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Random bisection and evolutionary walks

Part of: Artificial intelligence (68Txx) Special processes

Published online by Cambridge University Press:  14 July 2016

Eric Bach
Eric Bach*
Affiliation:
University of Wisconsin
*
∗ Postal address: Computer Sciences Department, University of Wisconsin, 1210 West Dayton St, Madison, WI 53706, USA. Email address: bach@cs.wisc.edu
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Abstract

As models for molecular evolution, immune response, and local search algorithms, various authors have used a stochastic process called the evolutionary walk, which goes as follows. Assign a random number to each vertex of the infinite N-ary tree, and start with a particle on the root. A step of the process consists of searching for a child with a higher number and moving the particle there; if no such child exists, the process stops. The average number of steps in this process is asymptotic, as N → ∞, to log N, a result first proved by Macken and Perelson. This paper relates the evolutionary walk to a process called random bisection, familiar from combinatorics and number theory, which can be thought of as a transformed Poisson process. We first give a thorough treatment of the exact walk length, computing its distribution, moments and moment generating function. Next we show that the walk length is asymptotically normally distributed. We also treat it as a mixture of Poisson random variables and compute the asymptotic distribution of the Poisson parameter. Finally, we show that in an evolutionary walk with uniform vertex numbers, the ‘jumps’, ordered by size, have the same asymptotic distribution as the normalized cycle lengths in a random permutation.

Keywords

immune responsemolecular evolutionlocal searchPoisson processPoisson-Dirichlet law

MSC classification

Primary: 60K99: None of the above, but in this section
Secondary: 68T20: Problem solving (heuristics, search strategies, etc.) 92D15: Problems related to evolution
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 582 - 596
Copyright
Copyright © Applied Probability Trust 2001 

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