  Networks are widely used to model the contact structure within a population
and in the resulting models of disease spread. While networks provide a high
degree of realism, the analysis of the exact model is out of reach and even
numerical methods fail for modest network size. Hence, mean-field models (e.g.
pairwise) focusing on describing the evolution of some summary statistics from
the exact model gained a lot of traction over the last few decades. In this
paper we revisit the problem of deriving triple closures for pairwise models
and we investigate in detail the assumptions behind some of the well-known
closures as well as their validity. Using a top-down approach we start at the
level of the entire graph and work down to the level of triples and combine
this with information around nodes and pairs. We use our approach to derive
many of the existing closures and propose new ones and theoretically connect
the two well-studied models of multinomial link and Poisson link selection. The
theoretical work is backed up by numerical examples to highlight where the
commonly used assumptions may fail and provide some recommendations for how to
choose the most appropriate closure when using graphs with no or modest degree
heterogeneity.
