Evolutionary games on isothermal graphs

Benjamin Allen,Martin A Nowak,Gabor Lippner

doi:10.1038/s41467-019-13006-7

Benjamin Allen, Martin A Nowak + Show 1 more

Open Access

https://doi.org/10.1038/s41467-019-13006-7

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Abstract

Population structure affects the outcome of natural selection. These effects can be modeled using evolutionary games on graphs. Recently, conditions were derived for a trait to be favored under weak selection, on any weighted graph, in terms of coalescence times of random walks. Here we consider isothermal graphs, which have the same total edge weight at each node. The conditions for success on isothermal graphs take a simple form, in which the effects of graph structure are captured in the ‘effective degree’—a measure of the effective number of neighbors per individual. For two update rules (death-Birth and birth-Death), cooperative behavior is favored on a large isothermal graph if the benefit-to-cost ratio exceeds the effective degree. For two other update rules (Birth-death and Death-birth), cooperation is never favored. We relate the effective degree of a graph to its spectral gap, thereby linking evolutionary dynamics to the theory of expander graphs. Surprisingly, we find graphs of infinite average degree that nonetheless provide strong support for cooperation.

Highlights

Population structure affects the outcome of natural selection
We find that the key quantity characterizing an isothermal graph is its effective degree ~κ, which we define as a weighted harmonic average of the graph’s Simpson degrees:
We prove in Supplementary Note 1 that strategy A is favored, for death-Birth updating on an isothermal graph under weak selection, if and only if σa þ b > c þ σd; with σ

Summary

Introduction

Population structure affects the outcome of natural selection These effects can be modeled using evolutionary games on graphs. Conditions were derived for a trait to be favored under weak selection, on any weighted graph, in terms of coalescence times of random walks. A condition was derived that determines which strategy is favored in any twoplayer, two-strategy game, on any weighted graph, under weak selection[30,31,32]. A weighted graph is called isothermal if the sum of edge weights is the same at each vertex (Fig. 1) This property has a natural interpretation: suppose that the edge weights represent the amount of time that two individuals interact with each other. The Isothermal Theorem[6,35] states that isothermal graphs neither amplify nor suppress the effects of selection for mutations of constant fitness effect

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