The stationary distribution in the antivoter model: exact sampling and approximations

Abstract

<!-- *** Custom HTML *** --> The antivoter model is a Markov chain on regular graphs which has a unique stationary distribution, but is not reversible. This makes the stationary distribution difficult to describe. Despite the fact that in general we know nothing about the stationary distribution other than it exists and is unique, we present a method for sampling exactly from this distribution. The method has running time $O(n^3 r / c)$, where $n$ is the number of nodes in the graph, $c$ is the size of the minimum cut in the graph, and $r$ is the degree of each node in the graph. We also show that the original chain has $O(n^3 r /c)$ mixing time. For the antivoter model on the complete graph we derive a closed form solution for the stationary distribution. Moreover we bound the total variation distance between the stationary distribution for the antivoter model on a multipartite graph and the stationary distribution on the complete graph, using Stein's method. Finally, we present computational experiments comparing the empirical Stein's method for estimating the stationary distribution to the classical frequency estimate.