Distributed consensus algorithms in sensor networks with imperfect communication: link failures and channel noise

• Authors: Soummya Kar;José M. F. Moura
• Venue: IEEE Transactions on Signal Processing
• Year: 2009

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 p~rsistence 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 î, 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 îp shorter runs (smaller î) 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 îp with high probability (1 - δ), (ε, δ)-consensus (ε residual bias). Our results hold under fairly general assumptions on the random link failures and communication noise.