Local Partitioning for Directed Graphs Using PageRank

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 a, and by performing a sweep over that vector, we can obtain a set of vertices S' with conductance FM(S') = O(root 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.