Evolving Voter Model on Dense Random Graphs

TL;DR

This paper studies a dynamic voter model on dense random graphs where agents can change opinions or rewire connections, revealing a phase transition in the network's fragmentation and opinion diversity depending on the parameter 2.

Contribution

The paper rigorously analyzes two variants of the evolving voter model on dense graphs, establishing phase transitions and long-term behavior of opinions and network structure.

Findings

Network splits into two communities for small 2

High 2 leads to sustained opinion diversity

Positive opinion fractions survive in rewire-to-random model

Abstract

In this paper we examine a variant of the voter model on a dynamically changing network where agents have the option of changing their friends rather than changing their opinions. We analyse, in the context of dense random graphs, two models considered in Durrett et. al.(Proc. Natl. Acad. Sci. 109: 3682-3687, 2012). When an edge with two agents holding different opinion is updated, with probability $\frac{\beta}{n}$, one agent performs a voter model step and changes its opinion to copy the other, and with probability $1-\frac{\beta}{n}$, the edge between them is broken and reconnected to a new agent chosen randomly from (i) the whole network (rewire-to-random model) or, (ii) the agents having the same opinion (rewire-to-same model). We rigorously establish in both the models, the time for this dynamics to terminate exhibits a phase transition in the model parameter $\beta$. For $\beta$ sufficiently small, with high probability the network rapidly splits into two disconnected communities with opposing opinions, whereas for $\beta$ large enough the dynamics runs for longer and the density of opinion changes significantly before the process stops. In the rewire-to-random model, we show that a positive fraction of both opinions survive with high probability.