Using PageRank to Locally Partition a Graph

Abstract

A local graph partitioning algorithm finds a cut near a specified starting vertex, with a running time that depends largely on the size of the small side of the cut, rather than the size of the input graph. In this paper, we present a local partitioning algorithm using a variation of PageRank with a specified starting distribution. We derive a mixing result for PageRank vectors similar to that for random walks, and we show that the ordering of the vertices produced by a PageRank vector reveals a cut with small conductance. In particular, we show that for any set $C$ with conductance $\Phi$ and volume $k$, a PageRank vector with a certain starting distribution can be used to produce a set with conductance $O(\sqrt{\Phi \log k})$. We present an improved algorithm for computing approximate PageRank vectors, which allows us to find such a set in time proportional to its size. In particular, we can find a cut with conductance at most $\phi$, whose small side has volume at least $2^{b}$, in time $O(2^{b} \log^2m/\phi^2)$ where $m$ is the number of edges in the graph. By combining small sets found by this local partitioning algorithm, we obtain a cut with conductance $\phi$ and approximately optimal balance in time $O(m \log^4 m/\phi^2)$.