Department of Mathematics, Emmanuel College, 400 The Fenway, Boston, MA, 02115, USA.
Department of Mathematics, Northeastern University, 360 Huntington Ave, Boston, MA, 02115, USA.
Nat Commun. 2019 Nov 8;10(1):5107. doi: 10.1038/s41467-019-13006-7.
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.
种群结构会影响自然选择的结果。这些影响可以通过图上的进化博弈来建模。最近，有人从随机游走的合并时间的角度，推导出了在任何加权图上，一个特征在弱选择下被选择的条件。在这里，我们考虑等温图，它们在每个节点处具有相同的总边权重。在等温图上成功的条件形式简单，其中图结构的影响体现在“有效度”——个体每个有效邻居的数量的度量。对于两种更新规则（死亡-出生和出生-死亡），如果收益-成本比超过有效度，则在大的等温图上合作行为会受到青睐。对于另外两种更新规则（出生-死亡和死亡-出生），合作永远不会受到青睐。我们将图的有效度与它的谱隙联系起来，从而将进化动力学与扩张图理论联系起来。令人惊讶的是，我们发现具有无限平均度的图尽管如此，仍为合作提供了有力支持。
