Correlation Analysis of Fitness Landscapes

Fitness landscapes underlie the dynamics of evolutionary processes and are a key concept of evolutionary theory. Recent research on molecular folding and on evolutionary algorithms has demonstrated that such landscapes are also important for understanding problems of chemistry and of combinatorial optimization. In these cases free energy or cost functions are used instead of biological fitness functions defined on genotypes. However, the image of a three dimensional landscape with many peaks and valleys turns out to be misleading. Genotypes tend to differ in numerous characteristics, resulting in multidimensional fitness landscapes. Properties of these landscapes are very different from those of low dimensional ones. The main intention of this study is to investigate how these features affect the duration of adaptive walks on such landscapes. For this purpose we focus on the Traveling Salesman Problem (TSP), which amounts to finding the shortest tour visiting a given set of locations. By comparing theoretical predictions for the duration of adaptive walks to the actual waiting times observed for an evolutionary algorithm we demonstrate that a sufficiently fine-grained correlation matrix succeeds in capturing essential structural features of the TSP fitness landscape. To test the performance of correlation-based predictions for a class of fitness landscapes with varying degree of neutrality, we have analyzed evolutionary waiting times on NKp fitness landscapes. We show that for low degrees of neutrality, correlation statistics again prove to be an excellent basis for predicting waiting times, while for very high degrees of neutrality, a population's drift along neutral networks turns out to require incorporation of additional information on network topologies.