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

Prof. J. J. Hunter
9 May 2012
In a finite m-state irreducible Markov chain with stationary probabilities {\pi_i} and mean first passage times m_{ij} (mean recurrence time when i=j) it was first shown, by Kemeny and Snell, that \sum_{j=1}^{m}\pi_jm_{ij} is a constant, K, not depending on i. This constant has since become known as Kemeny’s constant. We consider a variety of techniques for finding expressions for K, derive some bounds for K, and explore various applications and interpretations of theseresults. Interpretations include the expected number of links that a surfer on the World Wide Web located on a random page needs to follow before reaching a desired location, as well as the expected time to mixing in a Markov chain. Various applications have been considered including some perturbation results, mixing on directed graphs and its relation to the Kirchhoff index of regular graphs.
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