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

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…

💡 Research Summary

The paper tackles the problem of average consensus in sensor networks where communication links are intermittent and the channels are noisy—a setting that induces a classic bias‑variance dilemma. Running a conventional consensus algorithm for many iterations reduces the bias of the final estimate (i.e., brings the network state closer to the true average) but simultaneously amplifies the variance because the additive channel noise accumulates over time. To address this trade‑off, the authors propose two distinct algorithms, denoted A‑ND (Adaptive‑Noise‑Decaying) and A‑NC (Adaptive‑Noise‑Constant), and provide rigorous stochastic‑approximation based analyses that establish almost‑sure convergence and explicit mean‑square‑error (MSE) expressions for each method.

System Model.
The network is modeled as a time‑varying undirected graph G(i) whose edge set evolves according to an ergodic Markov chain. At each discrete time i, every node k holds a scalar state xₖ(i). The update rule incorporates a weight α(i) (or a constant α for A‑NC), the difference with its current neighbors, and an additive noise term vₖ(i) that is zero‑mean, independent across time and nodes, and has finite second moment σ². The Laplacian of the instantaneous graph is L(i); its expected value over the Markov chain is denoted (\bar L) with second smallest eigenvalue λ₂>0, which quantifies the average connectivity of the network.

A‑ND Algorithm.
A‑ND modifies the classic consensus iteration by imposing a persistence condition on the step‑size sequence:
\