Asymptotically optimal amplifiers for the Moran process

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 Ω ( 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.

Authors

• Leslie Ann Goldberg
• John Lapinskas
• Johannes Lengler
• Florian Meier
• Konstantinos Panagiotou
• Pascal Pfister

Keywords

• Strong amplifiers
• Moran process
• Fixation probability
• Extremal graph theory
• Markov chains