The Evolving Voter Model on Thick Graphs

In the evolving voter model, when an individual interacts with a neighbor having an opinion different from theirs, they will with probability $1-\alpha$ imitate the neighbor but with probability $ \alpha$ will sever the connection and choose a new neighbor at random (i) from the graph or (ii) from those with the same opinion. Durrett et al. used simulation and heuristics to study these dynamics on sparse graphs. Recently Basu and Sly have studied this system with $1-\alpha = \nu/N$ on a dense Erd\H{o}s-R\'{e}nyi graph $G(N,1/2)$ and rigorously proved that there is a phase transition from rapid disconnection into components with a single opinion to prolonged persistence of discordant edges as $\nu$ increases. In this paper, we consider the intermediate situation of Erd\H{o}s-R\'enyi random graphs with average degree $L=N^a$ where $0 < a < 1$. Most of the paper is devoted to a rigorous analysis of an approximation of the dynamics called the approximate master equation. Using ideas of \cite{LMR} and \cite{Silk} we are able to analyze these dynamics in great detail.