arXiv:1302.5366v3 [cs.DS] 10 Mar 2016 Testing whether the Uniform Distribution is a Stationary Distribution Sourav Chakraborty a , Akshay Kamath a , Rameshwar Pratap 1a a Chennai Mathematical Institute, Chennai, India. e-mail:{sourav,adkamath,rameshwar}@cmi.ac.in Abstract A random walk on a directed graph generates a Markov chain on the vertices of the graph. An important question that often arises in the context of Markov chains is, whether the uniform distribution on the vertices of the graph is a stationary distribution. A stationary distribution of a Markov chain is a global property of the graph. This leads to the belief that whether a particular distribution is a stationary distribution of a Markov chain depends on the global property of that Markov chain. In this paper for a directed graph whose underlying undirected graph is regular, we prove that 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 a directed edge, then out-degree(u) is equal to in-degree(v). This result also has an application to the problem of testing whether a given distribution is uniform or “far” from being uniform. 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 (orientations) of the input graph does one need to query in order to decide whether the distribution is uniform or “far” 1 from it? This is a problem of graph property testing, and we consider this problem in the orientation model. We reduce this problem to testing Eulerianity in the orientation model. Keywords: Markov Chain; Property Testing; Orientation Model; Stationary Distribution. 1. Introduction Spectral properties of undirected graphs have been well studied and well understood [2]. How- ever, there has been less success in the study of the same in the case of directed graphs, possibly due the non-symmetric structure associated with its adjaceny matrix. In this work, we attempt to understand the spectral properties of Markov chains obtained by a random walk on a directed graph. Markov chains are one of the most important structures in Theoretical Computer Science. The most significant characteristics of a Markov chain are its stationary distribution and mixing time. It is an 1 Corresponding author: This work done when the author was pursuing his PhD from Chennai Mathematical Institute. Present/corresponding address: 122/7 PAC Colony, Naini, Allahabad, UP, India. Contact Number: +91 9953842289 1 Here, farness does not imply any statistical distance between the distributions. Rather, it specifies the distance between the orientations - i.e. the minimum number of edges that need to be reoriented such that the stationary distribution obtained by a random walk on the resulting graph (obtained after reorientation of edges) is uniform. Preprint submitted to Elsevier October 30, 2018