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The stage of evolution is the population of reproducing individuals. Thestructure of the population is know to affect the dynamics and outcome ofevolutionary processes, but analytical results for generic random structureshave been lacking. The most general result so far, the isothermal theorem,assumes the propensity for change in each position is exactly the same, butrealistic biological structures are always subject to variation and noise. Weconsider a population of finite size $n$ under constant selection whosestructure is given by a wide variety of weighted, directed, random graphs;vertices represent individuals and edges interactions between individuals. Byestablishing a robustness result for the isothermal theorem and using largedeviation estimates to understand the typical structure of random graphs, weprove that for a generalization of the Erd\H{o}s-R\'{e}nyi model the fixationprobability of an invading mutant is approximately the same as that of a mutantof equal fitness in a well-mixed population with high probability. Simulationsof perturbed lattices, small-world networks, and scale-free networks behavesimilarly. We conjecture that the fixation probability in a well-mixedpopulation, $(1-r^{-1})/(1-r^{-n})$, is universal: for many random graphmodels, the fixation probability approaches the above function uniformly as thegraphs become large.

Universality of fixation probabilities in randomly structured populations