Fractal Fitness Landscapes

Sorkin introduced the notion of fractal fitness landscape of type h by the condition

E((f(x) - f(y))2) = d(x,y)2h

where d(x,y) is the genetic distance between the strings x and y, f(x) is the fitness of the string x, and E is the expected value. The precise form of the genetic distance depends on the problem being considered, but common choices are the Hamming distance (the number of sites on which x and y differ), and the minimum number of genetic operations (mutation, crossover, insertion, deletion) needed to go from x to y.

The autocorrelation function r(d) is defined as

r(d) = (E(f(x)*f(y)) - E(f(x)))2/s(f(x))2

where d = d(x,y) and s(f(x))2 is the variance of f(x). Sorkin showed for fractal landscapes

r(d) = 1 - d2h

By examining the autocorrelation function, Sorkin showed that if the strings have a maximum length, then the landscape becomes flat at the maximum fitness. If the current string length happens to be much shorter than the maximum length, then the landscape can appear to be fractal.

Landscapes with h > 1/2 give rise to persistent random walks, those with h < 1/2 to antipersistent random walks. However, the crossover operator can result in very large excursions, typically having self-similar jumps with probability inversely proportional to the size of the jump.

Experiments with populations of random sequences trying to reach a particular target configuration typically exhibit long periods of stasis, characterized by an approximately Gaussian random walk, punctuated by intermittent adaptations, Levy flights. The periods of stasis exhibit power law distributions, a characteritic of fractals.