Draft: yet to be submitted: An analysis of NK and generalized NK landscapes

Simulated landscapes have been used for several decades to evaluate search strategies whose goal is to find the landscape location with maximum fitness. Applications of simulated landscapes include modeling the capacity of enzymes to catalyze reactions or ligands to bind to proteins, and the clinical effectiveness of medical treatments. Understanding properties of landscapes is important for understanding search difficulty. This paper presents a novel and straightforward way of describing NK landscapes and studying their properties. We prove that NK landscapes can be represented by linear interaction models and derive the statistical properties of the model coefficients, thereby providing insight into how the NK algorithm parses importance to main effects and interactions. An important insight derived from the linear model representation is that the rank of the linear model defined by the NK algorithm is correlated with the number of local optima, a strong determinant of landscape complexity and search difficulty. We show that the maximal rank for an NK landscape is obtained through epistatic interactions that form balanced incomplete block designs. Finally, by representing NK models in matrix form, we are able to provide an analytic expression representing the expected number of local optima on the landscape.