Consensus Algorithms and the Decomposition-Separation Theorem

TL;DR

This paper investigates the convergence of consensus algorithms based on inhomogeneous Markov chains, establishing necessary and sufficient conditions for consensus, and generalizing existing results through an extension of the Kolmogorov-Doeblin decomposition.

Contribution

It extends the classical decomposition-separation theorem to inhomogeneous Markov chains and applies it to analyze consensus algorithms, providing a unified framework for existing results.

Findings

Necessary conditions for consensus are established.

Sufficient conditions are proven for chains of Class P*.

The approach generalizes and unifies previous consensus results.

Abstract

Convergence properties of time inhomogeneous Markov chain based discrete and continuous time linear consensus algorithms are analyzed. Provided that a so-called infinite jet flow property is satisfied by the underlying chains, necessary conditions for both consensus and multiple consensus are established. A recenet extension by Sonin of the classical Kolmogorov-Doeblin decomposition-separation for homogeneous Markov chains to the inhomogeneous case is then employed to show that the obtained necessary conditions are also sufficient when the chain is of Class P*, as defined by Touri and Nedic. It is also shown that Sonin's theorem leads to a rediscovery and generalization of most of the existing related consensus results in the literature.