Clusters in Markov Chains via Singular Vectors of Laplacian Matrices

TL;DR

This paper introduces a recursive algorithm that leverages singular vectors of the Laplacian matrix to identify clusters in Markov chains, supported by theoretical justification and numerical validation.

Contribution

It presents a novel clustering algorithm for Markov chains based on singular vectors of the Laplacian, with proven theoretical foundations and demonstrated effectiveness.

Findings

Algorithm successfully identifies clusters in Markov chains.

Theoretical results justify the use of singular vectors for clustering.

Numerical examples show the algorithm's practical performance.

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.