Stochastic Populations, Power Law and Fitness Aggregation in Genetic Algorithms

Abstract

Natural populations are dynamic in both time and space. In biological populations such as insects, spatial distribution patterns are often studied as the first step to characterize population dynamics. In nature, the spatial distribution patterns of insect populations are considered as the emergent expression property of individual behaviors at population levels and are fine-tuned or optimized by natural selection. This inspiration prompts us to investigate the possibly similar mechanisms in Genetic Algorithms GA populations. In this study, we introduce the mathematical models for the spatial distribution patterns of insect populations to GA with the conjecture that the emulation of biological populations in nature may lead to computational improvement. In particular, we introduce three modeling approaches from the research of spatial distribution patterns of insect populations: i probability distribution modeling approach, ii aggregation index approach, and iii Taylor's 1961, 1977 Power Law, Iwao's 1968, 1976 Mean Crowding Model and Ma's 1991c population aggregation critical density PACD, to characterize populations in GA. With these three approaches, we investigate four mappings from the research field of insect spatial distribution patterns to GA populations in order to search for possible counterpart mechanisms or features in GA. They are: i mapping insect spatial distribution patterns to GA populations or allowing GA populations to be controlled by stochastic distribution models that describe insect spatial distributions; ii mapping insect population distribution to GA population fitness distribution via Power Law and PACD modeling; iii mapping population aggregation dynamics to GA fitness progression across generations or fitness aggregation dynamics in GA via insect population aggregation index; iv mapping insect population sampling model to optimal GA population sizing. With regard to the mapping i, the experiment results show the significant improvements in GA computational efficiency in terms of the reduced fitness evaluations and associated costs. This prompts us to suggest using probability distribution models, or what we call stochastic GA populations, to replace the fixed-size population settings. We also found the counterpart for the second mapping, the wide applicability of Power Law and Mean Crowding model to the fitness distribution in GA populations. The testing of the third and fourth mappings is very preliminary; we use example cases to suggest two further research problems: the potential to use fitness aggregation dynamics for controlling the number of generations iterated in GA searches, and the possibility to use fitness aggregation distribution parameters [obtained in mapping ii] in determining the optimum population size in GA. A third interesting research problem is to investigate the relationship between mapping i and iii, i.e., the controlling of both population sizes and population generations.