Estimating some features of $NK$ fitness landscapes

Abstract

Kauffman and Levin introduced a class of models for the evolution of hereditary systems which they called $NK$ fitness landscapes. Inspired by spinglasses, these models have the attractive feature of being tunable, with regard to both overall size (through the parameter $N$) and connectivity (through $K$). There are $N$ genes, each of which exists in two possible alleles [leading to a system indexed by $\{0,1\}^{N}$]; the fitness score of an allele at a given site is determined by the alleles of $K$ neighboring sites. Otherwise the fitnesses are as simple as possible, namely i.i.d., and the fitnesses of different sites are simply averaged.

Much attention has been focused on these fitness landscapes as paradigms for investigating the interaction between size and complexity in making evolution possible. In particular, the effect of the interaction parameter $K$ on the height of the global maximum and the heights of local maxima has attracted considerable interest, as well as the behavior of a "hill-climbing" walk from a random starting point. Nearly all of this work has relied on simulations, not on rigorous mathematics.

In this paper, some asymptotic features of $NK$ fitness landscapes are reduced to questions about eigenvalues and Lyapunov exponents. When $K$ is fixed, the expected number of local maxima grows exponentially with $N$ at a rate depending on the top eigenvalue of a kernel derived from the distribution of the fitnesses, and the average height of a local maximum converges to a value determined by the corresponding eigenfunction.

The global maximum converges in probability as $N \to \infty$ to a constant given by the top Lyapunov exponent for a system of i.i.d. max-plus random matrices, and this constant is nondecreasing with $K$. Various such quantities are computed for certain special cases when $K$ is small, and these calculations can, in principle, be extended to larger $K$.