Statistical Dynamics of the Royal Road Genetic Algorithm

Metastability is a common phenomenon. Many evolutionary processes, both natural and artificial, alternate between periods of stasis and brief periods of rapid change in their behavior. In this paper, an analytical model for the dynamics of a simple genetic algorithm (GA) is developed that identifies a general mechanism that causes metastability in the evolutionary dynamics. The GA’s population dynamics is described in terms of flows in the space of fitness distributions. The trajectories through fitness distribution space are derived in closed form in the limit of infinite populations. We then show how finite populations induce metastibility, even in regions of “itness epochs”--periods of stasis in population fitness distributions--at finite population size and identifies the locations of these fitness epochs with the flow’s hyperbolic fixed points. This enables exact predictions of the metastable fitness distributions during the fitness epochs, as well as giving insight into the nature of periods of stasis and the innovations between them. All these results are obtained as closed form expressions in terms of the GA’s parameters. An analysis of the Jacobian matrices in the neighborhood of an epoch’s metastable fitness distribution allows for the calculation of its stable and unstable manifold dimensions and so reveals the state space’s topological structure. More general quantitative features of the dynamics--fitness fluctuation amplitudes, epoch stability, and speed of the innovations--are also determined from the Jacobian eigenvalues. The analysis shows how quantitative predictions for a range of dynamical behaviors, that are specific to the finite population dynamics, can be derived from the solution of the infinite population dynamics. The theoretical predictions are shown to agree very well with statistics from GA simulations. We also discuss the connections of our results with those from population genetics and molecular evolution theory.