Fixation and Consensus Times on a Network: A Unified Approach

We investigate a set of stochastic models of biodiversity, population genetics, language evolution, and opinion dynamics on a network within a common framework. Each node has a state $0<x_i<1$ with interactions specified by strengths $m_{ij}$. For any set of $m_{ij}$, we derive an approximate expression for the mean time to reach fixation or consensus (all $x_i=0$ or 1). Remarkably, in a case relevant to language change, this time is independent of the network structure.

Figure 1

Stochastic-copying dynamics between two islands $i$ and $j$ forming part of a larger network. After a death on $i$, an object is replaced either with a copy from (a) the same island $i$ or (b) a different island $j$ (rate $m_{ij}$).

Figure 2

Numerical solution for means ${\alpha }_i$ and ${\alpha }_j$ and (co)variances ${\sigma }_{ij}\equiv {\beta }_{ij}-{\alpha }_i{\alpha }_j$ on a fully connected network of $N=20$ sites. Note the $\alpha$’s converge at a time $T_0$ much less than the mean fixation time $T$ and at time $T_0$, ${\sigma }_{ij}\approx 0$ for $i\ne j$. These features become more pronounced as $N$ is increased.

Figure 3

Fixation times per speaker within the socially neutral utterance selection model (defined in the text) on various networks with $N$ nodes (see legend).