Toward a Theory of Markov Influence Systems and their Renormalization

TL;DR

This paper introduces Markov influence systems (MIS), analyzing their dynamics, bifurcations, and chaos, revealing that irreducible MIS are typically asymptotically periodic despite potential chaos.

Contribution

It generalizes Markov chain classification to time-varying graphs and analyzes the bifurcation and long-term behavior of MIS, including examples of hyper-torpid mixing.

Findings

MIS can exhibit chaotic dynamics.

Irreducible MIS are almost always asymptotically periodic.

Example of super-exponential mixing time.

Abstract

We introduce the concept of a Markov influence system (MIS) and analyze its dynamics. An MIS models a random walk in a graph whose edges and transition probabilities change endogenously as a function of the current distribution. This article consists of two independent parts: in the first one, we generalize the standard classification of Markov chain states to the time-varying case by showing how to "parse" graph sequences; in the second part, we use this framework to carry out the bifurcation analysis of a few important MIS families. We show that, in general, these systems can be chaotic but that irreducible MIS are almost always asymptotically periodic. We give an example of "hyper-torpid" mixing, where a stationary distribution is reached in super-exponential time, a timescale beyond the reach of any Markov chain.