Distributed Consensus Algorithms in Sensor Networks With Imperfect Communication: Link Failures and Channel Noise

The paper studies average consensus with random topologies (intermittent links) and noisy channels. Consensus with noise in the network links leads to the bias-variance dilemma-running consensus for long reduces the bias of the final average estimate but increases its variance. We present two different compromises to this tradeoff: the A - ND algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition (slowly decaying to zero;) and the A - NC algorithm where the weights are constant but consensus is run for a fixed number of iterations ^ (iota), then it is restarted and rerun for a total of ^ (p) runs, and at the end averages the final states of the ^ (p) runs (Monte Carlo averaging). We use controlled Markov processes and stochastic approximation arguments to prove almost sure convergence of A - ND to a finite consensus limit and compute explicitly the mean square error (mse) (variance) of the consensus limit. We show that A - ND represents the best of both worlds-zero bias and low variance-at the cost of a slow convergence rate; rescaling the weights balances the variance versus the rate of bias reduction (convergence rate). In contrast, A - NC, because of its constant weights, converges fast but presents a different bias-variance tradeoff. For the same number of iterations ^ (iota) ^ (p), shorter runs (smaller ^ (iota) ) lead to high bias but smaller variance (larger number ^ (p) of runs to average over. ) For a static nonrandom network with Gaussian noise, we compute the optimal gain for A - NC to reach in the shortest number of iterations ^ (iota) ^ (p), with high probability (1-delta), (epsiv, delta) -consensus (epsiv residual bias). Our results hold under fairly general assumptions on the random link failures and communication noise.