Evolutionary Dynamics on Graphs: Invited Talk

The Moran process [5], as adapted by Lieberman, Hauert and Nowak [4], is a discrete-time random process which models the spread of genetic mutations through populations. Individuals are modelled as the vertices of a graph. Each vertex is either infected or uninfected. The model has a parameter r > 0. Infected vertices have fitness r and uninfected vertices have fitness 1. At each step, an individual is selected to reproduce with probability proportional to its fitness. This vertex chooses one of its neighbours uniformly at random and updates the state of that neighbour (infected or not) to match its own. In the initial state, one vertex is chosen uniformly at random to be infected and the other vertices are uninfected. If the graph is strongly connected then the process will terminate with probability 1, either in the state where every vertex is infected (known as fixation) or in the state where no vertex is infected (known as extinction). The principal quantities of interest are the fixation probability (the probability of reaching fixation) and the expected absorption time (the expected number of steps before fixation or extinction is reached). In general, these depend on both the graph topology and the parameter r. We study three questions.