On the Influence of Noise in Randomized Consensus Algorithms

TL;DR

This paper analyzes how additive noise affects the accuracy of randomized consensus algorithms, providing explicit formulas and bounds for the mean square error, especially in systems modeled by Laplacian matrices derived from graph structures.

Contribution

It introduces a closed-form expression for noise-induced error in symmetric update matrices and links bounds to graph eigenvalues and effective resistance, enhancing understanding of noisy consensus.

Findings

Derived a closed-form mean square error expression.

Established bounds using Laplacian eigenvalues and effective resistance.

Validated bounds through numerical evaluations on example graphs.

Abstract

In this paper we study the influence of additive noise in randomized consensus algorithms. Assuming that the update matrices are symmetric, we derive a closed form expression for the mean square error induced by the noise, together with upper and lower bounds that are simpler to evaluate. Motivated by the study of Open Multi-Agent Systems, we concentrate on Randomly Induced Discretized Laplacians, a family of update matrices that are generated by sampling subgraphs of a large undirected graph. For these matrices, we express the bounds by using the eigenvalues of the Laplacian matrix of the underlying graph or the graph's average effective resistance, thereby proving their tightness. Finally, we derive expressions for the bounds on some examples of graphs and numerically evaluate them.