The Kemeny constant for finite homogeneous ergodic Markov chains

M. Catral, S. J. Kirkland, M. Neumann, Nung Sing Sze

Research output: Journal article publicationJournal articleAcademic researchpeer-review

19 Citations (Scopus)


A quantity known as the Kemeny constant, which is used to measure the expected number of links that a surfer on the World Wide Web, located on a random web page, needs to follow before reaching his/her desired location, coincides with the more well known notion of the expected time to mixing, i.e., to reaching stationarity of an ergodic Markov chain. In this paper we present a new formula for the Kemeny constant and we develop several perturbation results for the constant, including conditions under which it is a convex function. Finally, for chains whose transition matrix has a certain directed graph structure we show that the Kemeny constant is dependent only on the common length of the cycles and the total number of vertices and not on the specific transition probabilities of the chain.
Original languageEnglish
Pages (from-to)151-166
Number of pages16
JournalJournal of Scientific Computing
Issue number1-3
Publication statusPublished - 1 Oct 2010


  • Directed graphs
  • Group inverses
  • Markov chains
  • Mean first passage times
  • Nonnegative matrices
  • Stationary distribution vectors
  • Stochastic matrices

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Software
  • Engineering(all)
  • Computational Theory and Mathematics


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