Maximally 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 epistasis, k. This fact has been confirmed using both theoretical studies of landscape structure and empirical studies of evolutionary search. We show 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. That is, the highest peaks are found in the most rugged landscapes, scattered amongst masses of low-lying peaks. We find the asymptotic value of the global optimum as N approaches infinity for both the smooth and maximally rugged cases. In evolutionary search, the optima that are found reflect the local optima that exist in the landscape, the size of these optima -- which corresponds to the size of their basins of attraction, and the effort expended in the search process. Increasing the level of epistasis in an NK landscape stochastically introduces higher peaks, but renders them exponentially more difficult to find.