Evolutionary Dynamics on Graphs - the Effect of Graph Structure and Initial Placement on Mutant Spread

Abstract

We study the stochastic birth-death process in a finite and structured population and analyze how the fixation probability of a mutant depends on its initial placement. In particular, we study how the fixation probability depends on the degree of the vertex where the mutant is introduced, and which vertices are its neighbours. We find that within a fixed graph, the fixation probability of a mutant has a negative correlation with the degree of the starting vertex. For a general mutant fitness r, we give approximations of relative fixation probabilities in terms of the fixation probabilities of neighbours which will be useful for considering graphs of relatively simple structure but many vertices, for instance of the small world network type, and compare our approximations to simulation results. Further, we explore which types of graphs are conducive to mutant fixation and which are not. We find a high positive correlation between a fixation probability of a randomly placed mutant and the variation of vertex degrees on that graph.

Key Words:

• Birth-death process
• Fixation probability
• Heterogeneous graphs
• Small world networks