Ongoing Evolution on Networks Marcus Frean Victoria University of Wellington, New Zealand marcus.frean@vuw.ac.nz Gareth Baxter Departamento de F´ ısica, Universidade de Aveiro, Portugal Paul Rainey NZ Institute for Advanced Study, New Zealand In an evolving population, network structure can have striking effects on the sur- vival probability of a mutant allele and on the rate at which it spreads. In networks with ‘hubs’ (representing geographic or other constraints), the heightened probability of an initially rare mutant has led to the prediction that such networks act to amplify the effects of selection over drift. But selection and mutation interplay in a subtle way in such populations: hubs also slow the mutant’s rate of invasion, so that if multiple mutants are allowed to spread at the same time, more of them may be present. In other words it might be misleading to consider only the fixation probability, because new mutants spread at different rates in these networks. Instead of following a single mutation to fixation, we give a very simple model that allows for a stream of mutations, leading to a dynamic equilibrium. In this way we take account of ongoing evolution rather than simply following a single mutant to fixation. In this model the amplifier effect is largely reversed: for much of the parameter space, networks with hubs suppress rather than amplify selection. This is explained by considering the relative time scales of mutation and selection. 1 Introduction Evolution is the interplay between the sharpening force of selection and the diffusing force of variation. Both are required in order for evolution to proceed. When selection pressure is large but there is little or no variation introduced,