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 speeds up evolution - so called amplifiers of selection. It has been shown that for a structure to 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 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 the suppressor of selection. We also provide an estimate of mutation rate beyond which the mutation-selection dynamics on a graph deviates from the weak mutation rate approximation. This involves computing how the average fixation time scales with the population size for several graphs.