Correlated and uncorrelated fitness landscapes and how to tell the difference

The properties of multi-peaked “fitness landscapes” have attracted attention in a wide variety of fields, including evolutionary biology. However, relaively little attention has been paid to the properties of the landscapes themselves. Herein, we suggest a framework for the mathematical treatment of such landscapes, including an explicit mathematical model. A central role in this discussion is played by the autocorrelation of fitnesses obtained from a random walk on the landscape. Our ideas about average autocorrelations allow us to formulate a condition (satisfied by a wide class of landscapes we call AR(1) landscapes) under which the average autocorrelation approximates a decaying exponential. We then show how our mathematical model can be used to estimate both the globally optimal fitnesses of AR(1) landscapes and their local structure. We illustrate some aspects of our method with computer experiments based on a single family of landscapes (Kauffman's “N-k model”), that is shown to be a generic AR(1) landscape. We close by discussing how these ideas might be useful in the “tuning” of combinatorial optimization algorithms, and in modelling in the experimental sciences.

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