomes apparent that INDIGO is indeed a versatile
framework capable of providing an evaluation platform for a
myriad of algorithms and problems.
VI. C ONCLUSION
0.2
0
0
1
0.9
Relative Error (β)
network comprising of 49 nodes, arranged in a grid topology.
The key idea being that a node initially generates an estimate
of vector x using Bayesian ART to solve the LOP. It performs
performs gossip with neighbor(s) and obtains a new value of
x. This value is then used as a basis for computing the next
estimate of x and the process repeats.
Our objective is to show that by using distributed gossip
algorithms, the system converges to a solution obtained by
solving the centralized form of the same problem. For this
reason, we use the least square solution of the centralized
form of Ax = b, denoted by xgt as our ground truth. We
evaluate our results based on two parameters, η being the
relative residual and β being the relative error with respect
to the ground truth.
10
20
30
40
50
60
Iteration Number
70
80
90
100
Distributed Seismic Tomography relative residual(η)
Figure 11 depicts the error bar of the relative residual
value η for both the random and broadcast gossip experiments
each of which have performed 100 successful gossip updates.
Figure 12 depicts the error bar of the relative error value β
for both types of gossip, comprising of 100 successful gossip
updates.
This work focusses on the design, development and evaluation of INDIGO, a distributed gossip framework design for
sensor networks. Distributed gossip has been proposed as a
more efficient way for solving a global optimization problem
with respect to spatially co-related data. Distributed consensus
optimization employs gossip techniques to solve local optimization problems as a precursor to solving the global one.
We incorporate the random and broadcast gossip models in our
framework owing to their high suitability to our application
domain of seismic sensing. We present a practical framework
design which, realizes the true nature of asynchronous gossip
Number of Message
deployment.
Future work in this domain involves deeper analysis of
the effect of framework parameters on the convergence of
the optimization algorithms. We are also investigating the
extension of INDIGO to construct an asynchronous and purely
decentralized MPI like version for sensor network.
In conclusion it can be said that INDIGO is indeed an
efficient and robust gossip framework and can be applied
practically to any scenario which warrants asynchronous distributed consensus or distributed consensus optimization and
get reliable results.
4000
2000
0
2
4
6
X
2
4
6
Y
Number of Message
(a) Communication Cost Random Gossip
8000
6000
4000
2000
0
2
4
6
X
2
4
6
Y
(b) Communication Cost Broadcast Gossip
Fig. 13.
Distributed Seismic Tomography Communication Cost
and serves as a highly versatile setup for testing and evaluation
for distributed algorithms. We show that using INDIGO, we
could perform distributed consensus optimization to solve real
world practical problems in seismic domain. We characterize
the effect of our framework parameters on the chance of
success of gossip attempts before moving on to evaluation
of INDIGO in the domain of seismic sensing.
We apply INDIGO to solve two significant problems in
seismic sensing, event location and distributed tomography.
We demonstrate the flexibility of INDIGO by yielding concurrent results on both the system implementation as well as
the testbed set up. The results indicate a strong performance
of INDIGO even on a testbed comprising of low-powered
devices like the BBB and XBee. The results obtained on the
system implementation go on to show that INDIGO has the
capability to perform well even on a standard TCP/IP stack
based wireless network. By ensuring seamless portability of
algorithm between the two setups, INDIGO can be used in
a wide variety of ways starting from testing and evaluation
of different seismic algorithms all the way to actual field
R EFERENCES
[1] J. Ahrenholz, C. Danilov, T. Henderson, and J. Kim, “Core: A realtime network emulator,” in Military Communications Conference, 2008.
MILCOM 2008. IEEE, pp. 1–7, Nov 2008.
[2] D. Seither, A. Konig, and M. Hollick, “Routing performance of wireless
mesh networks: A practical evaluation of batman advanced,” in Local
Computer Networks (LCN), 2011 IEEE 36th Conference on, pp. 897–
904, Oct 2011.
[3] G. Kamath, L. Shi, and W.-Z. Song, “Component-average based distributed seismic tomography in sensor networks,” in Distributed Computing in Sensor Systems (DCOSS), 2013 IEEE International Conference
on, pp. 88–95, May 2013.
[4] L. Shi, W.-Z. Song, M. Xu, Q. Xiao, J. Lees, and G. Xing, “Imaging
seismic tomography in sensor network,” in Sensor, Mesh and Ad Hoc
Communications and Networks (SECON), 2013 10th Annual IEEE
Communications Society Conference on, pp. 327–335, June 2013.
[5] D. Anthony, W. Bennett, M. Vuran, M. Dwyer, S. Elbaum, A. Lacy,
M. Engels, and W. Wehtje, “Sensing through the continent: Towards
monitoring migratory birds using cellular sensor networks,” in Information Processing in Sensor Networks (IPSN), 2012 ACM/IEEE 11th
International Conference on, pp. 329–340, April 2012.
[6] J. Yick, B. Mukherjee, and D. Ghosal, “Wireless sensor network survey,”
Computer Networks, vol. 52, no. 12, pp. 2292 – 2330, 2008.
[7] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, “Randomized gossip algorithms,” Information Theory, IEEE Transactions on, vol. 52, pp. 2508–
2530, June 2006.
[8] T. Aysal, M. Yildiz, and A. Scaglione, “Broadcast gossip algorithms,”
in Information Theory Workshop, 2008. ITW ’08. IEEE, pp. 343–347,
May 2008.
[9] L. Geiger, “Probability method for the determination of earthquake
epicenters from the arrival time only,” Bull.St.Louis.Univ, vol. 8, pp. 60–
71, 1912.
[10] G. T. Herman, Fundamentals of Computerized Tomography: Image
Reconstruction from Projections. Springer Publishing Company, Incorporated, 2nd ed., 2009.
[11] A. Dimakis, S. Kar, J. Moura, M. Rabbat, and A. Scaglione, “Gossip
algorithms for distributed signal processing,” Proceedings of the IEEE,
vol. 98, pp. 1847–1864, Nov 2010.
[12] C. Li and H. Dai, “Efficient in-network computing with noisy wireless
channels,” Mobile Computing, IEEE Transactions on, vol. 12, pp. 2167–
2177, Nov 2013.
[13] N. M. Shapiro, M. Campillo, L. Stehly, and M. H. Ritzwoller, “Highresolution surface-wave tomography from ambient seismic noise,” Science, vol. 307, no. 5715, pp. 1615–1618, 2005.
[14] F.-C. Lin, D. Li, R. W. Clayton, and D. Hollis, “High-resolution
3d shallow crustal structure in long beach, california: Application of
ambient noise tomography on a dense seismic array,” GEOPHYSICS,
vol. 78, no. 4, pp. Q45–Q56, 2013.
[15] A. Dimakis, A. Sarwate, and M. Wainwright, “Geographic gossip:
efficient aggregation for sensor networks,” in Information Processing in
Sensor Networks, 2006. IPSN 2006. The Fifth International Conference
on, pp. 69–76, 2006.
[16] P. Denantes, F. Benezit, P. Thiran, and M. Vetterli, “Which distributed
averaging algorithm should i choose for my sensor network?,” in
INFOCOM 2008. The 27th Conference on Computer Communications.
IEEE, pp. –, April 2008.
[17] P. Braca, S. Marano, and V. Matta, “Running consensus in wireless
sensor networks,” in Information Fusion, 2008 11th International Conference on, pp. 1–6, June 2008.
[18] M. Kriegleder, R. Oung, and R. D’Andrea, “Asynchronous implementation of a distributed average consensus algorithm,” in Intelligent
Robots and Systems (IROS), 2013 IEEE/RSJ International Conference
on, pp. 1836–1841, Nov 2013.
[19] Y. Y. Jun and M. Rabbat, “Performance comparison of randomized
gossip, broadcast gossip and collection tree protocol for distributed averaging,” in Computational Advances in Multi-Sensor Adaptive Processing
(CAMSAP), 2013 IEEE 5th International Workshop on, pp. 93–96, Dec
2013.
[20] G. T. Herman, Fundamentals of Computerized Tomography: Image
Reconstruction from Projections. Springer Publishing Company, Incorporated, 2nd ed., 2009.
[21] K. Tsianos, S. Lawlor, and M. Rabbat, “Consensus-based distributed
optimization: Practical issues and applications in large-scale machine
learning,” in Communication, Control, and Computing (Allerton), 2012
50th Annual Allerton Conference on, pp. 1543–1550, Oct 2012.
[22] Z. Zhang, N. Rahbari-Asr, and M.-Y. Chow, “Asynchronous distributed
cooperative energy management through gossip-based incremental cost
consensus algorithm,” in North American Power Symposium (NAPS),
2013, pp. 1–6, Sept 2013.
[23] E. Lee, “Cyber physical systems: Design challenges,” in Object Oriented
Real-Time Distributed Computing (ISORC), 2008 11th IEEE International Symposium on, pp. 363–369, May 2008.
[24] W.-Z. Song, R. Huang, M. Xu, A. Ma, B. Shirazi, and R. LaHusen, “Airdropped sensor network for real-time high-fidelity volcano monitoring,”
in Proceedings of the 7th International Conference on Mobile Systems,
Applications, and Services, MobiSys ’09, (New York, NY, USA),
pp. 305–318, ACM, 2009.
[25] G. Werner-Allen, K. Lorincz, M. Welsh, O. Marcillo, J. Johnson,
M. Ruiz, and J. Lees, “Deploying a wireless sensor network on an active
volcano,” IEEE Internet Computing, vol. 10, pp. 18–25, Mar. 2006.
[26] G. Kamath, P. Ramanan, and W.-Z. Song, “Distributed randomized
kaczmarz and applications to seismic imaging in sensor network,” in
The 11th International Conference on Distributed Computing in Sensor
Systems (DCOSS), (Fortaleza, Brazil), 2015.
[27] M. Rabbat and R. Nowak, “Distributed optimization in sensor networks,”
in Information Processing in Sensor Networks, 2004. IPSN 2004. Third
International Symposium on, pp. 20–27, April 2004.
[28] V. Shnayder, M. Hempstead, B.-r. Chen, G. W. Allen, and M. Welsh,
“Simulating the power consumption of large-scale sensor network applications,” in Proceedings of the 2Nd International Conference on
Embedded Networked Sensor Systems, SenSys ’04, (New York, NY,
USA), pp. 188–200, ACM, 2004.
[29] G. J. Pottie and W. J. Kaiser, “Wireless integrated network sensors,”
Commun. ACM, vol. 43, pp. 51–58, May 2000.
[30] W. Shi, Q. Ling, G. Wu, and W. Yin, “EXTRA: an exact first-order
algorithm for decentralized consensus optimization,” SIAM Journal on
Optimization, vol. 25, no. 2, pp. 944–966, 2015.
[31] K. Tsianos and M. Rabbat, “Distributed consensus and optimization under communication delays,” in Communication, Control, and Computing
(Allerton), 2011 49th Annual Allerton Conference on, pp. 974–982, Sept
2011.