Asymptotically Optimal Amplifiers for the Moran Process

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

Published 2016-11-13Version 1

We study the Moran process as adapted by Lieberman, Hauert and Nowak. A family of directed graphs is said to be strongly amplifying if the extinction probability tends to 0 when the Moran process is run on graphs in this family. The most-amplifying known family of directed graphs 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 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.