Power-Delay Analysis of Consensus Algorithms

on Wireless Networks with Interference

S. Vanka, V. Gupta and M. Haenggi∗

May 2, 2008

Abstract

We study the convergence of the average consensus algorithm in wire-

less networks in the presence of interference. Regular lattices with peri-

odic boundary conditions and hierarchical networks with a backbone of

communication nodes supporting randomly placed sensor nodes are ana-

lyzed. We derive an optimal MAC protocol that maximizes the rate of

convergence of consensus algorithms on these networks, and provide an-

alytical upper and lower bounds for convergence rate. Our results show

that forming long-range communication links improves convergence even

in an interference-limited scenario.

1 Introduction

Consensus in general, and average consensus in particular, has become an areaof increasing research focus in recent years (e.g. see [1, 14, 12, 11] and thereferences therein). Many applications including distributed estimation [17, 16,2], motion coordination [13] and load balancing in multiple processes [4] havebeen analyzed in this framework.

Given n nodes each with a scalar value and a possibly time-varying intercon-nection graph de�ned on these nodes, such algorithms specify the updating rulethat every node should follow. The updated value of each node at every timestep depends on the value held by itself and its neighbors at the previous timestep. Initial results about such algorithms proved that the values held by thenodes converge to a common number, provided that the interconnection graphssatisfy some connectivity constraints. Lately, the focus has shifted to analyzingthe convergence properties in the face of realistic communication constraintsimposed by the channels between the nodes. Thus, e�ects such as quantiza-tion [10], packet erasures [2, 6], additive channel noise [7, 8], and delays [9] havebegun to gain attention.

∗Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN.

{svanka,vgupta2,mhaenggi}@nd.edu

1

Such works typically assume that the communication channels between eachpair of nodes are uncoupled. However, typical applications of consensus algo-rithms would involve nodes communicating over wireless channels. Models withindependent channel realizations are not suitable for wireless networks. Wirelesschannels are inherently coupled due to their broadcast nature and the presenceof interference. Moreover, in a wireless network, any two nodes can communi-cate by spending enough energy. Long range interconnections lead to smallergraph diameter, but also to interference with more nodes. The communicationtopology in wireless networks thus depends on the network protocols and is, infact, a design parameter. In this work, we take the �rst steps towards analyzingthe e�ect of communication constraints on the consensus algorithms and design-ing the communication parameters for the consensus problem. In particular, weconsider the rate of convergence of the average consensus algorithm while ex-plicitly accounting for interference. We identify scheduling algorithms that areoptimal with respect to the rate of convergence of the consensus algorithms. Wealso provide an analytical understanding of the impact of transmission poweron the rate of convergence.

The paper is organized as follows. We begin by formulating the problemand introducing our notation. We concentrate on two speci�c arrangementsof nodes: the case of nodes being arranged physically on a grid with periodicboundary conditions (Section 3) and a class of networks in which the positionsof nodes are stochastic (Section 4). Some avenues for future work are presentedin Section 5.

2 Problem Formulation

Average Consensus Algorithm: In this paper, we will concentrate exclu-sively on the average consensus algorithm. Consider n nodes that aim to reachconsensus with the �nal value being the average of their initial scalar values.Denote the value held by the i-th node at time k as xi(k). Also denote by x(k)the n-dimensional vector obtained by stacking the values of all the nodes in acolumn vector.

We describe in brief the average consensus algorithm de�ned with a giveninterconnection topology among the nodes. The topology can be described by agraph, with an edge present between two nodes if and only if they can exchangeinformation. Denote the neighbor set of node i at time k by Ni(k), wherethe argument k is included to model dynamic interconnection topolgies. Aniteration consists of every node i exchanging its state variable xi(k) with allnodes in Ni(k). Assuming this exchange happens in a single time step, thestate of the system evolves as

xi(k + 1) = xi(k)− h∑

j∈Ni(k)

(xi(k)− xj(k)), (1)

where h is a scalar constant designed to ensure convergence of the algorithm.Note that in practice, a number of transmissions are necessary for information

2

exchange. Denote the interconnection graph at time k by G(k). The systemthus evolves according to the discrete time equation

x(k + 1) = (I − hL(k))x(k), x(0) = x0, (2)

where L(k) denotes the Laplacian matrix of the graph G(k). It can easily beshown (see, e.g., [14]) that under proper connectivity assumptions, if the pa-rameter h is small enough, consensus is achieved with each node assuming theaverage value xav = 1

n

∑i xi(0). Throughout our presentation, we will assume

that h is �xed and has a value h < 12dmax

where dmax is the maximum degreecorresponding to any node in the graph over all time. To ensure that the nodesconverge to the average of x0, it is also essential that the graph at every timestep be balanced. The protocols we consider below will ensure that the graphis symmetric, which satis�es this condition.

The rate of convergence of the value of the nodes is a function of graphtopology. In the case of a static graph topology (i.e., G(k) = G for all timek), it can be easily shown (see, e.g., [5, 14, 15]) that the convergence of theconsensus protocol is geometric, with the rate being governed by the secondlargest eigenvalue modulus (SLEM) of the matrix I−hL. In general, a consensusalgorithm on a graph with smaller SLEM converges more quickly. If the matrixL is symmetric, its SLEM can be written as its norm restricted to the subspaceorthogonal to 1 = [1 1 1...1]T .

Communication Protocols: Usual treatments of the average consensus al-gorithm assume the communication or interconnection graph as given. In typicalapplications, however, the nodes communicate over wireless channels. In suchsituations, any two nodes can potentially communicate by expending enoughpower or by lowering the transmission rate. Moreover, the wireless channelis inherently multicast. Finally, the interference from other nodes that are si-multaneously transmitting also needs to be accounted for. The e�ect of suchfeatures on the average consensus algorithm has not been studied previously.

In particular, we consider a situation in which the physical locations of thenodes are given, which is a reasonable assumption in many sensing environments.Each node then decides on the power with which it transmits. This powerdetermines the communication radius of the node according to the relation

P = P0rαc , (3)

where P0 is a normalization constant, α is the path-loss exponent (typically2 ≤ α ≤ 5), P is the transmission power and rc is the communication radius.All nodes at a distance smaller than rc can receive the transmitted message.

Similar to the communication radius, we can also de�ne an interferenceradius ri. A node at position x can receive a message successfully from a node atposition y only if ||y−x|| < rc (noise constraint), and there is no node at positionz that is simultaneously transmitting, such that ||z − x|| < ri (interferenceconstraint). In this paper, for simplicity, we assume rc = ri. The results can begeneralized to other cases.

3

Given the above condition for successful transmission, we require a mediumaccess control (MAC) protocol for the nodes. We focus on scheduling basedMAC protocols in this paper, rather than random access protocols. These pro-tocols assure successful communication by scheduling transmissions such thatmessages do not interfere. They demonstrate better throughput than colli-sion based MAC protocols, at the expense of greater synchronization and co-ordination requirements among the nodes [18, 19].

Problem Formulation: The operation of the average consensus protocol canbe divided into two phases that are repeated at every update of the node values.In the �rst phase, the nodes exchange their values, possibly through multipletransmissions. We consider each transmission to consume one time step. Thee�ective communication graph at each update is, thus, composed of edges (i, j)such that node j has received the value of node i during the previous commu-nication phase. In the second phase, the nodes update their values accordingto the equation (1). As in the standard model, this step is assumed to be in-stantaneous. Therefore, due to multiple transmissions to set up the consensusgraph, in our model, the state update does not occur at every time step. Infact, assuming that each communication phase is completed in T time steps,

x(kT + T ) = (I − hL)x(kT ) (4)

Therefore the e�ect of �nite communication time, possibly due to interference,is to slow down the convergence rate.

We are interested in the following problem: Given a set of nodes M atknown locations and a desired consensus graph G, what is the MAC protocol thatminimizes the number of time steps needed for communication (thus maximizingthe update rate)? In other words, this MAC protocol should ensure the fastestconvergence of a consensus algorithm on a graph G when used by nodes inM.

We derive the MAC protocol for two physical distributions of the nodes: aregular grid with a periodic boundary condition in which every node is a sensorand part of the communication network, and a hierarchical network in whicha backbone of nodes regularly placed on a torus serve as the communicationnetwork for sensor nodes that are stochastically distributed according to a bi-nomial point process on a torus. We make the following further assumptions inthis paper:

• We limit the transmission policy to be time-invariant.

• At the time of an update of the values of the nodes, we demand that thee�ective communication graph be undirected, i.e., for any two nodes i, jin the network, j ∈ Ni ⇔ i ∈ Nj . Note that this is slightly strongerthan the necessary and su�cient condition for convergence of the averageconsensus algorithm that the graph be balanced [14].

• We do not assume routing of values through nodes.

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• We assume half-duplex operation of the nodes, and assume that packetsthat su�er collisions cannot be decoded.

Under these assumptions, we are able to prove the following results:

• We present the optimal MAC scheduling protocol and the rate of conver-gence of the average consensus algorithm if such a protocol is followed.

• We prove that the rate of convergence increases monotonically in the trans-mission power even when one accounts for interference. The interferenceconstraint means that increased transmission power reduces the numberof nodes that are allowed to communicate in a given time step - the con-sensus graph is formed in a greater number of time slots. However, thise�ect can be potentially o�set by the resulting greater connectivity of theconsensus graph. We show that the latter e�ect dominates the formerwith increasing transmission power.

In the next section, we present the MAC protocol and the analysis for thedistribution of nodes on a regular grid with periodic boundary conditions.

3 Analysis of a ring and a 2D torus

We begin by considering nodes placed on a regular grid with periodic boundaryconditions.

3.1 Ring Topology

Consider n nodes placed uniformly on a circle of radius r centered at the origin,as shown in Figure 1. Suppose that the transmission power is such that everynode can transmit information tom of its nearest neighbors on either side. As anexample, in Figure 1, m = 1. If we de�ne Pm, m ≤ bn2 c as the transmit powerthat enables a node to form error-free links with 2m neighbours (m nearestneighbours on either side). Due to the geometry, we see that the transmissionradius is 2r sin(mπn ), and

Pm ∝ (2r sin (mπ

n))α, (5)

where α ≥ 2 is the path-loss exponent. As stated above, for simplicity, we willassume that the interference radius of each node is also 2r sin(mπn ).

If the wireless channel could support simultaneous transmissions by everynode, the system would evolve according to (2), with I − hL an n× n circulantmatrix with the �rst row given by1− 2h

m times︷ ︸︸ ︷−h − h · · · − h 0 · · · 0

m times︷ ︸︸ ︷−h − h · · · − h

. (6)

5

Figure 1: Schematic of nodes placed along a ring.

For future reference, denote this matrix by F1,m. The MAC protocol that wepropose guarantees that the system evolves according to this matrix. However,the communication phase occurs over multiple steps. We begin by bounding thenumber of steps required for this. Towards this end, we �rst prove the ratherintuitive result that equal power allocation per node is optimal in terms of usingthe smallest number of time slots to construct F1,m.

Lemma 1. Consider the set-up described above where each node has a trans-mission power constraint P ≤ Pmax, where Pmax is the power needed transmit tom′ ≥ m neighbors and m ≤ bn/2c. Let T denote the number of communicationsteps to form F1,m. Then T is minimized by allowing all nodes to transmit atpower Pm, where Pm is the transmit power that enables a node to form error-freelinks with 2m neighbours (m nearest neighbours on either side).

Proof. We need to form 2mn edges in T time slots. Consequently, the averagenumber of edges formed per slot is

Nav =2mnT

=1T

T∑t=1

Nt, (7)

where Nt is the number of edges that are formed in every slot. A MAC protocol

6

minimizes T if and only if it maximizes Nav. Without loss of generality, let{1, 2, . . . ,K} be the set of transmitting nodes in time slot t, with power allo-cations to reach {i1, i2, . . . , iK} neighbors on each side, respectively. Assumingeach node transmits at Pik ≤ Pmax, the number of edges formed is given by

Nt ≤K∑k=1

2ik, (8)

where equality is achieved if and only if there are no interfering transmissions(since colliding packets cannot be decoded). Since there is no routing:

• Any transmission scheme that leads to collisions is suboptimal.

• A node should broadcast its message to its m nearest neighbors in eitherdirection. In other words ik ≥ m for all k.

• Only m ≤ m′ nearest neighbor nodes exchange messages - so the numberof useful edges formed is always 2m per transmitting node for all P ≥ Pm.

These facts allow us to write

Nt =K∑k=1

2m = 2mK. (9)

Also, the total number of nodes that are either transmitting or receiving packetsin any given slot cannot exceed n. Thus,

K∑k=1

(2ik + 1) = (2iav + 1)K ≤ n⇒ K ≤ b n

2iav + 1c, (10)

where we have de�ned

iav ,1K

K∑k=1

ik. (11)

This implies

Nt ≤ 2mb n

2iav + 1c. (12)

The individual node power constraints m ≤ ik ≤ m′ imply m ≤ iav ≤ m′. Sincethe right hand side of (12) is a decreasing function of iav, we conclude that

Nt ≤ Nmaxt , 2mb n

2m+ 1c. (13)

Since ik ≥ m, iav is minimized when ik = m for all k.

The expression for Nmaxt above suggests a simple transmission schedule that

we describe below. We show that this schedule achieves the upper bound onNt in every time slot, and is therefore optimal in that it constructs F1,m in thesmallest time possible among all MAC protocols.

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Lemma 2. Consider the set-up described above, where the matrix F1,m is tobe constructed in the minimum number of time steps. If each node transmitsat power Pm, any transmission schedule that is used to construct F1,m mustconsume at least 2m + 1 time slots. Moreover, it is always possible to form aschedule that consumes at most 4m+ 1 time slots.

Proof. Without loss of generality, suppose node 1 transmits at Pm. Then, if mnodes on either side should receive its message, then

• none of these 2m neighbors can transmit at this time (half-duplex con-straint);

• the nearest node(s) that can transmit at the same time are 2m+1 mod n(interference constraint).

Therefore, the maximum number of simultaneous transmissions possible is b n2m+1c.

Extending this argument, in 2m+ 1 time slots, at most (2m+ 1)b n2m+1c nodes

can transmit. In other words, maximizing the number of transmitting nodes pertime slot requires that all nodes (2m+ 1) nodes apart should transmit, as longas the half-duplex and interference constraints are satis�ed. Thus after 2m+ 1time slots,

n− (2m+ 1)b n

2m+ 1c = rem(n, 2m+ 1) (14)

nodes are yet to transmit. Since all nodes that are 2m + 1 apart have alreadytransmitted, the remaining nodes are at most 2m nodes apart. This impliesthat at most one node can be scheduled for transmission in any slot. Thereforewe need rem(n, 2m + 1) more time slots to construct F1,m. The upper boundcan be derived by observing that rem(N, 2m+ 1) ≤ 2m.

Scheduling Alogrithm: We describe the MAC protocol for a given Pm. Letthe nodes be numbered {0, 1, . . . n − 1}. Denote the set of nodes that are yetto transmit by S, and the set of transmitters in time slot t by Qt. The MACalgorithm is as follows:

1. Initialize: S = {0, 1, . . . n− 1} and t = 1.

2. For every t ≥ 1, while S is non-empty:

(a) Pick any node i ∈ S. Form a set of nodes Qt = {j | j = i+ s(2m+1), 0 ≤ s ≤ b n

2m+1c − 1}.

(b)Qt ← S ∩QtS ← S \ Qtt← t+ 1

As proved above, the �nal value of t will T = 2m + 1 + rem(n, 2m + 1). TheMAC protocol that we denote by P1 consists of picking one element of the set

8

{Q1,Q2, . . .QT } at a time, in T time slots. Notice that P1 achieves the upperbound on the number of edges formed in every time slot:

Nt = 2mb n

2m+ 1c = Nmax

t . (15)

From Lemma 1 we conclude that this protocol consumes the minimum numberof time steps for the exchange of information, so that at every update stepthe system can update according to F1,m. Therefore, P1 is optimal in that itwill result in the fastest rate of convergence for a given F1,m. Since F1,m isthe graph with the highest connectivity that can be formed given a particulartransmission power Pm, the protocol P1 achieves fastest rate of convergence fora given constraint on the maximum transmission power for any particular node.

We are now in a position to present the rate of convergence of the averageconsensus protocol for the regular grid with interference constraints.

Theorem 3. Consider the problem set-up described above. If each node trans-mits at power Pm and follows a scheduling-based MAC protocol, the error vectorε(k) = x(k)− 1xav converges geometrically to zero with the rate of decay β thatis bounded as

ρ1

2m+11 ≤ β ≤ ρ

14m+11 (16)

where ρ1 = 1−h(2m+1)+hS(m,n)1 , with S

(m,n)p ,

sin((2m+1)πp

n )

sin(πn ) , p = 0, 1, . . . N−1.

Proof. The communication graph at each update step is balanced and con-nected. Thus, the node values converge to the average of their initial valueswith the decay rate as the modulus of the second largest eigenvalue of F1,m [14].Since F1,m is circulant, its eigenvalues can be calculated as

ρk = 1− 2mh+ h

m∑l=1

e−j2πkn + h

l=n−m∑l=n−1

e−j2πkn

= 1− 2mh+ 2hl=m∑l=1

cos (2πlkn

)

= 1− (2m+ 1)h+ h(1 + 2l=m∑l=1

cos (2πlkn

))

= 1− (2m+ 1)h+ hsin ( (2m+1)πk

n )

sin (πkn )︸ ︷︷ ︸,S(m,n)

k

= 1− (2m+ 1)h+ hS(m,n)k , k = 0, 1, · · · , n− 1.

We can con�rm that ρ0 = 1. The second largest eigenvalue is given by ρ1 =ρn−1. However, the eigenvectors corresponding to these eigenvalues are linearly

9

independent due to the symmetry of F1,m. From Lemma 2, the system updatesits values every T = (2m+ 1) + rem(2m+ 1) time steps, so that 2m+ 1 ≤ T ≤4m + 1. Thus we obtain that the convergence rate is lower bounded by ρ

12m+11

and upper bounded by ρ1

4m+11 .

Figure 2: Variation of the convergence rate with the transmission power for aring.

Remarks:

1. The upper bound for the decay can be achieved by using the transmissionschedule P1, provided 2m+ 1 divides n. If this condition is not true, theupper bound is strict.

2. For any given transmission power Pm, we see that the MAC constraintsreduce the rate by a factor of T where 2m+ 1 ≤ T ≤ 4m+ 1.

3. The rate of convergence is an increasing function inm, and hence in Pm. Anumerical illustration of this fact is provided in Figure 2. For the purposeof the plot, we have plotted the time taken for the error norm to becomehalf, as a function of transmission power for 31 nodes arranged regularly ona ring of radius 1 unit. We have assumed α = 2, h = 1

2n and the constant

10

of proportionality in equation (5) as unity. This is somewhat counter-intuitive since it indicates that rate reduction due to a larger number ofsteps in the communication phase is always compensated by the increase inrate due to higher connectivity. That forming long range communicationlinks would lead to faster convergence even in networks with interferencewas not a priori evident.

4. The e�ect of increasing the transmission power becomes more prominentas Pm increases. This can again be seen from Figure 2. For large n,approximating

sin(pπn

)≈ pπ

n− 1

3

(pπn

)3

, (17)

we can express the spectral gap SG , 1− ρ1 as

SG = h(2m+ 1)− hsin ( (2m+1)π

n )

sin (πn )

≈ h(2m+ 1)− h(2m+1)π

n − π3

3(2m+1)3

n3

πn −

π3

3n3

Simplifying this expression

SG ≈hπ2

3n2 2m(2m+ 1)(2m+ 2)

1− π2

3n2

. (18)

This suggests that for large n, the spectral gap increases with m3. For a�xed m, SG scales as O( 1

n2 ) for large n.

3.2 Torus

The above results can be generalized to higher dimensions. We present thecase when nodes are placed in a regular lattice in two dimensions with periodicboundary conditions, i.e., on a 2-D torus as in Figure 3.

As before, de�ne Pm to be the transmit power that enables a node to formerror-free links with m neighbors in the axial directions. For simplicity, weassume that the torus is formed by making a square grid periodic. Thus, thereare√n agents in either of the axial directions. Thus, we see that

Pm ∝(m√n

)α, (19)

where α ≥ 1 is the path-loss exponent. As before, the objective is to constructthe Laplacian matrix of the nearest m-neighbor consensus graph. We denotethis matrix by F2,m. F2,m is an n×n block circulant matrix with its �rst l rowsdescribed by[

F0

m times︷ ︸︸ ︷F1 F2 · · · Fm 0 0 · · · 0

m times︷ ︸︸ ︷Fm Fm−1 · · · F1

]l×n

, (20)

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Figure 3: Schematic of nodes placed along a 2-dimensional torus.

where the l× l matrices F0, F1, . . ., Fm are also circulant, with the �rst row ofmatrix Fk being[

dk

Nm(k) times︷ ︸︸ ︷−1− 1 · · · − 1 0 0 0 · · ·

Nm(k) times︷ ︸︸ ︷−1− 1 · · · − 1

]l×1

, (21)

with

Nm(k) , |{l | |l| ≤√m2 − k2, l ∈ Z}|

d0 = (2m+ 1)2 − 1dk = −1, ∀k ≥ 1.

We can easily bound the number of time steps required to form the matrixF2,m.

Lemma 4. If each node transmits at power Pm, any transmission schedule thatis used to construct F2,m has at least (2m+1)2 time slots. Moreover, it is alwayspossible to form a schedule that has at most 2(2m+ 1)2 − 1 time slots.

Proof. Using arguments similar to those used in Lemma 2, a maximum ofb n

(2m+1)2 c simultaneous transmissions can be scheduled per time slot. After

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(2m+ 1)2time slots,

n− b n

(2m+ 1)2c(2m+ 1)2 = rem(n, (2m+ 1)2) (22)

nodes are yet to transmit. Therefore we need at least (2m + 1)2 time slotsto construct the consensus graph. It is always possible to form a transmissionschedule to complete the rest of the transmissions in

rem(n, (2m+ 1)2) ≤ (2m+ 1)2 − 1 (23)

time slots.

Bounding the rate of decay of the error poses some problems in tori ofdimensions two or higher. Essentially, this is related to the fact that the nodesthat can receive data from a particular node are speci�ed through circular disks.While in a one-dimensional ring such disks can cover the entire ring, in higherdimensions, such coverage is not possible. For our purpose, we lower and upperbound such discs by squares of suitable side length that cover the entire region.To this end, we begin with the following preliminary result.

Lemma 5. Let T2(n) denote a set of n = l2 nodes uniformly placed on a unit

2D torus [0, 1]2at (ri, rj) = ( i√n, j√

n) for i, j = 0, 1, . . .

√n− 1. Let G(m)

p be the

consensus graph formed over T2(n) by placing edges between each node i withall other nodes j 6= i satisfying

Lp(ri, rj) ≤m√n, 1 ≤ m ≤ b

√n

2c. (24)

Also denote the Laplacian of G(m)p by L

(m)∞ and its maximum degree by dmax

so that the corresponding consensus matrix is de�ned as F(m)p = I − hL(m)

∞ forsome 0 ≤ h ≤ 1

2dmax. The following hold:

1. G(m)2 is a subgraph of G

(m)∞ .

2. The eigenvalues of the F(m)∞ are

λ∞a,b = 1− h(2m+ 1)2 + hS(m,l)a S

(m,l)b . (25)

Proof. The �rst result follows from the de�nition of Lp norm: Any node j forwhich L2(ri, rj) ≤ m√

nnecessarily satis�es max

k∈{1,2}(|ri,k|, |rj,k|) ≤ m√

n. Therefore,

any edge in G(m)2 is also present in G

(m)∞ , which means G

(m)2 is a subgraph of

G(m)∞ .Given the standing assumption h < 1

2dmax, this implies that a consensus

algorithm on G(m)∞ must converge at least as fast as one on G

(m)2 , provided the

iterations occur at the same rate. This allows us to lower bound the convergence

rate of consensus algorithms on G(m)2 as follows.

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To begin with, note that the Laplacian matrix L(m)∞ of G

(m)∞ is an n×n block

circulant matrix with its l rows as[F0

m times︷ ︸︸ ︷F1 F1 · · · F1 0 0 · · · 0

m times︷ ︸︸ ︷F1 F1 · · · F1

]l×n

. (26)

Here the l × l matrices F0 and F1 are also circulant, with their �rst row being[dk

m times︷ ︸︸ ︷−1− 1 · · · − 1 0 0 0 · · ·

m times︷ ︸︸ ︷−1− 1 · · · − 1

]l×1

, (27)

with d0 = (2m + 1)2 − 1 and d1 = −1. Given this structure, we can usethe properties of block circulant matrices and the fact that Fks are themselvescirculant (and consequently will share the same eigenvectors) to compute the

eigenvalues of L(m)∞ as

µ∞r,s =l−1∑t=0

ηr,te−j 2πst

l (28)

where ηr,t is the rth eigenvalue of Ft ∀r, s = 0, 1, . . . , l − 1.

Now, from the 1D torus calculation, it is apparent that

ηr,t = dt −m∑k=1

2 cos(

2πrkl

). (29)

Using this result, and making use of the fact that Ft = F1 ∀t ≥ 1, we obtainthat

µ∞r,s = d0 − 2m∑k=1

cos(

2πrkl

)+ 2

m∑t=1

(d1 − 2

m∑k′=1

cos(

2πrk′

l

))cos(

2πstl

).

(30)Plugging in the values of d0 and d1 and simplifying, we obtain

µ∞r,s = (2m+ 1)2 − (1 + 2V (r,l)m )(1 + 2V (s,l)

m ), (31)

where we have de�ned V(r,l)m =

∑mk=1 cos

(2πkrl

)for notational convenience.

Now, we note that

1 + 2V (r,l)m =

sin ( (2m+1)πrl )

sin (πrl ), S(m,l)

r . (32)

Thus the eigenvalues of F(m)∞ = I − hL(m)

∞ are given by

λ∞a,b = 1− hµ∞a,b = 1− h(2m+ 1)2 + hS(m,l)a S

(m,l)b . (33)

We are now in a position to bound the rate of decay for the case of the torus.

14

Theorem 6. If each node transmits at Pm (m ≤ b√n

2 c), there exists a trans-mission schedule such that the state covariance δ(k) = x(k)−1xav converges tozero at a rate β that is lower bounded as

λ1

(2m+1)2

1 < β < λ1

2(2m+1)2−12 (34)

where

λ1 =(

1− h(2m+ 1)2 + hS(m,√n)

1

)λ2 =

(1− h(2m̃+ 1)2 + hS

(m̃,√n

1

)m̃ = b m√

2c.

Proof. Without loss of generality, assume that node 0, located at the origin,transmits �rst with power Pm. Assuming there are no collisions, a node atposition r can successfully receive its packet only if

L2(r) ≤ m√n, (35)

where L2(.) denotes the familiar L2 norm. Denoting by Sm the set of all suchnodes, we may write

Sm = {l|L2(r) ≤ m√n}. (36)

Using Lemma 5, the consensus graph formed will always be a subgraph of

G(m)∞ = (V,E(m)

∞ ) where V is the set of all nodes in the torus and E∞ is the setof all edges formed using the L∞neighbourhood:

E(m)∞ = {(i, j)|L∞(ri, rj) ≤

m√n}. (37)

Since the convergence of G(m)2 cannot be faster than G

(m)∞ , letting n = l2we can

use the above result to work with the eigenvalues of F(m)∞ :

ρpq = 1− h(2m+ 1)2 + hS(m,l)p S(q,l)

q , (38)

for p, q = 0, 1, ...,√n − 1. It is easy to see that the maximum eigenvalue is 1.

The second largest eigenvalue is corresponds to p = 1, q = 0 and is given by

ρ1,0 = 1− h(2m+ 1)2 + hS(m,l)1 , λ1. (39)

Similar to G(m)∞ we can de�ne G

(m̃)∞ whose vertex set consists of all nodes on the

torus but whose edges E(m̃)∞ is de�ned as

E(m̃)∞ = {(i, j)|L∞(ri, rj) ≤

m̃√n}. (40)

15

But for all nodes i, j on G(m̃)∞

max(|ri|, |rj |) ≤ m̃√n

⇒ `2(ri, rj) ≤√

2b m√2c 1√

n<

m√n.

Geometrically, this can be understood as stating that a square of side 2b m√2c is

completely contained in a concentric circle (i.e., having the same centroid) ofradius m.

Hence we have proved that G(m̃)∞ ⊂ G(m)

2 - so the convergence on G(m̃)∞ cannot

be faster than that on G(m)2 for the same update rate. Substituting m with m̃

in the result of Lemma 5 the second largest eigenvalue turns out to be equalto λ2 as de�ned above. Using arguments similar to those in Theorem 3, we seethat

λ1

(2m+1)2

1 < β < λ1

2(2m+1)2−12 . (41)

4 Hierarchical networks

Since calculation of the rates of convergence for average consensus for arbitrarygraphs is not possible even without the interference constraints, we do not expectto be able to extend our results for arbitrary graphs. In this section, we consideranother class of useful graphs that allow us to state analytical results. Weconsider a variation on the random geometric graphs by adding a backbone ofdedicated (long-distance) communication nodes. Thus we consider a hierarchicalnetwork with N sensing nodes uniformly randomly placed on a torus in [0, 1]d,and Kd identical regularly spaced backbone nodes on the torus, as shown inFigure 4 for d = 1.

Figure 4: Schematic of backbone nodes placed along a 1-dimensional torus. Theith backbone node is placed at ri =

(i− 1

2

)1K from the origin, for i = 1, 2, . . . ,K.

We assume that the backbone nodes do not participate in sensing, and onlycommunicate with each other in the network. Each backbone node has a �xed

16

exclusive region of coverage, i.e., it (alone) collects data from all the sensingnodes within a sphere of radius r = 1/2K. Initially, each backbone node collectsand averages the data from all the sensing nodes in its region of coverage. Allthe backbone nodes now run an average consensus algorithm among themselveswith their respective averaged data as initial values. It is possible to pass onthis (global) average back to all the nodes within their coverage regions in O(1)steps. Therefore for analyzing the rate of convergence, it is su�cient to analyzethe time taken for collecting data by the backbone nodes and the time taken toreach consensus among these nodes.

We begin by characterizing the number of sensors reporting to each backbonenode.

Lemma 7. For large N and if the number of backbone nodes scales as o( NlogN )

w.h.p. the number of sensors per coverage area is n = Nπd/2

Γ(1+d/2)(2K)d, where K

is the number of nodes per dimension.

Proof. This can be proved by a a variation of the argument used in the theoryof random geometric graphs to show regularity. Consider a sphere S of radiusr centered at point P on the torus. Let random variable Xk (1 ≤ k ≤ N) bede�ned as:

Xk = 1S(k) (42)

where 1(.) is the indicator function. Since the sensing nodes are placed onthe torus uniformly and independent of each other, {Xk} are iid Bernoulli withsuccess probability

p =Vol. of sphere

Vol. of torus=

πd/2

Γ(1 + d/2)(2K)d. (43)

The number of sensors inside the sphere is thus a binomial random variable

N∑k=1

Xk , Y, (44)

with µ , EY = Np. We can now use the Cherno� bound:

P(|Y − µ| > µδ) ≤ 2 exp(−µδ

2

2

). (45)

For 0 < δ < 1, since the number of nodes Kd = o( Nlog N ) we can always choose

δ =

√4logNN

· γKd = o(1), (46)

for large enough N and some γ > 0. Plugging in this value into the bound, wesee that

P(Y /∈ [µ(1− δ), µ(1 + δ)]) ≤ 1N2ηγ

≤ 1N3

, (47)

17

where η = πd/2

2dΓ(1+d/2)and γ is chosen such that ηγ > 2. So w.h.p.,

Y = Np(1± o(1)). (48)

Thus, the probability that any coverage area does not have n(1±o(1)) nodesis

P(∪k{Yk /∈ [µ(1− δ), µ(1 + δ)]}) ≤ ∪kP(Yk /∈ [µ(1− δ), µ(1 + δ)])

= Kd 1N3

< N1N3

=1N2

,

where we use the union bound. The result now follows readily.

Since each of the Kd backbone nodes has n sensors in its coverage regionw.h.p. whenN is large, a total of nKd sensing nodes will be covered by the back-bone network. Therefore for large N , it is always possible to achieve consensusover a positive fraction

α =nKd

N=

πd/2

2dΓ(1 + d/2)(49)

of the sensing nodes, independent of N and K.We will now study speci�c cases for d = 1 and 2. To begin with, we note

that for d = 1,

α1 =π1/2

2Γ(3/2)= 1, (50)

and for d = 2,α2 =

π

4Γ(2)=π

4≈ 78.5%. (51)

As stated above, we assume that each backbone node collects data by pollingall the sensing nodes in its coverage region. This is done in parallel over allbackbone nodes, by a suitable choice of transmit power. Assuming each nodetransmits in one time slot, we need n time slots to initialize the consensusalgorithm, where w.h.p.

n =αN

Kd. (52)

During the consensus phase, the network topology is identical to the (regular)ring and torus topologies discussed previously. As before, let each node transmitat �xed power Pmto reach m ≤ bK/2c neighbors per direction per dimension.

Using the results in Theorems 3 and 6, we obtain that for Kd = o(

NlogN

),

w.h.p.,(1− h(2m+ 1) + hS

(m,K)1

) 14m+1 ≥ β

≥(

1− h(2m+ 1) + hS(m,K)1

) 12m+1

18

for the ring or 1D torus and

(1− h(2m̃+ 1)2 + hS

(m̃,K)1

) 12(2m+1)2−1

> β >(1− h(2m+ 1)2 + hS

(m,K)1

) 1(2m+1)2

for the 2D torus, where m̃ = b m√2c as de�ned in Theorem 6.

5 Conclusions and Future Work

In this paper, we introduced a framework for considering the e�ects of real-istic communication networks on average consensus algorithms. In particular,we considered the issue of designing the transmission power of nodes and themedium access control scheduling algorithm to optimize the rate of convergence.For two classes of graphs - lattices with periodic boundary conditions and hier-archical graphs with a backbone of communication nodes supporting randomlyplaced sensing nodes - we analytically characterized the e�ect of these commu-nication parameters on the distributed algorithm performance. We showed thatincreasing transmission power and forming long-range links increases the rateof convergence, even in the presence of interference.

The work can be extended in many directions. An immediate extensionis the consideration of other classes of graphs. Cayley graphs and expandergraphs have been shown to lead to good convergence properties for consensusalgorithms [3]. It will be useful to extend our framework to such graphs. An-other direction is to consider the e�ect of data loss through e�ects other thaninterference. Due to fading, stochastic packet losses are unavoidable in wire-less channels. While the impact of such losses on average consensus algorithmshas begun to be addressed [2, 6], it will be interesting to see the e�ect in ourframework with interference explicitly being considered.

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