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Suppose that $T$ is a stochastic matrix. We propose an algorithm foridentifying clusters in the Markov chain associated with $T$. The algorithm isrecursive in nature, and in order to identify clusters, it uses the signpattern of a left singular vector associated with the second smallest singularvalue of the Laplacian matrix $I-T.$ We prove a number of results that justifythe algorithm's approach, and illustrate the algorithm's performance withseveral numerical examples.

Clusters in Markov Chains via Singular Vectors of Laplacian Matrices