Preserving Bifurcations through Moment Closures

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.