  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.
