Exploiting Quotients of Markov Chains to Derive Properties of the Stationary Distribution of the Markov Chain Associated to an Evolutionary Algorithm

  • Boris Mitavskiy
  • Jonathan E. Rowe
  • Alden Wright
  • Lothar M. Schmitt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4247)


In this work, a method is presented for analysis of Markov chains modeling evolutionary algorithms through use of a suitable quotient construction. Such a notion of quotient of a Markov chain is frequently referred to as “coarse graining” in the evolutionary computation literature. We shall discuss the construction of a quotient of an irreducible Markov chain with respect to an arbitrary equivalence relation on the state space. The stationary distribution of the quotient chain is “coherent” with the stationary distribution of the original chain. Although the transition probabilities of the quotient chain depend on the stationary distribution of the original chain, we can still exploit the quotient construction to deduce some relevant properties of the stationary distribution of the original chain. As one application, we shall establish inequalities that describe how fast the stationary distribution of Markov chains modelling evolutionary algorithms concentrates on the uniform populations as the mutation rate converges to 0. Further applications are discussed.


Markov Chain Evolutionary Algorithm Stationary Distribution Coarse Graining Markov Chain Model 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Boris Mitavskiy
    • 1
  • Jonathan E. Rowe
    • 2
  • Alden Wright
    • 3
  • Lothar M. Schmitt
    • 4
  1. 1.Academic Unit of Mathematical Modelling and Genetic Epidemiology, Division of Genomic Medicine, School of MedicineUniversity of SheffieldSheffield
  2. 2.School of Computer ScienceUniversity of BirminghamBirminghamGreat Britain
  3. 3.Computer ScienceUniversity of MontanaMissoulaUSA
  4. 4.School of Computer Science and EngineeringThe University of AizuAizu-Wakamatsu City, Fukushima PrefectureJapan

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