Asymptotically Optimal Amplifiers for the Moran Process

TL;DR

This paper investigates the efficiency of certain graph structures in amplifying advantageous mutations in the Moran process, establishing optimal bounds for extinction probabilities across various graph families.

Contribution

It proves the optimality of megastars as amplifiers and introduces new graph families called dense and sparse incubators with proven optimal extinction probabilities.

Findings

Megastars are optimal up to logarithmic factors for strongly-connected digraphs.

Dense incubators achieve near-optimal extinction probability of O(n^(-1/3)).

Sparse incubators have extinction probability O(n/m), optimal up to constants.

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 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.