In this paper we introduce a notion of spectral approximation for directed graphs. While there are many potential ways one might define approximation for directed graphs, most of them are too strong to allow sparse approximations in general. In contrast, we prove that for our notion of approximation, such sparsifiers do exist, and we show how to compute them in almost linear time. Using this notion of approximation, we provide a general framework for solving asymmetric linear systems that is broadly inspired by the work of [Peng-Spielman, STOC`14]. Applying this framework in conjunction with our sparsification algorithm, we obtain an almost linear time algorithm for solving directed Laplacian systems associated with Eulerian Graphs. Using this solver in the recent framework of [Cohen-Kelner-Peebles-Peng-Sidford-Vladu, FOCS`16], we obtain almost linear time algorithms for solving a directed Laplacian linear system, computing the stationary distribution of a Markov chain, computing expected commute times in a directed graph, and more. For each of these problems, our algorithms improves the previous best running times of $O((nm^{3/4} + n^{2/3} m) log^{O(1)} (n kappa epsilon^{-1}))$ to $O((m + n2^{O(sqrt{log{n}loglog{n}})}) log^{O(1)} (n kappa epsilon^{-1}))$ where $n$ is the number of vertices in the graph, $m$ is the number of edges, $kappa$ is a natural condition number associated with the problem, and $epsilon$ is the desired accuracy. We hope these results open the door for further studies into directed spectral graph theory, and will serve as a stepping stone for designing a new generation of fast algorithms for directed graphs.
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