Invasion of mutants in an evolutionary process on a graph.

We study the Moran process (MP) on an undirected graph. The MP has been studied on the complete graph and lattices extensively. Remarkably the fixation probability of the mutants is the same for all undirected degree-regular graphs. However, this is not true for degree-heterogeneous graphs. Some graphs can enhance the fixation probabilities over the degree-regular graphs. We derive fixation probabilities for general graphs when the mutants have a small advantage over the resident population. For the general bias case, we derive exact results for the bipartite graphs and discuss the structure of graphs where such a result can be applicable.