Local Partitioning for Directed Graphs Using PageRank

Abstract

A local partitioning algorithm finds a set with small conductance near a specified seed vertex. In this paper, we present a generalization of a local partitioning algorithm for undirected graphs to strongly connected directed graphs. In particular, we prove that by computing a personalized PageRank vector in a directed graph, starting from a single seed vertex within a set $S$ that has conductance at most $\alpha$, and by performing a sweep over that vector, we can obtain a set of vertices $S'$ with conductance $\Phi_{M}(S')= O(\sqrt{\alpha \log |S|})$. Here, the conductance function $\Phi_{M}$ is defined in terms of the stationary distribution of a random walk in the directed graph. In addition, we describe how this algorithm may be applied to the PageRank Markov chain of an arbitrary directed graph, which provides a way to partition directed graphs that are not strongly connected.