Toward a Theory of Markov Influence Systems and their Renormalization
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.
Keywords
Cite
@article{arxiv.1802.01208,
title = {Toward a Theory of Markov Influence Systems and their Renormalization},
author = {Bernard Chazelle},
journal= {arXiv preprint arXiv:1802.01208},
year = {2019}
}