Journal of Scientific Computing

, Volume 45, Issue 1, pp 151–166

The Kemeny Constant for Finite Homogeneous Ergodic Markov Chains


DOI: 10.1007/s10915-010-9382-1

Cite this article as:
Catral, M., Kirkland, S.J., Neumann, M. et al. J Sci Comput (2010) 45: 151. doi:10.1007/s10915-010-9382-1


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.


Nonnegative matricesGroup inversesDirected graphsMarkov chainsStationary distribution vectorsStochastic matricesMean first passage times

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • M. Catral
    • 1
  • S. J. Kirkland
    • 2
  • M. Neumann
    • 3
  • N.-S. Sze
    • 4
  1. 1.Department of Mathematics and Computer ScienceXavier UniversityCincinnatiUSA
  2. 2.Hamilton InstituteNational University of Ireland MaynoothMaynooth, Co. KildareIreland
  3. 3.Department of MathematicsUniversity of ConnecticutStorrsUSA
  4. 4.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHong KongChina