A Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains

TL;DR

This paper generalizes the Perron-Frobenius theorem to time-varying stochastic matrices, establishing conditions for unique, positive absolute probability sequences applicable in various dynamic network processes.

Contribution

It introduces a set of connectivity conditions for time-varying matrices, extending the Perron-Frobenius theorem to dynamic networks and proving the positivity and uniqueness of absolute probability sequences.

Findings

Absolute probability sequences are uniformly positive under the conditions.

The results apply to both discrete-time and continuous-time systems.

Applications include non-Bayesian learning and opinion dynamics.

Abstract

We derive a generalization of the Perron-Frobenius theorem to time-varying row-stochastic matrices as follows: using Kolmogorov's concept of absolute probability sequences, which are time-varying analogs of principal eigenvectors, we identify a set of connectivity conditions that generalize the notion of irreducibility (strong connectivity) to time-varying matrices (networks), and we show that under these conditions, the absolute probability sequence associated with a given matrix sequence is (a) uniformly positive and (b) unique. Our results apply to both discrete-time and continuous-time settings. We then discuss a few applications of our main results to non-Bayesian learning, distributed optimization, opinion dynamics, and averaging dynamics over random networks.