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

TL;DR

This paper introduces faster algorithms for key random walk computations on directed graphs, significantly improving efficiency and breaking previous time complexity barriers, with implications for directed spectral graph theory.

Contribution

The paper presents novel algorithms that compute stationary distributions, personalized PageRank, hitting times, and escape probabilities in sub-quadratic time, advancing directed Laplacian system solutions.

Findings

Achieves $ ilde{O}(m^{3/4}n+mn^{2/3})$ time for key computations on directed graphs.

Breaks the $O(n^{2})$ barrier for sparse graphs, improving over previous methods.

Provides a new approach to solving directed Laplacian systems efficiently.

Abstract

In this paper, we provide faster algorithms for computing various fundamental quantities associated with random walks on a directed graph, including the stationary distribution, personalized PageRank vectors, hitting times, 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; previous results either invoke a general purpose linear system solver on a $n\times n$ 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 a running time of $\tilde{O}(n^{7/4})$, breaking the $O(n^{2})$ barrier of the best running time 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 the well studied symmetric or undirected Laplacian systems. We show how to solve such systems in time $\tilde{O}(m^{3/4}n+mn^{2/3})$, and efficiently reduce a broad range of problems to solving $\tilde{O}(1)$ directed Laplacian systems on Eulerian graphs. We hope these results and our analysis open the door for further study into directed spectral graph theory.