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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: [email protected]
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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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References

Aarts, E., and Lenstra, J. K. (eds) (1997). Local Search in Combinatorial Optimization. John Wiley, New York.Google Scholar
Arratia, R., Barbour, A. D. and Tavaré, S. (1997). Random combinatorial structures and prime factorizations. Notices Amer. Math. Soc. 44, 903–910.Google Scholar
Bach, E. (1988). How to generate factored random numbers. SIAM J. Comput. 17, 179–193.CrossRefGoogle Scholar
Billingsley, P. (1972). On the distribution of large prime divisors. Periodica Math. Hungarica 2, 283–289.CrossRefGoogle Scholar
Condon, E. U., and Breit, G. (1936). The energy distribution of neutrons slowed by elastic impacts. Phys. Rev. 49, 229–231.CrossRefGoogle Scholar
Erdõs, P., and Kac, M. (1940). The Gaussian law of errors in the theory of additive number theoretic functions. Amer. J. Math. 62, 738–742.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Gontcharoff, W. (1942). Sur la distribution des cycles dans les permutations. C. R. (Doklady) Acad. Sci. URSS (N.S.) 35, 267–269.Google Scholar
Griffiths, R. C. (1988). On the distribution of points in a Poisson Dirichlet process. J. Appl. Prob. 25, 336–345.CrossRefGoogle Scholar
Halmos, P. R. (1944). Random alms. Ann. Math. Statist. 15, 182–189.CrossRefGoogle Scholar
Kauffman, S. A. (1993). The Origins of Order: Self Organization and Selection in Evolution. Oxford University Press.CrossRefGoogle Scholar
Kauffman, S. A., and Levin, S. (1987). Towards a general theory of adaptive walks on rugged landscapes. J. Theoret. Biol. 128, 11–45.CrossRefGoogle ScholarPubMed
Kingman, J. F. C. (1977). The population structure associated with the Ewens sampling formula. Theoret. Popn Biol. 11, 274–283.CrossRefGoogle ScholarPubMed
Knuth, D. E. (1997). The Art of Computer Programming, Vol. 1, 3rd edn. Addison-Wesley, Reading.Google Scholar
Knuth, D., and Trabb Pardo, L. (1976). Analysis of a simple factorization algorithm. Theoret. Comput. Sci. 3, 321–348.CrossRefGoogle Scholar
Macken, C. A., Hagan, P. S., and Perelson, A. S. (1991). Evolutionary walks on rugged landscapes. SIAM J. Appl. Math. 51, 799–827.CrossRefGoogle Scholar
Macken, C. A., and Perelson, A. S. (1989). Protein evolution on rugged landscapes. Proc. Nat. Acad. Sci. USA 86, 6191–6195.CrossRefGoogle ScholarPubMed
Macken, C. A., and Perelson, A. S. (1991). Affinity maturation on rugged landscapes. In Molecular Evolution on Rugged Landscapes, eds Perelson, A. and Kauffman, S. Addison-Wesley, Reading.Google Scholar
Pólya, G. and Szegö, G. (1972). Problems and Theorems in Analysis I. Springer, New York.Google Scholar
Tovey, C. A. (1985). Hill climbing with multiple local optima. SIAM J. Algebraic Discrete Meth. 6, 384–393.CrossRefGoogle Scholar
Weinberger, E. (1988). A more rigorous definition of some properties of uncorrelated fitness landscapes. J. Theoret. Biol. 134, 125–129.CrossRefGoogle Scholar