A Probabilistic Model of Size Drift

Abstract

Drift towards growing size of genotypes is oneoutstanding and constantly disputed invariant in anoverwhelming number of applications of evolutionaryalgorithms with variable-size structures. In contrastto previous work to reveal its fundamentals, weprobabilistically analyse genotype growth by buildingon the idea of a 'representation-less' model by Banzhafand Langdon. Our model, called the fitness-size model,corresponds to a simple evolutionary algorithm usingoverproduction selection and mutation working onabstract objects retaining only fitness and sizeinformation.The probalistic analysis offer some surprises counterto present credence. The analysis predicts that averageeffective and noneffective lengths (and thus overallsize) tend to be invariant over time. The same is truefor the variance of the effective length. In contrast,the variance of the noneffective size featuresincreases linearly in time, and its variation shows thetrademark of a diffusion process.Drift to increasingsize manifest s if search biases favour boundaryconditions. We present experimental results with boththe implementation of the theoretical model and astandard genetic programming algorithm. Statisticalresults with the two implementations are similar andfit the theoretical predictions.