Fast fixation with a generic network structure.

We investigate the dynamics of a broad class of stochastic copying processes on a network that includes examples from population genetics (spatially structured Wright-Fisher models), ecology (Hubbell-type models), linguistics (the utterance selection model), and opinion dynamics (the voter model) as special cases. These models all have absorbing states of fixation where all the nodes are in the same state. Earlier studies of these models showed that the mean time when this occurs can be made to grow as different powers of the network size by varying the degree distribution of the network. Here we demonstrate that this effect can also arise if one varies the asymmetry of the copying dynamics while holding the degree distribution constant. In particular, we show that the mean time to fixation can be accelerated even on homogeneous networks when certain nodes are very much more likely to be copied from than copied to. We further show that there is a complex interplay between degree distribution and asymmetry when they may covary, and that the results are robust to correlations in the network or the initial condition.