A Study of NK Landscapes' Basins and Local Optima Networks

TL;DR

This paper introduces a network-based approach to analyze NK landscape fitness landscapes, revealing they are small-world networks with non-random structures and exploring basin and local maxima relationships.

Contribution

It adapts inherent network concepts to NK landscapes, providing a novel network characterization and analysis of their structural properties.

Findings

NK landscape maxima networks are small-worlds.

Maxima graphs have high clustering coefficients.

Degree distributions are close to exponential.

Abstract

We propose a network characterization of combinatorial fitness landscapes by adapting the notion of inherent networks proposed for energy surfaces (Doye, 2002). We use the well-known family of $NK$ landscapes as an example. In our case the inherent network is the graph where the vertices are all the local maxima and edges mean basin adjacency between two maxima. We exhaustively extract such networks on representative small NK landscape instances, and show that they are 'small-worlds'. However, the maxima graphs are not random, since their clustering coefficients are much larger than those of corresponding random graphs. Furthermore, the degree distributions are close to exponential instead of Poissonian. We also describe the nature of the basins of attraction and their relationship with the local maxima network.