The Kemeny Constant for Finite Homogeneous Ergodic Markov Chains Article

First Online: 02 July 2010 Received: 12 February 2009 Revised: 22 February 2010 Accepted: 25 May 2010 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 Abstract 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.

Keywords Nonnegative matrices Group inverses Directed graphs Markov chains Stationary distribution vectors Stochastic matrices Mean first passage times This paper is dedicated to the memory of Professor David Gottlieb 1944–2008.

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Authors and Affiliations 1. Department of Mathematics and Computer Science Xavier University Cincinnati USA 2. Hamilton Institute National University of Ireland Maynooth Maynooth, Co. Kildare Ireland 3. Department of Mathematics University of Connecticut Storrs USA 4. Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong China