Faster Algorithms for Computing the Stationary Distribution, Simulating Random Walks, and More

In this paper we provide faster algorithms for computing various fundamental quantities associated with the random walk on a directed graphs, including the stationary distribution, personalized PageRank vectors, the hitting time between vertices, and escape probabilities. In particular, on a directed graph with n vertices and m edges. We show how to compute each quantity in time \tilde{O}(m^{3/4}n+mn^{2/3}) where the \tilde{O} notation suppresses polylogarithmic factors in n, the desired accuracy, and the appropriate condition number (i.e. the mixing time or restart probability). Our result, improves upon the previous fastest running times for these problems which either invoke a general purpose linear system solver on n\times n matrix with m non-zero entries or depend polynomially on the desired error or natural condition number associated with the problem (i.e. the mixing time or restart probability). For sparse graphs we obtain running time of \tilde{O}(n^{7/4}), which breaks the O(n^{2}) barrier that would be the best that one could hope to achieve using fast matrix multiplication. We achieve our result by providing a similar running time improvement for solving directed Laplacian systems, a natural directed or asymmetric analog of well studied symmetric or undirected Laplacian systems. We show how to solve such systems in \tilde{O}(m^{3/4}n+mn^{2/3}) and efficiently reduce all of these problems to solving a \otilde(1) number of directed Laplacian systems with Eulerian graphs. We hope these results and our analysis open the door for further study into directed spectral graph theory.