Rugged NK landscapes contain the highest peaks

NK models provide a family of tunably rugged fitness landscapes used in a wide range of evolutionary computation studies. It is well known that the average height of local optima regresses to the mean of the landscape with increasing ruggedness, K. This fact has been confirmed with both theoretical studies of landscape structure and empirical studies of evolutionary search. We show mathematically that the global optimum behaves quite differently: the expected value of the global maximum is highest in the maximally rugged case. Furthermore, we demonstrate that this expected value increases with K, despite the fact that the average fitness of the local optima decreases. We find the asymptotic value of the global optimum as N approaches infinity for both the smooth and maximally rugged cases. We interpret these results in the context of evolutionary search, and describe the relationship between the global optimum, local optima and found optima as search effort is exponentially increased. Index Terms NK landscapes, rugged landscapes, combinatorial landscapes, search, optimisation. I. THE NK FITNESS LANDSCAPE The NK landscape model was designed to develop intuitions about the structure of complex combinatorial fitness landscapes [1], [2]. Over the last fifteen years, the NK model has become an important tool for modelling biological systems that incorporate combinatorial interactions, such as the evolution of gene networks [3] and the development of the immune system [4]. The NK landscape has also been widely used by the evolutionary computation community as a generator of epistatic landscapes that serve as important test functions for search and optimisation techniques [5]–[7]. An NK system is defined by two parameters, N , the number of components and K, the number of epistatic interactions between components. Each component has a binary value so an NK system can be represented by a bit string of length N in which the bit at locus i marks the value of component i. Each locus makes a B. Skellett, N. Geard and J. Wiles are with the School of Information Technology and Electrical Engineering, University of Queensland, 4072, Australia (e-mail: [email protected]). J. Wiles is also with the School of Psychology, University of Queensland. B. Tonkes was with the School of Information Technology and Electrical Engineering, University of Queensland at the time this paper was written and is now with the ARC Centre for Autonomous Systems, School of Computer Science and Engineering, University of New South Wales B. Cairns is with the Department of Mathematics, University of Queensland.