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.
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References
Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)
Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)
Brauer, A.: Limits for the characteristic roots of a matrix. IV: Applications to stochastic matrices. Duke Math. J. 19, 75–91 (1952)
Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear Transformations. Dover, New York (1991)
Cho, G.E., Meyer, C.D.: Markov chain sensitivity measured by mean first passage times. Linear Algebra Appl. 316, 21–28 (2000)
Dietzenbacher, E.: Perturbations of the Perron vector: applications to finite Markov chains and demographic population models. Environ. Plan. 22, 747–761 (1990)
Fiedler, M., Pták, V.: Diagonally dominant matrices. Czech. Math. J. 17, 420–433 (1967)
Grinstead, C.M., Snell, J.L.: Introduction to Probability. American Mathematical Society, Providence (1997)
Higham, D.: A matrix perturbation view of the small world phenomenon. SIAM J. Matrix Anal. Appl. 25, 429–444 (2003)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, New York (1985)
Hunter, J.J.: Mixing times with applications to perturbed Markov chains. Linear Algebra Appl. 417, 108–123 (2006)
Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Springer, New York (1960)
Kirkland, S.J.: The group inverse associated with an irreducible periodic nonnegative matrix. SIAM J. Matrix Anal. Appl. 16, 1127–1134 (1995)
Kirkland, S.J., Neumann, M.: On group inverses of M-matrices with uniform diagonal entries. Linear Algebra Appl. 296, 153–170 (1999)
Langville, A.N., Meyer, C.D.: Deeper inside PageRank. Internet Math. 1, 335–400 (2004)
Levene, M., Loizou, G.: Kemeny’s constant and the random surfer. Am. Math. Mon. 109, 741–745 (2002)
Metzler, L.: A multiple-county theory of income transfer. J. Political Econ. 59, 14–29 (1951)
Meyer, C.D.: The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev. 17, 443–464 (1975)
Seneta, E.: Non-Negative Matrices and Markov Chains. Springer, New York (1981)
Serra-Capizzano, S.: Jordan canonical form of the Google matrix: a potential contribution to the Pagerank computation. SIAM J. Matrix Anal. Appl. 27, 305–312 (2005)
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This paper is dedicated to the memory of Professor David Gottlieb 1944–2008.
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Catral, M., Kirkland, S.J., Neumann, M. et al. The Kemeny Constant for Finite Homogeneous Ergodic Markov Chains. J Sci Comput 45, 151–166 (2010). https://doi.org/10.1007/s10915-010-9382-1
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DOI: https://doi.org/10.1007/s10915-010-9382-1