Dynamics of a distance-based population diversity measure

We study a class of steady-state genetic algorithms where, at each time step, two parents are selected to produce a child which then replaces one member of the population at the next time step. We consider the finite-population case. A general crossover and mutation operation are defined, as well as a genomic distance between individuals. Certain specific properties are required to hold for such operations and distance functions, and we present examples of crossover operations, mutation operations, and distance functions which meet the requirements. We then define the sum over all pairwise population distances as a measure of the diversity of a population and consider the time evolution of the expected diversity of a population. We show conditions where, under uniform, independent selection of parents and the individual to be replaced, the expected diversity monotonically approaches a fixed point. For this case we calculate an explicit formula for the expected diversity at each time step