The Role of Kemeny's Constant in Properties of Markov Chains
Speaker:\n\nProf. J. J. Hunter\n\n\nAbstract:\n\nIn 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}\tis a constant, K, not depending on i. This constant has since become known as Kemeny\u2019s 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.