Dirichlet Eigenvalues, Local Random Walks, and Analyzing Clusters in Graphs.

A cluster S in a massive graph G is characterised by the property that its corresponding vertices are better connected with each other, in comparison with the other vertices of the graph. Modeling, finding and analyzing clusters in massive graphs is an important topic in various disciplines. In this work we study local random walks that always stay in a cluster S. Moreover, we initiate the study of the local mixing time and the almost stable distribution, by analyzing Dirichlet eigenvalues in graphs. We prove that the Dirichlet eigenvalues of any connected subset S can be used to bound the epsilon-uniform mixing time, which improves the previous best-known result. We further present two applications of our results. The first is a polynomial-time algorithm for finding clusters with an improved approximation guarantee, while the second is the significance ordering of vertices in a cluster.