Global optima results for the Kauffman NK model

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.

[1]  Weinberger,et al.  Local properties of Kauffman's N-k model: A tunably rugged energy landscape. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[2]  Rémi Monasson,et al.  Statistical mechanics methods and phase transitions in optimization problems , 2001, Theor. Comput. Sci..

[3]  Pitu B. Mirchandani,et al.  Shortest distance and reliability of probabilistic networks , 1976, Comput. Oper. Res..

[4]  Douglas R. Shier,et al.  A factoring approach for the stochastic shortest path problem , 1991, Oper. Res. Lett..

[5]  H. Robbins,et al.  Maximally dependent random variables. , 1976, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Calyampudi R. Rao,et al.  1 Order statistics: An introduction , 1998, Order statistics.

[7]  O. Martin,et al.  The Stochastic Traveling Salesman Problem: Finite Size Scaling and the Cavity Prediction , 1998, cond-mat/9802295.

[8]  E. D. Weinberger,et al.  A more rigorous derivation of some properties of uncorrelated fitness landscapes , 1988 .

[9]  Flyvbjerg,et al.  Coevolution in a rugged fitness landscape. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[10]  A. Perelson,et al.  Evolutionary walks on rugged landscapes , 1991 .

[11]  J. J. Martin.  Distribution of the Time Through a Directed, Acyclic Network , 1965 .

[12]  Weinberger,et al.  RNA folding and combinatory landscapes. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  R. Durrett,et al.  Rigorous results for the N K model , 2003 .

[14]  Jane N. Hagstrom,et al.  Computing the probability distribution of project duration in a PERT network , 1990, Networks.

[15]  Sheldon H. Jacobson,et al.  New global optima results for the Kauffman NK model: handling dependency , 2006, Math. Program..

[16]  Yong Gao,et al.  An Analysis of Phase Transition in NK Landscapes , 2002, J. Artif. Intell. Res..

[17]  Colm O Riordan,et al.  Analysis of the performance of Genetic Algorithms and their Operators using Kauffman’s NK Model , 2002 .

[18]  A. Perelson,et al.  Protein evolution on partially correlated landscapes. , 1994, Proceedings of the National Academy of Sciences of the United States of America.

[19]  Vidyadhar G. Kulkarni,et al.  Shortest paths in stochastic networks with ARC lengths having discrete distributions , 1993, Networks.

[20]  Robin Pemantle,et al.  More rigorous results on the Kauffman–Levin model of evolution , 2003, math/0308282.

[21]  Alan M. Frieze.  On Random Symmetric Travelling Salesman Problems , 2004, Math. Oper. Res..

[22]  S. Kauffman,et al.  Towards a general theory of adaptive walks on rugged landscapes. , 1987, Journal of theoretical biology.

[23]  Daniel A. Levinthal.  Adaptation on rugged landscapes , 1997 .

[24]  Andrew W. Shogan.  Bounding distributions for a stochastic pert network , 1977, Networks.

[25]  M. Evans.  Statistical Distributions , 2000 .

[26]  George L. Vairaktarakis,et al.  Managerial Insights into the E ® ects of Interactions on Replacing Members of a Team , 2001 .

[27]  Daniel Solow,et al.  On the Expected Performance of Systems with Complex Interactions Among Components , 2000, Complex Syst..

[28]  M. Eigen,et al.  The molecular quasi-species , 2007 .

[29]  B. Derrida.  Random-energy model: An exactly solvable model of disordered systems , 1981 .

[30]  Stuart A. Kauffman,et al.  ORIGINS OF ORDER , 2019, Origins of Order.

[31]  Bajis M. Dodin,et al.  Bounding the Project Completion Time Distribution in PERT Networks , 1985, Oper. Res..

[32]  Steven N. Evans,et al.  ESTIMATING SOME FEATURES OF NK FITNESS LANDSCAPES , 2002 .

[33]  B. Arnold,et al.  Bounds on Expectations of Linear Systematic Statistics Based on Dependent Samples , 1979 .