Voter models on heterogeneous networks

We study simple interacting particle systems on heterogeneous networks, including the voter model and the invasion process. These are both two-state models in which in an update event an individual changes state to agree with a neighbor. For the voter model, an individual “imports” its state from a randomly chosen neighbor. Here the average time $T_N$ to reach consensus for a network of $N$ nodes with an uncorrelated degree distribution scales as $N{\mu }_1^2∕{\mu }_2$, where ${\mu }_k$ is the $k\text{th}$ moment of the degree distribution. Quick consensus thus arises on networks with broad degree distributions. We also identify the conservation law that characterizes the route by which consensus is reached. Parallel results are derived for the invasion process, in which the state of an agent is “exported” to a random neighbor. We further generalize to biased dynamics in which one state is favored. The probability for a single fitter mutant located at a node of degree $k$ to overspread the population—the fixation probability—is proportional to $k$ for the voter model and to $1∕k$ for the invasion process.