Self-loops in evolutionary graph theory: Friends or foes?

Evolutionary dynamics in spatially structured populations has been studied for a long time. More recently, the focus has been to construct structures that amplify selection by fixing beneficial mutations with higher probability than the well-mixed population and lower probability of fixation for deleterious mutations. It has been shown that for a structure to substantially amplify selection, self-loops are necessary when mutants appear predominately in nodes that change often. As a result, for low mutation rates, self-looped amplifiers attain higher steady-state average fitness in the mutation-selection balance than well-mixed populations. But what happens when the mutation rate increases such that fixation probabilities alone no longer describe the dynamics? We show that self-loops effects are detrimental outside the low mutation rate regime. In the intermediate and high mutation rate regime, amplifiers of selection attain lower steady-state average fitness than the complete graph and suppressors of selection. We also provide an estimate of the mutation rate beyond which the mutation-selection dynamics on a graph deviates from the weak mutation rate approximation. It involves computing average fixation time scaling with respect to the population sizes for several graphs.Author summary: Evolutionary and ecological dynamics is strongly affected by the underlying population structure. Evolutionary graph theory considers networks in which individuals are placed on the nodes and replace each other via the links. Amplifiers and suppressors of selection are particularly intriguing structures that can effectively change the selective advantage of a mutant compared to unstructured populations. For very low mutation rates, strong amplification requires that mutants can replace their parents via self-loops. We show that this beneficial role of self-loops is reversed when the mutation rate is increased: In this case, self looped-graphs have a lower average fitness in mutation-selection balance. More generally, we show that suppressors of fixation—structures that reduce the fixation of mutants regardless of their relative fitness—can increase the fitness in mutation selection balance both for weak mutation and for strong mutation. This calls for a closer investigation of structures other than the amplifiers of selection.