Evolutionary games on isothermal graphs   

Allene et al., 2019.

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.

Isothermal graphs and their effective degrees. A graph is isothermal if the sum of edge weights is the same for each vertex. The effective degree 𝜅̃  of the graph, defined in Eq. (3), determines the outcome of evolutionary game dynamics. a An asymmetric isothermal graph; weights are shown for each edge. b A wheel graph, with one hub and 𝑛 wheel vertices. All connections with the hub have weight 1/𝑛. All connections in the periphery have weight (𝑛−1)/2𝑛. As 𝑛→∞, the effective degree approaches 2. A formula for arbitrary 𝑛 is derived in Supplementary Note 3. c A 30-vertex graph generated with preferential attachment⁶² and linking number 𝑚=3. Isothermal edge weights are obtained by quadratic programming (see Methods). The effective degree, 𝜅̃ ≈2.47, is less than the average topological degree, 𝑘¯=5.6. d An island model, with edges of weight 𝛼≪1 between each inter-island pair of vertices. Shown here are two islands: a 𝑘1-regular graph of size 𝑁1, and a 𝑘2-regular graph of size 𝑁2

https://www.nature.com/articles/s41467-019-13006-7 

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