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Global optima results for the Kauffman NK model

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Abstract

The Kauffman NK model has been used in theoretical biology, physics and business organizations to model complex systems with interacting components. Recent NK model results have focused on local optima. This paper analyzes global optima of the NK model. The resulting global optimization problem is transformed into a stochastic network model that is closely related to two well-studied problems in operations research. This leads to applicable strategies for explicit computation of bounds on the global optima particularly with K either small or close to N. A general lower bound, which is sharp for K = 0, is obtained for the expected value of the global optimum of the NK model. A detailed analysis is provided for the expectation and variance of the global optimum when K = N−1. The lower and upper bounds on the expectation obtained for this case show that there is a wide gap between the values of the local and the global optima. They also indicate that the complexity catastrophe that occurs with the local optima does not arise for the global optima.

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References

  1. Arnold, B.C., Groeneveld, R.A.: Bounds on expectations of linear systematic statistics based on dependent samples. Ann. Statist. 7(1), 220–223 (1979)

    MathSciNet  Google Scholar 

  2. Balakrishna, N., Rao, C.R.: Order Statistics - An Introduction. Order Statistics - Theory and Methods, N. Balakrishna, C.R. Rao (eds.), Elsevier Science B.V., 1998, pp. 3–24

  3. Corea, G.A., Kulkarni, V.G.: Shortest paths in stochastic networks with arc lengths having discrete distributions. Networks 23, 175–183 (1993)

    MATH  MathSciNet  Google Scholar 

  4. David, H.A.: Order statistics. Second edition. John Wiley & Sons, 1981

  5. Derrida, B.: Random-energy model - An exactly solvable model of disordered systems. Phys. Rev. B, Condensation Matter 24, 2613–2626 (1981)

    MathSciNet  Google Scholar 

  6. Dodin, B.: Bounding the project completion time distribution in PERT networks. Operations Research 33, 862–8881 (1985)

    MATH  Google Scholar 

  7. Durrett, R., Limic, V.: Rigorous results for the NK model. Ann. Probab. 31(4), 1713–1753 (2003)

    Article  MathSciNet  Google Scholar 

  8. Durrett, R., Solow, D.: Personal Communication, 2003

  9. Eigen, M., McCaskill, J., Schuster, P.: The molecular quasispecies. Adv. Chem. Phys. 75, 149–171 (1989)

    Google Scholar 

  10. Evans, M., Hastings, N., Peacock, J.B.: Statistical Distributions, 3rd edition. Wiley, 2000

  11. Evans, S.N., Steinsaltz, D.: Estimating some features of NK fitness landscapes. Annals of Applied Probability 12, 1299–1321 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Flyvbjerg, H., Lautrup, B.: Evolution in a rugged landscape. Phys. Rev. A, At. Mol. Opt. Phys. 46, 6714–6723 (1992)

    Google Scholar 

  13. Fontana, W., Stadler, P.F., Bornberg-Bauer, E.G., Griesmacher, T., Hofacker, I.L., Tacker, M., Tarazona, P., Weinberger, E.D., Schuster, P.: RNA folding and combinatory landscapes. Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 47, 2083–2099 (1993)

    Google Scholar 

  14. Frieze, A.: On random symmetric travelling salesman problems. Math. Oper. Res. 29, 878–890 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gao, Y., Culberson, J.: An analysis of phase transition in NK landscapes. J. Artificial Intelligence Res. 17, 309–332 (electronic) (2002)

    MATH  MathSciNet  Google Scholar 

  16. Geard, N., Wiles, J., Halliman, J., Tonkes, B., Skellet, B.: A comparison of neutral landscapes - NK, NKp, NKq. Preprint, University of Queensland, Brisbane, Australia, 2003

  17. Hagstorm, J.N.: Computing the probability distribution of project duration in a PERT network. Networks 20, 231–244 (1990)

    MathSciNet  Google Scholar 

  18. Hayhurst, K.J., Shier, D.R.: A factoring approach for the stochastic shortest path problem. Operations Research Letters 10, 329–334 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hill, S., O'Riordan, C.: Genetic Algorithms, their Operators and the NK Model. Preprint, National University of Ireland, Galway, 2001

  20. Hill, S., O'Riordan, C.: Analysis of the performance of Genetic Algorithms and their Operators using Kauffman's NK Model. Preprint, National University of Ireland, Galway, 2002

  21. Kauffman, S.A.: The Origins of Order. Oxford University Press, Oxford, 1993

  22. Kauffman, S.A., Levin, S.: Towards a general theory of adaptive walks on rugged landscapes. Journal of Theoretical Biology 128, 11–45 (1987)

    MathSciNet  Google Scholar 

  23. Kauffman, S.A., Weinberger, E.D., Perelson, A.S.: Maturation of the immune response via adaptive walks on affinity landscapes. Theoretical Immunology, Part I, SFI studies in the Sciences of Complexity, A.S. Perelson (ed.), Addison-Wesley, 1988, pp. 349–382

  24. Kaul, H., Jacobson, S.H.: New Global Optima Results for the Kauffman NK Model: Handling Dependency. Mathematical Programming, accepted for publication, 2005

  25. Lai, T.L., Robbins, H.: Maximally dependent random variables. Proc. Nat. Acad. Sci. U.S.A. 73, 286–288 (1976)

    Article  MATH  Google Scholar 

  26. Levinthal, D.A.: Adaptation on rugged landscapes. Management Science 43, 934–950 (1997)

    MATH  Google Scholar 

  27. Limic, V., Pemantle, R.: More rigorous results on the Kauffman-Levin model of evolution. The Annals of Probability 32, 2149–2178 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  28. Macken, C.A., Hagan, P.S., Perelson, A.S.: Evolutionary walks on rugged landscapes. SIAM J. Appl. Math. 51, 799–827 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  29. Martin, J.J.: Distribution of the time through a directed acyclic network. Oper. Res. 13, 46–66 (1965)

    Article  MATH  Google Scholar 

  30. Martin, O.C., Monasson, R., Zecchina R.: Statistical mechanics methods and phase transitions in optimization problems. Theoretical Computer Science 265, 3–67 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. Mirchandani, P.B.: Shortest distance and reliability of probabilistic networks. Comput. Oper. Res. 3, 347–355 (1976)

    Article  Google Scholar 

  32. Percus, A.G., Martin, O.C.: The stochastic traveling salesman problem. J. Stat. Phys. 94, 739–758 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  33. Perelson, A.S., Macken, C.A.: Protein evolution on partially correlated landscapes. Proc. National Academy of Science USA 92, 9657–9661 (1995)

    Article  MATH  Google Scholar 

  34. Shogan, A.W.: Bounding distributions for a stochastic PERT network. Networks 7, 359–381 (1977)

    MATH  MathSciNet  Google Scholar 

  35. Solow, D., Burnetas, A., Tsai, M., Greenspan, N.S.: On the expected performance of systems with complex interactions among components. Complex Systems 12, 423–456 (2000)

    MathSciNet  Google Scholar 

  36. Solow, D., Vairaktarakis, G., Pideritt, S., Tsai, M.: Managerial insights into the effects of interactions on replacing members of a team. Management Science 48, 1060–1073 (2002)

    Article  Google Scholar 

  37. Weinberger, E.D.: A more rigorous derivation of some properties of uncorrelated fitness landscapes. J. Theoretical Biology 134, 125–129 (1988)

    Article  MathSciNet  Google Scholar 

  38. Weinberger, E.D.: Local properties of Kauffman's NK model: A tunably rugged energy landscape. Phys. Rev. A, At. Mol. Opt. Phys. 44, 6399–6413 (1991)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Hemanshu Kaul

  2. Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL, 61801, USA

    Sheldon H. Jacobson

Authors
  1. Hemanshu Kaul
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  2. Sheldon H. Jacobson
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Correspondence to Sheldon H. Jacobson.

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Kaul, H., Jacobson, S. Global optima results for the Kauffman NK model. Math. Program. 106, 319–338 (2006). https://doi.org/10.1007/s10107-005-0609-0

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