Consensus and Products of Random Stochastic Matrices: Exact Rate for Convergence in Probability

TL;DR

This paper derives the exact exponential rate of convergence in probability for products of random symmetric stochastic matrices in distributed systems, providing a computable method for common models and applications in sensor networks.

Contribution

It establishes the precise exponential convergence rate for symmetric i.i.d. stochastic matrix products and links it to a min-cut problem for practical models.

Findings

Exact rate I is derived for symmetric i.i.d. matrices.

Rate I can be computed via a min-cut problem.

Application to sensor power allocation in distributed detection.

Abstract

Distributed consensus and other linear systems with system stochastic matrices $W_k$ emerge in various settings, like opinion formation in social networks, rendezvous of robots, and distributed inference in sensor networks. The matrices $W_k$ are often random, due to, e.g., random packet dropouts in wireless sensor networks. Key in analyzing the performance of such systems is studying convergence of matrix products $W_kW_{k-1}... W_1$. In this paper, we find the exact exponential rate $I$ for the convergence in probability of the product of such matrices when time $k$ grows large, under the assumption that the $W_k$'s are symmetric and independent identically distributed in time. Further, for commonly used random models like with gossip and link failure, we show that the rate $I$ is found by solving a min-cut problem and, hence, easily computable. Finally, we apply our results to optimally allocate the sensors' transmission power in consensus+innovations distributed detection.