  Moment systems arise in a wide range of contexts and applications, e.g. in
network modeling of complex systems. Since moment systems consist of a high or
even infinite number of coupled equations, an indispensable step in obtaining a
low-dimensional representation that is amenable to further analysis is, in many
cases, to select a moment closure. A moment closure consists of a set of
approximations that express certain higher-order moments in terms of
lower-order ones, so that applying those leads to a closed system of equations
for only the lower-order moments. Closures are frequently found drawing on
intuition and heuristics to come up with quantitatively good approximations. In
contrast to that, we propose an alternative approach where we instead focus on
closures giving rise to certain qualitative features, such as bifurcations.
Importantly, this fundamental change of perspective provides one with the
possibility of classifying moment closures rigorously in regard to these
features. This makes the design and selection of closures more algorithmic,
precise, and reliable. In this work, we carefully study the moment systems that
arise in the mean-field descriptions of two widely known network dynamical
systems, the SIS epidemic and the adaptive voter model. We derive conditions
that any moment closure has to satisfy so that the corresponding closed systems
exhibit the transcritical bifurcation that one expects in these systems coming
from the stochastic particle model.
