On Ergodicity, Infinite Flow and Consensus in Random Models

TL;DR

This paper investigates the relationship between ergodicity, infinite flow, and consensus in random linear models driven by independent stochastic matrices, establishing a fundamental theorem linking infinite flow to ergodicity.

Contribution

It introduces the infinite flow phenomenon and proves its equivalence to ergodicity in certain random models with common steady states and feedback, providing a deterministic characterization.

Findings

Infinite flow is necessary and sufficient for ergodicity in the studied models.

The results apply to consensus and average consensus over random graphs.

A new fundamental theorem linking infinite flow and ergodicity is established.

Abstract

We consider the ergodicity and consensus problem for a discrete-time linear dynamic model driven by random stochastic matrices, which is equivalent to studying these concepts for the product of such matrices. Our focus is on the model where the random matrices have independent but time-variant distribution. We introduce a new phenomenon, the infinite flow, and we study its fundamental properties and relations with the ergodicity and consensus. The central result is the infinite flow theorem establishing the equivalence between the infinite flow and the ergodicity for a class of independent random models, where the matrices in the model have a common steady state in expectation and a feedback property. For such models, this result demonstrates that the expected infinite flow is both necessary and sufficient for the ergodicity. The result is providing a deterministic characterization of the ergodicity, which can be used for studying the consensus and average consensus over random graphs.