Testing Uniformity of Stationary Distribution

TL;DR

This paper establishes a local graph property that determines when the uniform distribution is stationary for a regular directed graph's Markov chain, and applies this to analyze the complexity of testing uniformity in property testing.

Contribution

It proves a local criterion for uniform stationary distribution in regular directed graphs and connects this to the complexity of testing uniformity in graph property testing.

Findings

The uniform distribution is stationary iff outdegree(u) = indegree(v) for edges (u,v).

Reduces uniformity testing to Eulerian graph testing in the orientation model.

Provides bounds on query complexity for testing stationarity of the uniform distribution.

Abstract

A random walk on a directed graph gives a Markov chain on the vertices of the graph. An important question that arises often in the context of Markov chain is whether the uniform distribution on the vertices of the graph is a stationary distribution of the Markov chain. Stationary distribution of a Markov chain is a global property of the graph. In this paper, we prove that for a regular directed graph whether the uniform distribution on the vertices of the graph is a stationary distribution, depends on a local property of the graph, namely if (u,v) is an directed edge then outdegree(u) is equal to indegree(v). This result also has an application to the problem of testing whether a given distribution is uniform or "far" from being uniform. This is a well studied problem in property testing and statistics. If the distribution is the stationary distribution of the lazy random walk on a directed graph and the graph is given as an input, then how many bits of the input graph do one need to query in order to decide whether the distribution is uniform or "far" from it? This is a problem of graph property testing and we consider this problem in the orientation model (introduced by Halevy et al.). We reduce this problem to test (in the orientation model) whether a directed graph is Eulerian. And using result of Fischer et al. on query complexity of testing (in the orientation model) whether a graph is Eulerian, we obtain bounds on the query complexity for testing whether the stationary distribution is uniform.