Computing Heat Kernel Pagerank and a Local Clustering Algorithm

Heat kernel pagerank is a variation of Personalized PageRank given in an exponential formulation. In this work, we present a sublinear time algorithm for approximating the heat kernel pagerank of a graph. The algorithm works by simulating random walks of bounded length and runs in time O( log( ) logn 3 log log( 1) ), assuming performing a random walk step and sampling from a distribution with bounded support take constant time. The quantitative ranking of vertices obtained with heat kernel pagerank can be used for local clustering algorithms. We present an e cient local clustering algorithm that nds cuts by performing a sweep over a heat kernel pagerank vector, using the heat kernel pagerank approximation algorithm as a subroutine. Speci cally, we show that for a subset S of Cheeger ratio , many vertices in S may serve as seeds for a heat kernel pagerank vector which will nd a cut of conductance O( p ).