# Evolutionary Games on Isothermal Graphs

## TL;DR

This paper demonstrates that on isothermal graphs, the success of cooperation depends on the benefit-to-cost ratio relative to the graph's effective degree, simplifying the understanding of evolutionary dynamics in structured populations.

## Contribution

It introduces a simple criterion for evolutionary success on isothermal graphs using effective degree, linking graph structure to cooperation promotion.

## Key findings

- Success depends on benefit-to-cost ratio exceeding effective degree
- Effective degree relates to the spectral gap of the graph
- Graphs with infinite average degree can still promote cooperation

## Abstract

Population structure affects the outcome of natural selection. Static population structures can be described by graphs, where individuals occupy the nodes, and interactions occur along the edges. General conditions for evolutionary success on any weighted graph were recently derived, for weak selection, in terms of coalescence times of random walks. Here we show that for a special class of graphs, the conditions for success take a particularly simple form, in which all effects of graph structure are described by the graph's "effective degree"---a measure of the effective number of neighbors per individual. This result holds for all weighted graphs that are isothermal, meaning that the sum of edge weights is the same at each node. Isothermal graphs encompass a wide variety of underlying topologies, and arise naturally from supposing that each individual devotes the same amount of time to interaction. Cooperative behavior is favored on a large isothermal graph if the benefit-to-cost ratio exceeds the effective degree. We relate the effective degree of a graph to its spectral gap, thereby providing a link between evolutionary dynamics and the theory of expander graphs. As a surprising example, we report graphs of infinite average degree that are nonetheless highly conducive for promoting cooperation.