Amplifiers for the Moran Process
Andreas Galanis, Andreas Göbel, Leslie Ann Goldberg, John Lapinskas, David Richerby
The Moran process, as studied by Lieberman, Hauert and Nowak, is a stochastic process modelling the spread of genetic mutations in populations. It has an underlying graph in which vertices correspond to individuals. Initially, one individual (chosen uniformly at random) possesses a mutation, with fitness r>1. All other individuals have fitness 1. At each step of the discrete-time process, an individual is chosen with probability proportional to its fitness, and its state (mutant or non-mutant) is passed on to an out-neighbour chosen u.a.r. If the underlying graph is strongly connected, the process will eventually reach fixation (all individuals are mutants) or extinction (no individuals are mutants). We define an infinite family of directed graphs to be strongly amplifying if, for every r>1, the extinction probability tends to 0 as the number n of vertices increases. Strong amplification is a rather surprising property – the initial mutant only has a fixed selective advantage, independent of n, which is “amplified” to give a fixation probability tending to 1. Strong amplifiers have received quite a bit of attention. Lieberman et al. proposed two potential families of them: superstars and metafunnels. It has been argued heuristically that some infinite families of superstars are strongly amplifying. The same has been claimed for metafunnels. We give the first rigorous proof that there is a strongly amplifying family of directed graphs which we call “megastars”. We show that the extinction probability of n-vertex graphs in this family of megastars is roughly n−1/2, up to logarithmic factors, and that all infinite families of superstars and metafunnels have larger extinction probabilities as a function of n. Our analysis of megastars is tight, up to logarithmic factors. We also clarify the literature on the isothermal theorem of Lieberman et al.