Abstract
The fixation probability is the probability that a new mutant introduced in a homogeneous population eventually takes over the entire population. The fixation probability is a fundamental quantity of natural selection, and known to depend on the population structure. Amplifiers of natural selection are population structures which increase the fixation probability of advantageous mutants, as compared to the baseline case of well-mixed populations. In this work we focus on symmetric population structures represented as undirected graphs. In the regime of undirected graphs, the strongest amplifier known has been the Star graph, and the existence of undirected graphs with stronger amplification properties has remained open for over a decade. In this work we present the Comet and Comet-swarm families of undirected graphs. We show that for a range of fitness values of the mutants, the Comet and Comet-swarm graphs have fixation probability strictly larger than the fixation probability of the Star graph, for fixed population size and at the limit of large populations, respectively.
Highlights
It is well-known that population structure affects the evolutionary dynamics[5, 12,13,14,15,16,17,18,19,20,21,22,23,24]
The generalized Moran process on a graph is identical to the Moran process on well-mixed populations, with the exception that each offspring can only replace a neighbor of the reproducing individual
The well-mixed population follows as a special case of the generalized Moran process, where the individuals are spread on the vertices of a Clique KN
Summary
Andreas Pavlogiannis[1], Josef Tkadlec[1], Krishnendu Chatterjee1 & Martin A. Amplifiers of natural selection are population structures which increase the fixation probability of advantageous mutants, as compared to the baseline case of well-mixed populations. A graph of N vertices GN is said to amplify selection[5], if the fixation probability ρ(r, GN) of a randomly placed initial mutant on GN is larger than the fixation probability on a well-mixed population of the same size (i.e., if ρ(r, GN) > ρ(r, KN)). We focus on the most commonly studied case, where the population structure is modeled as an undirected graph, and the initial mutant arises with uniform probability on each vertex. In this work we present graphs with higher fixation probability than that on the Star graph, both for finite populations and at the limit of large populations, for some values of r. The counterexample is a simple and natural extension of the Star family
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