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Biological Cybernetics

, Volume 63, Issue 5, pp 325–336 | Cite as

Correlated and uncorrelated fitness landscapes and how to tell the difference

  • E. Weinberger
Article

Abstract

The properties of multi-peaked “fitness landscapes” have attracted attention in a wide variety of fields, including evolutionary biology. However, relaively little attention has been paid to the properties of the landscapes themselves. Herein, we suggest a framework for the mathematical treatment of such landscapes, including an explicit mathematical model. A central role in this discussion is played by the autocorrelation of fitnesses obtained from a random walk on the landscape. Our ideas about average autocorrelations allow us to formulate a condition (satisfied by a wide class of landscapes we call AR(1) landscapes) under which the average autocorrelation approximates a decaying exponential. We then show how our mathematical model can be used to estimate both the globally optimal fitnesses of AR(1) landscapes and their local structure. We illustrate some aspects of our method with computer experiments based on a single family of landscapes (Kauffman's “N-k model”), that is shown to be a generic AR(1) landscape. We close by discussing how these ideas might be useful in the “tuning” of combinatorial optimization algorithms, and in modelling in the experimental sciences.

Keywords

Mathematical Model Autocorrelation Optimization Algorithm Random Walk Combinatorial Optimization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Binder K, Young A (1986) Spinglasses: experimental facts, theoretical concepts, and open questions. Rev Mod Phys 54:801Google Scholar
  2. Brady R (1985) Optimization strategies gleaned from biological evolution. Nature 317:804Google Scholar
  3. Breiman L (1968) Probability. Addison-Wesley, Reading, MassGoogle Scholar
  4. David H (1970) Order statistics. Wiley, New YorkGoogle Scholar
  5. Eigen M, McCaskill J, Schuster P (1988) Molecular quasi-species. J Phys Chem 92:6881Google Scholar
  6. Feller W (1965) An introduction to probability theory and its applications, vol 1, 3rd edn. Wiley, New YorkGoogle Scholar
  7. Feller W (1972) An introduction to probability theory and its applications, vol 2, 2nd edn. Wiley, New YorkGoogle Scholar
  8. Garey M, Johnson D (1979) Computers and intractability: a guide to the theory of incomputability. Freeman, San FranciscoGoogle Scholar
  9. Gumbel E (1985) Statistics of extremes. Colorado University, New YorkGoogle Scholar
  10. Herstein I (1964) Topics in algebra. Blaisdell, New YorkGoogle Scholar
  11. Holland J (1981) Genetic algorithms and adaptation. Technical Report # 34, University of Michigan Cognitive Sciences Department Ann Arbor, MichGoogle Scholar
  12. Karlin S, Taylor H (1975) A first course in stochastic processes. Academic Press, New YorkGoogle Scholar
  13. Karlin S, Taylor H (1981) A second course in stochastic processes. Academic Press, New YorkGoogle Scholar
  14. Kauffman S, Levin S (1987) Towards a general theory of adaptive walks on rugged landscapes. J Theor Biol 128:11–45Google Scholar
  15. Kauffman S, (1989) Complex systems. In: Stein DL (eds) Proceedings of the 1988 Summer School on Complex Systems in Santa Fe, NM. Santa Fe Institute Studies in the Sciences of Complexity, Addison-Wesley, Reading, MassGoogle Scholar
  16. Kauffman S, Weinberger E, Perelson A (1988) Maturation of the immune response via adaptive walks on affinity landscapes. In: Perelson AS (eds) Theoretical immunology, part I. Santa Fe Institute Studies in the Sciences of Complexity. Addison-Wesley, Reading, MassGoogle Scholar
  17. Kirkpatrick S, Gelatt C, Vecci M (1983) Optimization by simulated annealing. Science 220:671–680Google Scholar
  18. Kirkpatrick S, Toulouse G (1985) Configuration space analysis of Travelling Salesman Problems. J Phys 46:1277Google Scholar
  19. Macken, C, Perelson, A (1989) Protein evolution on rugged landscapes. Proc Natl Acad Sci USA 86:6191–6195Google Scholar
  20. Mandelbrot B (1982) The fractal geometry of nature. Freeman, New YorkGoogle Scholar
  21. Palmer RG (1989) Complex systems. In: Stein DL (eds) Proceedings of the 1988 Summer School on Complex Systems in Santa Fe, NM. Santa Fe Institute Studies in the Sciences of Complexity. Addison-Wesley, Reading, MassGoogle Scholar
  22. Papoulis A (1965) Probability, random variables, and stochastic processes. McGraw-Hill, New YorkGoogle Scholar
  23. Priestley M (1981) Spectral analysis and time series. Academic Press, LondonGoogle Scholar
  24. Smith JM (1970) Natural selection and the concept of a protein space. Nature 225:563Google Scholar
  25. Stein DL (1989) Complex systems. In: Stein DL (ed) Proceedings of the 1988 Summer School on Complex Systems in Santa Fe, NM. Santa Fe Institute Studies in the Sciences of Complexity. Addison-Wesley, Reading, MassGoogle Scholar
  26. Weinberger E (1987a) A stochastic generalization of Eigen's model of natural selection. PhD Thesis, Courant Institute of Mathematical Sciences, New York, NY Available from University Microfilms, Ann Arbor, MichGoogle Scholar
  27. Weinberger E (1987b) A model of natural selection that exhibits a dynamic phase transition. J Stat Phys 49:1011–1028Google Scholar
  28. Weinberger E (1988) A more rigorous derivation of some results on rugged fitness landscapes. J Theor Biol 134:125–129Google Scholar
  29. White SR (1984) Concepts of scale in simulated annealing. Proc ICCDGoogle Scholar
  30. Wright S (1932) The roles of mutation, inbreeding, crossbreeding and selection in evolution. In: Proceedings 6th Congress on Genetics 1:356Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • E. Weinberger
    • 1
  1. 1.Department of Biochemistry and BiophysicsUniversity of PennsylvaniaPhiladelphiaUSA

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