Martingales and the fixation probability of high-dimensional evolutionary graphs

A principal problem of evolutionary graph theory is to find the probability that an initial mutant population will fix on a graph, i.e. that the mutants will eventually replace the indigenous population. This problem is particularly difficult when the dimensionality of a graph is high. Martingales can yield compact and exact expressions for the fixation probability of an evolutionary graph. Crucially, the tractability of martingales does not necessarily depend on the dimensionality of a graph. We will use martingales to obtain the exact fixation probability of graphs with high dimensionality, specifically k-partite graphs (or 'circular flows') and megastars (or 'superstars'). To do so, we require that the edges of the graph permit mutants to reproduce in one direction and indigenous in the other. The resultant expressions for fixation probabilities explicitly show their dependence on the parameters that describe the graph structure, and on the starting position(s) of the initial mutant population. In particular, we will investigate the effect of funneling on the fixation probability of k-partite graphs, as well as the effect of placing an initial mutant in different partitions. These are the first exact and explicit results reported for the fixation probability of evolutionary graphs with dimensionality greater than 2, that are valid over all parameter space. It might be possible to extend these results to obtain fixation probabilities of high-dimensional evolutionary graphs with undirected or directed connections. Martingales are a formidable theoretical tool that can solve fundamental problems in evolutionary graph theory, often within a few lines of straightforward mathematics.