Asymptotically optimal amplifiers for the Moran process

Abstract

We study the Moran process as adapted by Lieberman, Hauert and Nowak. This is a model of an evolving population on a graph or digraph where certain individuals, called “mutants” have fitness r and other individuals, called “non-mutants” have fitness 1. We focus on the situation where the mutation is advantageous, in the sense that $r>1$. A family of digraphs is said to be strongly amplifying if the extinction probability tends to 0 when the Moran process is run on digraphs in this family. The most-amplifying known family of digraphs is the family of megastars of Galanis et al. We show that this family is optimal, up to logarithmic factors, since every strongly-connected n-vertex digraph has extinction probability $\mathrm{\Omega }(n^{-1/2})$. Next, we show that there is an infinite family of undirected graphs, called dense incubators, whose extinction probability is $O(n^{-1/3})$. We show that this is optimal, up to constant factors. Finally, we introduce sparse incubators, for varying edge density, and show that the extinction probability of these graphs is $O(n/m)$, where m is the number of edges. Again, we show that this is optimal, up to constant factors.