Using PageRank to Locally Partition a Graph
Reid Andersen, Fan Chung, Kevin Lang · Internet Mathematics · 2007
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 Φ and volume _k_, a PageRank vector with a certain starting distribution can be used to produce a set with conductance . 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 _ϕ_, whose small side has volume at least 2 b , in time _O_(2 b log2_m/ϕ_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 _ϕ_ and approximately optimal balance in time _O_(_m_ log4_m/ϕ_2).