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arXiv · 1208.4716

The Role of Kemeny's Constant in Properties of Markov Chains

Jeffrey J. Hunter
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Abstract

In a finite state irreducible Markov chain with stationary probabilities π_i and mean first passage times m_(ij) (mean recurrence time when i = j) it was first shown by Kemeny and Snell (1960) that \sum_j π_j m_(ij) is a constant K, not depending on i. This constant has since become known as Kemeny's constant. A variety of techniques for finding expressions and various bounds for K are derived. The main interpretation focuses on its role as the expected time to mixing in a Markov chain. Various applications are considered including perturbation results, mixing on directed graphs and its relation to the Kirchhoff index of regular graphs.

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Jeffrey J. Hunter. 2012-08-23. The Role of Kemeny's Constant in Properties of Markov Chains. https://doi.org/10.1080/03610926.2012.741742

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