Evolutionary graph theory models the effects of natural selection and random drift on structured populations of  mutant and non-mutant individuals. Recent studies have shown that fixation times , which determine the rate of evolution, often have right-skewed distributions. Little is known, however, about how these distributions and their skew depend on mutant fitness. Here we calculate the fitness dependence of the fixation-time distribution for the Moran Birth-death process in populations modeled by two extreme networks: the complete graph and the one-dimensional ring lattice, each of which admits an exact solution in the limit of large network size. We find that with non-neutral fitness, the Moran process on the ring has normally distributed fixation times, independent of the relative fitness of mutants and non-mutants. In contrast, on the complete graph, the fixation-time distribution is a weighted convolution of two Gumbel distributions , with a weight depending on the relative fitness . When fitness is neutral , however, the Moran process has a highly skewed fixation-time distribution  on both the complete graph and the ring. In this sense, the case of neutral fitness is singular. Even on these simple network structures , the fixation-time distribution exhibits rich fitness dependence, with discontinuities and regions of universality. Applications of our methods to a multi-fitness Moran model , times to partial fixation, and evolution on random networks are discussed.
Evolutionary graph theory models the effects of natural selection and random drift on structured populations of competing mutant and non-mutant individuals. Recent studies have found that fixation times in such systems often have right-skewed distributions. Little is known, however, about how these distributions and their skew depend on mutant fitness. Here we calculate the fitness dependence of the fixation-time distribution for the Moran Birth-death process in populations modeled by two extreme networks: the complete graph and the one-dimensional ring lattice, obtaining exact solutions in the limit of large network size. We find that with non-neutral fitness, the Moran process on the ring has normally distributed fixation times, independent of the relative fitness of mutants and non-mutants. In contrast, on the complete graph, the fixation-time distribution is a fitness-weighted convolution of two Gumbel distributions . When fitness is neutral the fixation-time distribution jumps discontinuously and becomes highly skewed on both the complete graph and the ring. Even on these simple networks , the fixation-time distribution exhibits rich fitness dependence, with discontinuities and regions of universality. Extensions of our results to two-fitness Moran models , times to partial fixation, and evolution on random networks are discussed.
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competing
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shown
found
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, which determine the rate of evolution,
in such systems
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each of which admits an exact solution
obtaining exact solutions
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weighted
fitness-weighted
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, with a weight depending on the relative fitness
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, however, the Moran process has a highly skewed
the
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jumps discontinuously and becomes highly skewed
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In this sense, the case of neutral fitness is singular.
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network structures
networks
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Applications of our methods to a multi-fitness Moran model
Extensions of our results to two-fitness Moran models