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

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.

💡 Research Summary

The paper investigates the asymptotic convergence behavior of products of random, symmetric, doubly‑stochastic matrices (W_k) that are independent and identically distributed (i.i.d.) over time. Such matrix products arise in a variety of distributed algorithms—consensus, gossip, and consensus‑plus‑innovations—where the underlying communication graph may be random because of protocol randomness or link failures. The authors focus on the probability that the spectral norm of the deviation from the averaging matrix (J=\frac1N\mathbf 1\mathbf 1^\top) exceeds a fixed threshold (\epsilon>0): \