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Directed forests and the constancy of Kemeny’s constant

  • Steve KirklandEmail author
Article
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Abstract

Consider a discrete–time, time–homogeneous Markov chain on states \(1, \ldots , n\) whose transition matrix is irreducible. Denote the mean first passage times by \(m_{jk}, j,k=1,\ldots , n,\) and stationary distribution vector entries by \(v_k, k=1, \ldots , n\). A result of Kemeny reveals that the quantity \(\sum _{k=1}^n m_{jk}v_k\), which is the expected number of steps needed to arrive at a randomly chosen destination state starting from state j,  is–surprisingly–independent of the initial state j. In this note, we consider \(\sum _{k=1}^n m_{jk}v_k\) from the perspective of algebraic combinatorics and provide an intuitive explanation for its independence on the initial state j. The all minors matrix tree theorem is the key tool employed.

Keywords

Markov chain Kemeny’s constant All minors matrix tree theorem 

Notes

Acknowledgements

The author’s research is supported in part by NSERC Discovery Grant RGPIN–2019–05408. The author is grateful to an anonymous referee, whose comments improved this article.

References

  1. 1.
    Bini, D., Hunter, J., Latouche, G., Meini, B., Taylor, P.: Why is Kemeny’s constant a constant? J. Appl. Probab. 55, 1025–1036 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chaiken, S.: A combinatorial proof of the all minors matrix tree theorem. SIAM J. Algebra. Disc. Methods 3, 319–329 (1982)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Doyle, P.: The Kemeny constant of a Markov chain (2009). arXiv:0909.2636v1 [math.PR]
  4. 4.
    Leighton, F., Rivest, R.: Estimating a probability using finite memory. IEEE Trans. Inf. Theory 32, 733–742 (1986)CrossRefGoogle Scholar
  5. 5.
    Grinstead, C., Snell, J.: Introduction to Probability. AMS, Providence (1997)zbMATHGoogle Scholar
  6. 6.
    Kemeny, J., Snell, J.: Finite Markov Chains. Springer, New York (1976)zbMATHGoogle Scholar
  7. 7.
    Kirkland, S., Zeng, Z.: Kemeny’s constant and an analogue of Braess’ paradox for trees. Electron. J. Linear Algebr. 31, 444–464 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada

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