  Adaptive voter models (AVMs) are simple mechanistic systems that model the
emergence of mesoscopic structure from local networked processes driven by
conflict and homophily. AVMs display rich behavior, including a phase
transition from a fully-fragmented regime of "echo-chambers" to a regime of
persistent disagreement governed by low-dimensional quasistable manifolds. Many
extant methods for approximating the behavior of AVMs are either restricted in
scope, expensive in computation, or inaccurate in predicting important
statistics. In this work, we develop a novel, second-order moment closure
approximation method for binary-state rewire-to-random and rewire-to-same model
variants. We incorporate a small amount of noise via a random mutation term,
which renders the system ergodic. Using ergodicity, we then approximate the
voting process, which is non-Markovian in the second moments of the system,
with a Markovian term near the phase transition. This approximation exploits an
asymmetry between different classes of voting events. The resulting scheme
enables us to predict the location of the phase transition and the active edge
density in the regime of persistent disagreement, across the entire space of
parameters and opinion densities. Numerically, our results are nearly exact for
the rewire-to-random model, and competitive with other current approaches for
the rewire-to-same model. Moreover, our computations display constant scaling
in the mean degree, enabling approximations for denser systems than previously
possible. We conclude with suggestions for model refinements and extensions.
