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

TL;DR

This paper investigates consensus algorithms in sensor networks with unreliable links and noisy channels, proposing two methods to balance bias and variance, and analyzing their convergence and error properties.

Contribution

It introduces two novel consensus algorithms, $ ext{A-ND}$ and $ ext{A-NC}$, that effectively handle link failures and noise, with rigorous convergence proofs and explicit error analysis.

Findings

$ ext{A-ND}$ achieves asymptotic unbiasedness and explicit variance calculation.

$ ext{A-NC}$ offers a practical compromise with fixed iterations and Monte Carlo averaging.

The algorithms balance bias and variance in noisy, intermittent network conditions.

Abstract

The paper studies average consensus with random topologies (intermittent links) \emph{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 $\mathcal{A-ND}$ algorithm modifies conventional consensus by forcing the weights to satisfy a \emph{persistence} condition (slowly decaying to zero); and the $\mathcal{A-NC}$ algorithm where the weights are constant but consensus is run for a fixed number of iterations $\hat{\imath}$, then it is restarted and rerun for a total of $\hat{p}$ runs, and at the end averages the final states of the $\hat{p}$ runs (Monte Carlo averaging). We use controlled Markov processes and stochastic approximation arguments to prove almost sure convergence of $\mathcal{A-ND}$ to the desired average (asymptotic unbiasedness) and compute explicitly the m.s.e. (variance) of the consensus limit. We show that $\mathcal{A-ND}$ represents the best of both worlds--low bias and low variance--at the cost of a slow convergence rate; rescaling the weights...