Reachability of Consensus and Synchronizing Automata

TL;DR

This paper presents a polynomial-time algorithm to determine if a sequence of stochastic matrices can drive a consensus system to reach agreement, by linking automata theory with consensus dynamics.

Contribution

It introduces a novel connection between automata theory and stochastic matrix products to efficiently decide consensus reachability.

Findings

Polynomial-time algorithm for consensus reachability

Generalization of automata theory results to stochastic matrices

Efficient decision procedure for consensus systems

Abstract

We consider the problem of determining the existence of a sequence of matrices driving a discrete-time consensus system to consensus. We transform this problem into one of the existence of a product of the transition (stochastic) matrices that has a positive column. We then generalize some results from automata theory to sets of stochastic matrices. We obtain as a main result a polynomial-time algorithm to decide the existence of a sequence of matrices achieving consensus.