Kemeny's constant

In probability theory, Kemeny’s constant is the expected number of time steps required for a Markov chain to transition from a starting state i to a random destination state sampled from the Markov chain's stationary distribution. Surprisingly, this quantity does not depend on which starting state i is chosen.[1] It is in that sense a constant, although it is different for different Markov chains. When first published by John Kemeny in 1960 a prize was offered for an intuitive explanation as to why quantity was constant.[2][3]

Definition

For a finite ergodic Markov chain[4] with transition matrix P and invariant distribution π, write mij for the mean first passage time from state i to state j (denoting the mean recurrence time for the case i = j). Then

is a constant and not dependent on i.[5]

Prize

Kemeny wrote, (for i the starting state of the Markov chain) “A prize is offered for the first person to give an intuitively plausible reason for the above sum to be independent of i.”[2] Grinstead and Snell offer an explanation by Peter Doyle as an exercise, with solution “he got it!”[6][7]

In the course of a walk with Snell along Minnehaha Avenue in Minneapolis in the fall of 1983, Peter Doyle suggested the following explanation for the constancy of Kemeny's constant. Choose a target state according to the fixed vector w. Start from state i and wait until the time T that the target state occurs for the first time. Let Ki be the expected value of T. Observe that

and hence

By the maximum principle, Ki is a constant. Should Peter have been given the prize?

References

  1. ↑ Crisostomi, E.; Kirkland, S.; Shorten, R. (2011). "A Google-like model of road network dynamics and its application to regulation and control". International Journal of Control. 84 (3): 633. doi:10.1080/00207179.2011.568005.
  2. 1 2 Kemeny, J. G.; Snell, J. L. (1960). Finite Markov Chains. Princeton, NJ: D. Van Nostrand. (Corollary 4.3.6)
  3. ↑ Catral, M.; Kirkland, S. J.; Neumann, M.; Sze, N.-S. (2010). "The Kemeny Constant for Finite Homogeneous Ergodic Markov Chains" (PDF). Journal of Scientific Computing. 45: 151. doi:10.1007/s10915-010-9382-1.
  4. ↑ Levene, Mark; Loizou, George (2002). "Kemeny's Constant and the Random Surfer" (PDF). The American Mathematical Monthly. Mathematical Association of America. 109 (8): 741–745. doi:10.2307/3072398. JSTOR 3072398.
  5. ↑ Hunter, Jeffrey J. (2012). "The Role of Kemeny's Constant in Properties of Markov Chains". Communications in Statistics - Theory and Methods. 43: 1309–1321. arXiv:1208.4716 [math.PR]. doi:10.1080/03610926.2012.741742. Cite uses deprecated parameter |class= (help)
  6. ↑ Grinstead, Charles M.; Snell, J. Laurie. Introduction to Probability (PDF).
  7. ↑ "Two exercises on Kemeny's constant" (PDF). Retrieved 1 March 2013.
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