Clusters in Markov chains via singular vectors of Laplacian matrices

Abstract

Suppose that T is a stochastic matrix. We propose an algorithm for identifying clusters in the Markov chain associated with T. The algorithm is recursive in nature, and in order to identify clusters, it uses the sign pattern of a left singular vector associated with the second smallest singular value of the Laplacian matrix I−T. We prove a number of results that justify the algorithm's approach, and illustrate the algorithm's performance with several numerical examples.