Fast fixation without fast networks

TL;DR

This paper explores how the mean fixation time in stochastic copying processes on networks can be significantly affected by asymmetry in copying dynamics, not just network degree distribution, across various models.

Contribution

It demonstrates that asymmetry in copying dynamics can accelerate fixation times even on homogeneous networks, extending previous understanding.

Findings

Asymmetry in copying can speed up fixation times.

Degree distribution and asymmetry interact complexly.

Results are robust to network correlations and initial conditions.

Abstract

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 the degree distribution of the network. Here we demonstrate that this effect can also arise if one varies the asymmetry of the copying dynamics whilst 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 co-vary; and that the results are robust to correlations in the network or the initial condition.