Testing uniformity of stationary distribution

A random walk on a directed graph gives a markov chain on the vertices on 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 a stationary distribution for the markov chain. Testing whether a distribution is uniform or “far” from being uniform is a well studied problem in property testing and statistics. But in most of the cases the distribution is accessed by drawing random samples. In this paper we consider the case when the distribution is the stationary distribution for the random walk on a directed graph with the uniform distribution as the initial distribution. The distribution is specified using the graph. Random samples cannot be drawn from the distribution. The distribution is accessed by querying the structure of the graph. In this paper we consider the case where the underlying undirected graph structure is known in advance and the orientation of the edges have to queried. This model for testing graph properties is called the orientation model (introduced by Halevy et al). We prove that testing whether the stationary distribution is uniform can be reduced to testing a Eulerian in graphs. Fischer et al studied the problem of testing Eulerianness in the orientation model. Using their results we obtain bounds on the query complexity for testing whether the stationary distribution is uniform.