Computing the stationary distribution of a Markov Chain is one of the most fundamental problems in optimization. It lies at the heart of numerous computational tasks including computing personalized PageRank vectors, evaluating the utility of policies in Markov decision process, and solving asymmetric diagonally dominant linear systems. Despite the ubiquity of these problems, until recently the fastest known running times for computing the stationary distribution either depended polynomially on the mixing time or appealed to generic linear system solving machinery, and therefore ran in super-quadratic time.

In this talk I will present recent results showing that the stationary distribution and related quantities can all be computed in almost linear time. I will present new iterative methods for extracting structure from directed graphs and and show how they can be tailored to achieve this new running time. Moreover, I will discuss connections between this work and recent developments in solving Laplacian systems and optimizing over directed graphs.

This talk reflects joint work with Michael B. Cohen (MIT), Jonathan Kelner (MIT), John Peebles (MIT), Richard Peng (Georgia Tech) and Adrian Vladu (MIT).