Abstract

The graph Moran process is an interacting particle system, somewhat similar to a randomised version of bootstrap percolation, introduced in 2005 as a way to model the spread of mutations in evolutionary biology. Individuals are modelled as vertices on a connected graph, allowing only adjacent individuals to interact. Initially, a single uniformly random vertex is a “mutant” and the remaining vertices are “non-mutants”. The process evolves as a Markov chain, with vertices copying their states to their neighbours (“reproducing”) at intervals. Mutants are either more or less likely to reproduce than non-mutants, corresponding to a beneficial or harmful initial mutation. Eventually, the entire graph will be filled with either mutants (“fixation”) or non-mutants (“extinction”). This framing naturally lends itself to extremal questions, such as: How high can fixation probability be? How long can the process take to absorb? And how small a “jump” can there be between the performance of harmful and beneficial mutations? In this talk, I will survey a number of recent advances on these questions and others.