  Moment-closure approximations are an important tool in the analysis of the
dynamics on both static and adaptive networks. Here, we provide a broad survey
over different approximation schemes by applying each of them to the adaptive
voter model. While already the simplest schemes provide reasonable qualitative
results, even very complex and sophisticated approximations fail to capture the
dynamics quantitatively. We then perform a detailed analysis that identifies
the emergence of specific correlations as the reason for the failure of
established approaches, before presenting a simple approximation scheme that
works best in the parameter range where all other approaches fail. By combining
a focused review of published results with new analysis and illustrations, we
seek to build up an intuition regarding the situations when existing approaches
work, when they fail, and how new approaches can be tailored to specific
problems.
