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
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ND
algorithm modifies conventional consensus by forcing the weights to satisfy a
persistence
condition (slowly decaying to zero;) and the
A
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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
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ND
to a finite consensus limit and compute explicitly the mean square error (mse) (variance) of the consensus limit. We show that
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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
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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.