Binary Consensus Algorithm

MOHAMED SEIFCOMMUNICATIONS AND ELECTRONICS DEPARTMENT

IntroductionCertain

Observation(Primary User)

Cooperative Sensing

(Secondary Users)

Global Decision(H0 / H1)

Iteration for ‘K’ time

steps

System ModelFully Connected Graph g=(V,E)

V={0,1,…..,M-1}

E=M x M

-We want to make a virtual fusion center

So;

Consensus Algorithm

Binary Consensus Algorithm(Overview)

Local decision of node ‘i’

Bi(0)

Sending Local Decisions to each other

Applying Consensus Algorithm

AnalysisFirst Phase (initial decision) for each agent

Second Phase (Sending Decisions to each other)

-nj,i(k) : a zero-mean Gaussian random variable with variance of σn2.

Analysis(Cont’d)Each agent will update its vote based on the received in information as fellows : R

So, the Decision Rule :

Analysis(Cont’d )

Markov Chain Representation

-Pi,j will have a binomial distribution as follows :

-ki : represents the probability that any agent votes 1 in the next time step, S(k)=I, (i.e current state =i)

Markov Chain Representation(Cont’d)Some Important Properties of P(Transition Matrix):

-By Applying Perron’s theorem, we will have :

1. λo =1, as a simple eigenvalue

2.|λi|< 1 for i ≠ 0

-We are interested in calculating the second largest eigenvalue λ1, which is important for time convergence analysis

Steady State Behavior

A. Case of σn = 0

In this ideal case, the network will reach an accurate consensus in one time step (all ones / all zeros)

B. Case of σn ≠ 0

The network will reach a steady state asymptotically and is independent of the initial state

Second Largest Eigenvalue Illustration of its importance :

Second Largest Eigenvalue(Cont’d)

Transient Part

Steady State Part

Second Largest Eigenvalue(Cont’d)

Results

Results(Cont’d)

ConclusionCensoring should beat this algorithm in two considerations :

1-The asymptotic behavior in case of σn ≠ 0

2-The time convergence should be smaller than the original case

Thank you