978-1-4673-0024-7/10/$26.00 ©2012 IEEE 2143

2012 9th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2012)

Broadcast Gossip Algorithm with Quantization

Na Wang School of Science

Anhui University of Science and Technology Huainan, China

Dequan Li, Zhixiang Yin

School of Science Anhui University of Science and Technology

Huainan, China

Abstract—The paper studies the problem of asynchronous broadcast gossip average consensus with quantized information communication constraints. How quantization affects the evolution of the broadcast gossip average consensus algorithm is investigated. We show that the agents’ states converge to a random variable that deviates from the average of the agents’ initial states. We give result of the mean square error of the states, which depends on the quantized precision and the network parameters.

Keywords-Broadcast Gossip algorithm;Average consensus; Quantization;Distributed

I. INTRODUCTION Distributed agreement and average-consensus problems

have been gained much attention in recent years. Many distributed control and estimation strategies are designed based on consensus algorithms ([1], [2]). Distributed consensus is a fundamental problem in ad hoc network applications, including distributed synchronization problems [6]–[8], [10], distributed coordination of mobile autonomous agents [4], [5], and distributed data fusion in sensor networks [3], [9], [11]. It is also a central topic for load balancing (with divisible tasks) in parallel computers [12]. Vicsek et al. provided a variety of simulation results which demonstrate that the simple distributed algorithms allow all nodes to eventually agree on a parameter [6]. The work in [13] provided the theoretical explanation for behavior observed in these reported simulation studies. To overcome the drawbacks of the standard packet based gossip algorithms, a broadcast-based gossiping consensus algorithm for wireless sensor networks was also recently studied [16].

The broadcast-based consensus algorithm is one typical randomized algorithm, which has been recently studied in [16]. According to this algorithm, at every step, one node is chosen randomly with a certain probability to broadcast its state to its neighbors. If the nodes within this node’s transmission range have received the broadcasted value then they update their own state value, and the remaining nodes out of this node’s transmission range sustain their state value. Thus, the evolution matrix for this algorithm is asymmetric and varies randomly at each step, which means that information topologies need not be undirected at each step. Also, this algorithm does not preserve the node total sum invariance in each round. However, from a practical point of view, this broadcast-based consensus algorithm is very amenable to implementation, for it exploits the broadcast nature of the wireless communication environment and does not require the typical bidirectional

communication among nodes, thus it obviates the need for sophisticated underlying media access control mechanisms. Further, [16] shows that the broadcast-based consensus algorithm yields consensus with probability one, and the random consensus value attained, in expectation, is equal to the average of the initial node states. Our work in this paper will focus exclusively on this type of consensus algorithm.

It is well known that in real digital networks, communication channels have a finite channel capacity, thus, at each time steps, agents can only transmit a finite amount of information to their neighbors. The communication between different agents can be viewed as such a process: At each time step, the sender encodes the quantized state and sends out the code. When the neighbors receive the code, they use a decoding algorithm to obtain an estimate of the sender’s state. Thus, quantization plays an important role in information exchange among agents. The question that motivates this paper is investigating if it is possible to avoid the partner selection process altogether, analyzing a broadcast communication protocol where each random transmission triggers an update by all nodes within range, without a mechanism of reply in place to maintain the network average [15]. In this paper, we provide study of broadcast gossip algorithms’ speed of convergence and mean squared error characteristics.

This paper is organized as follows: Section II introduces some notations and preliminaries on graph theory, and gives an overview of standard broadcast gossip average algorithm. Section III presents the broadcast gossip average estimation algorithm with quantized communication. Simulation results are provided in section IV, and finally we conclude with some discussion and future directions in Section V.

II. FORMATTING INSTRUCTIONS In the following, we describe briefly the distributed average

consensus problem along with the proposed consensus algorithm.

A. Graph Model We model our wireless sensor network as a graph ( ),G N R= with N vertices or nodes distributed in the plane

at locations { }: 1, ,iR i N= in 2R . The N ---node topology of G at time step t is represented by the N N× adjacency

matrix ( )ijA A= , where1,0,

iij

j NA

otherwise∈⎧

= ⎨⎩

. Moreover, we define

{ }{ }1,2, , : 0i ijN j N A= ∈ ≠ , ( )1D diag A= . Finally, the Laplacian

matrix of a graph is defined as L D A= - . For example, we may

This work is partially supported by NSFC Grant # 61073102, 60973050, 61170172, 61170059, and Anhui Provincial Natural Science Foundation Grant # 090412251.

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take G to be the random geometric graph, where the N sensor locations are chosen uniformly and independently in a unit square area.

B. Average Consensus At time slot 0t ≥ , each node 1, ,i N= has an estimate

( )ix t of the global average, and we use ( )x t to denote the N -vector of these estimates. The ultimate goal is to use the minimal amount of communication to drive the estimate ( )x t

as close as possible to the average vector ( )0 1x , where

[ ]1 1,1, ,1 T nR= ∈ and

( ) ( )1

10 0N

ii

x xN =

= ∑ (1)

Because our algorithms are randomized, the quantity ( )x t for 0t > is a random vector even though we assume ( )0x is deterministic.

C. Time Model We use the asynchronous time model, which is

well---matched to the distributed nature of sensor networks [5], [14]. In this model, each sensor node is assumed to have a clock which ticks independently according to a rate μ Poisson process. Consequently, the inter-tick times are exponentially distributed and independent across nodes and over time. This process is equivalent to a single clock whose ticking times form a Poisson process of rate Nμ . Let tZ be the arrival times of this global process. In expectation, there are approximately Nμ clock ticks per unit of absolute time but we will always measure time in number of ticks of this (virtual) global clock. We therefore think of time as discretized with the interval [ )1;t tZ Z + corresponds to the t -th timeslot. We can adjust time units relative to the communication time so that only one broadcast event occurs in the network at each time slot with high probability.

D. Probabilistic Quantization The probabilistic quantization :Q R R→ is defined as

follows: suppose x R∈ is bounded to a finite interval [ ],I I− , and the interval is equally divided into 1M − sub-intervals with quantization points defined by the set { }1 2, , , Mθ θ θ θ= , where 1 , MI Iθ θ= − = . Denote the interval as 1i iθ θ+Δ = − , for

{ }1, 2, , 1i M∈ − . Then, for [ )1,i ix θ θ+∈ , ( )Q x is a random variable defined by

( ) ( )( )

1

1

,

,i i

i i

with probability xQ x

with probability x

θ θθ θ

+

+

− Δ⎧⎪= ⎨− Δ⎪⎩

The following lemma gives two important properties of the probabilistic quantizer.

Lemma1. For every [ )1,i ix θ θ+∈ ,

( )E Q x x=⎡ ⎤⎣ ⎦ , ( )( )2

2

4E Q x x Δ⎡ ⎤− ≤⎢ ⎥⎣ ⎦

(2)

Note that ( )Q x is an unbiased uniform quantizer, that is, the quantized data ( )Q x is an unbiased representation of x .

E. Broadca t Consensus Ptotocol The asynchronous broadcast gossip algorithm is described

as follows: Suppose node i ’s clock is the t th that ticked. Then, node i broadcasts its own state value which is received by all neighboring nodes within distance R from it. Once the broadcasted value is received, the neighboring nodes set their values equal to the (weighted) average of their current value and the value broadcasted by the node i . Formally, node i activates and the following events occur: onode i broadcasts wirelessly its current state value ( )ix t ;

othe broadcasted value is successfully received by the nodes that are within the radius R ; oall nodes in the set of node i ’s neighbors iN receive the

broadcasted value ( )ix t , and update their state values according

to the following equation: ( ) ( ) ( ) ( )1 1 ,k k i ix t rx t r x t k N+ = + − ∀ ∈ (3)

Where ( )0,1r ∈ denoting the mixing parameter;

othe remaining nodes in the network, including i , update their state values as:

( ) ( )1 ,k k ix t x t k N+ = ∀ ∉ (4) This procedure takes place at every clock tick.

Let ( )x t denote the vector of values at the end of the t

-th ticking event. Then ( ) ( ) ( )1x t W t x t+ = (5)

Where the random matrix ( )W t , with probability 1 N is

(assuming that the t -th clock ticks)

( )

1, ,, ,

1 , ,0,

i

i ijk

i

j N k jr j N k j

Wr j N k i

elsewhere

∉ =⎧⎪ ∈ =⎪= ⎨ − ∈ =⎪⎪⎩

(6)

Where ( )iW denotes the weight matrix corresponding to the case where node i ’s clock ticks. For the quantized version, the states evolve according to the following equation

( ) ( ) ( ) ( ) ( )( )( )

ˆ ˆ1i

i i ij i jj N t

x t x t W t x t x t∈

+ = − −∑ (7)

Let ( )x̂ t denote the vector of values at the end of the t

-th ticking event. Then

( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )

ˆ1

ˆ

x t x t W t I x t

x t W t I x t x t W t I x t

W t x t W t I e t

+ = + −

= + − − + −

= + −

(8)

Where ( ) ( ) ( )ˆe t x t x t= − . Furthermore, according to lemma 1, there holds

( ) 0E e t =⎡ ⎤⎣ ⎦ , ( ) ( )2 2

, ,4 4

TE e t e t diag⎛ ⎞Δ Δ⎡ ⎤ ≤ ⎜ ⎟⎣ ⎦ ⎝ ⎠

.

The states ( )1x t + from the different iterations of the consensus algorithm are expressed as

( ) ( ) ( ) ( ) ( )( ) ( )0

1 ,0 0 ,t

ix t t x t i W i I e iϕ ϕ

=

+ = + −∑ (9)

2145

Where ( ) ( ) ( ) ( ), 1t i W t W t W iϕ = − . This equation can include that the states vector of values will converges to the initial values.

III. ANALYSIS OF THE ALGORITHM We now turn to the analysis of the algorithm. In subsection

A: we provide some properties of the averaging matrices. In subsection B: we prove that the algorithm converges in the expectation.

A. Averaging Matrix Properties The following results reveal important properties regarding

the random weight matrices.

Lemma 1: The weight matrices ( ){ }: 1,2, , 1iW i N= satisfy the

following: i) 1 is a right eigenvector of all ( )iW , i.e., ( )1 1 , ;iW i= ∀

ii) 1T is not a left eigenvector of any ( )iW ，i.e., ( )1 1 ,iT TW i≠ ∀

. Lemma 2: The average weight matrix W is given by

( ) { }1 1 11 =r r rW E W t I diag A A I LN N N− − −= = − + −⎡ ⎤⎣ ⎦ (10)

And for all r , W satisfies the following conditions: ( )1 1,1 1 , 1T TW W W Jρ= = − < . (11)

Where ( )ρ ⋅ denotes the spectral radius of its argument and

1 11TJN

= .

Remark: If ( )1 1,1 1 , 1T TW W W Jρ= = − < , from [4] ，we have

1lim 11t T

tW

N→∞= .

B. Converges in the Expectation We consider the convergence in expectation of the

broadcasting gossip algorithm. We consider the initial state as deterministic, and hence all expectations are averaging the mixing matrices only.

Proposition 1: The limiting of random vector in expectation with quantization is given by

( ) ( )1lim 1 11 0T

tE x t x

N→∞⎡ ⎤+ =⎣ ⎦ (12)

Proof: From Lemma 2 and (9), we have

( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )

( ) ( )

0

0

1

lim 1 lim ,0 0 ,

lim ,0 0 lim ,

1lim 0 11 0

t

t t i

t

t t i

t T

t

E x t E t x t i W i I e i

E t x E t i W i I e i

W x xN

ϕ ϕ

ϕ ϕ

→∞ →∞=

→∞ →∞=

+

→∞

⎡ ⎤⎛ ⎞⎡ ⎤+ = + −⎢ ⎥⎜ ⎟⎣ ⎦ ⎝ ⎠⎣ ⎦⎡ ⎤= + −⎡ ⎤ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= =

∑

∑

This proposition shows that the states of agents converge in expectation to the average of the agents’ initial states. Proposition 2: Let kλ is the k -th smallest eigenvalue of L . Then there holds

( ) ( )( )

22 2 2

22

1 1lim 1 11 04 1 1

T N

tE x t x

N Nη λ

ηλ→∞

⎧ ⎫⎡ ⎤ Δ⎪ ⎪+ − ≤⎢ ⎥⎨ ⎬⎡ ⎤− −⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭ ⎣ ⎦

Proof: From Proposition 1, we have the following expression:

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )( ) ( ) ( )

2

2

2

1lim 1 11 0

1 1lim 1 11 0 1 11 0

1lim 1 11 0 1

1 1lim 1 11 0 11 0

lim

T

t

TT T

t

TT

t

T T T

t

t

E x t xN

E x t x x t xN N

E x t x x tN

E x t x xN N

→∞

→∞

→∞

→∞

→

⎧ ⎫⎡ ⎤⎪ ⎪+ −⎢ ⎥⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞ ⎛ ⎞= + − + −⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞= + − +⎢ ⎥⎨ ⎬⎜ ⎟

⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭⎡ ⎤⎛ ⎞− + −⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

= ( ){ } ( )( ) ( )2 11 0 11 0T TE x t x x

N∞⎡ ⎤+ −⎢ ⎥⎣ ⎦

(13)

Then from (9),

( ){ }( ) ( ) ( ) ( )( ) ( )

( ) ( ){ }( ) ( )( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) ( )( )

2

2

0

2

0

0

lim 1

lim ,0 0 ,

lim ,0 0

lim ,0 0 ,

lim , ,0 0

lim

t

t

t i

t

tT

t i

Tt

t i

t

E x t

E t x t i W i I e i

E t x

E t x t i W i I e i

E t i W i I e i t x

ϕ ϕ

ϕ

ϕ ϕ

ϕ ϕ

→∞

→∞ =

→∞

→∞ =

→∞ =

→

⎡ ⎤+⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤⎪ ⎪= + −⎢ ⎥⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

⎡ ⎤= ⎢ ⎥⎣ ⎦

⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪+ −⎨ ⎬⎢ ⎥⎜ ⎟⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪+ −⎢ ⎥⎨ ⎬⎜ ⎟⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

+

∑

∑

∑

( ) ( )( ) ( )

( )( ) ( ) ( ) ( )( ) ( )

2

0

2

0

,

1 0 11 0 lim ,

t

i

tT T

t i

E t i W i I e i

x x E t i W i I e iN

ϕ

ϕ

∞ =

→∞ =

⎧ ⎫⎡ ⎤⎪ ⎪−⎢ ⎥⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

⎧ ⎫⎪ ⎪= + −⎨ ⎬⎪ ⎪⎩ ⎭

∑

∑

(14)

By (13) and (14), we have

( ) ( )

( ) ( )( ) ( )

2

2

0

1 1lim 1 11 0

1lim ,

T

t

t

t i

E x t xN N

E t i W i I e iN

ϕ

→∞

→∞ =

⎧ ⎫⎡ ⎤⎪ ⎪+ −⎢ ⎥⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

⎧ ⎫⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭∑

(15)

Since ( ) ( ) ( ) ( ), 1t i W t W t W iϕ = − and ( )E W t W=⎡ ⎤⎣ ⎦ , then

( )( ) ( ) ( ) ( )( ) 1, 1 t iE t i E W t W t W i Wϕ − += − = and

( ) 1 rE W i I W I L LN

η−− = − = − = −⎡ ⎤⎣ ⎦ , Where 1 rN

η −= .

Since L and ( )E W t W I Lη= = −⎡ ⎤⎣ ⎦ are symmetric and doubly

stochastic, thus they can be diagonalized by same

orthogonal-matrix for the reason that they have the same

eigenvectors. Let kP denote the orthogonal eigenvector of L ,

so 1Tk kP P = .Now the eigen-decomposition of W yields

( )1

1N

Tk k k

lW P Pηλ

== −∑ .

Then the following expression holds:

2146

( )( ) ( )( ) ( )( ) ( )( )

( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )

2 2 2

2 2 2 2

1

2 2 2 2

1

e e

e 1 e

1 e e

t iTT

NT t i T

k k k ki

Nt i T T

k k k kk

E i W i I W W i I E i

E i P P E i

E i P P i

ηλ ηλ

ηλ ηλ

− +

− −

=

− −

=

− −

⎛ ⎞= − −⎜ ⎟⎝ ⎠

= − −

∑

∑

(16)

Due to independence assumption, we have ( ) ( )e e 0TE j i⎡ ⎤ =⎣ ⎦

when i j≠ , then

( ) ( )( ) ( )

( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )

2

0

2 2 2 2

1

2 2 2 2

0 1

,

1 e e

1 e e

t

i

Nt i T T

k k k kkt N

t i T Tk k k k

i k

E t i W i I e i

E i P P i

traceE i P P i

ϕ

ηλ ηλ

ηλ ηλ

=

− −

=

− −

= =

⎡ ⎤−⎢ ⎥

⎢ ⎥⎣ ⎦

= − −

= − −

∑

∑

∑∑

(17)

Recalling that 1Tk kP P = , Utilizing the fact that

( )2 1 1 2T Ttrace f f f f= , therefore we obtain

( ) ( )( )2

e e4

T Tk ktraceE i P P i Δ≤ , with this

( ) ( )( ) ( )

( ) ( )

2

0

22 2 2 2

0 1

,

14

t

i

t Nt i

k ki k

E t i W i I e iϕ

ηλ ηλ

=

− −

= =

⎡ ⎤−⎢ ⎥

⎢ ⎥⎣ ⎦Δ≤ − −

∑

∑ ∑ (18)

Since kλ is the k -th smallest eigenvalue of L , then

10 Nλ λ≤ ≤ ≤ , we obtain

( ) ( )( ) ( )

( ) ( )

( )

( ) ( )

( ) ( )( )

2

2

2

2

0

22 2 2 2

0 1

2 22 2 2

0 1

2 22 2 2

20

22 22

22

,

14

14

11

4

1 1 14 1 1

t

i

t Nt i

k ki k

t Nt i

k ki k

tt iN

it

N

E t i W i I e i

N

N

ϕ

η λ η λ

η λ η λ

η λη λ

η λ η λη λ

=

− −

= =

− −

= =

− −

=

⎡ ⎤−⎢ ⎥

⎢ ⎥⎣ ⎦Δ≤ − −

Δ= −

Δ −≤ −

Δ − − −= ⋅

− −

∑

∑ ∑

∑ ∑

∑

. (19)

Combining (15) with (19), we have

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

2 2

2

2

0

22 2 2 22

2 22 2

1 1lim 1 11 0

1lim ,

1 1 1lim

4 1 1 4 1 1

T

t

t

t i

tN N

t

E x t xN N

E t i W i I e iN

N

ϕ

η λ ηλ η ληλ ηλ

→∞

→∞ =

→∞

⎧ ⎫⎡ ⎤⎪ ⎪+ −⎢ ⎥⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

⎧ ⎫⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭

Δ − − − Δ≤ ⋅ =⎡ ⎤− − − −⎣ ⎦

∑ .

The proof of proposition 2 is complete, and this proposition shows that the mean square error have an upper bound. This upper bound clearly demonstrates the dependence on the quantization resolution, the second and last smallest eigenvalue of Laplacian matrix.

IV. SIMULATION RESULT We consider a network with 30 agents randomly

distributed on the unit square [ ] [ ]0,1 0,1× in the simulation, and any two nodes connect if the distance between them is less than 0.35 and there exists a link between them. Quantization length l 2 =1,5,8I= , that is the quantization bins ( )2 2 1lIΔ = − . Fig1shows the evolution of mean square error obtained by using the broadcast Gossip algorithm with quantized information communication constraints in this paper.

Fig1 Evolution of mean square error.

V. CONCLUSTION In this work we presented the conditions on the weights

matrices that would guarantee convergence to a consensus, and showed that the broadcast gossip algorithm achieves consensus with quantized information communication constraints. Moreover, the random consensus value is, in expectation, equal to the average of the initial node states. We analyzed the statistical performance of the algorithm and obtained an upper bound for the mean squared error of the broadcasting gossip algorithm. This upper bound clearly demonstrates the dependence on the quantization resolution, the second and last smallest eigenvalue of Laplacian matrix.

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