Mean-field models of dynamics on networks via moment closure: An automated procedure.

Abstract

In the study of dynamics on networks, moment closure is a commonly used method to obtain low-dimensional evolution equationsamenable to analysis. The variables in the evolution equationsare mean counts of subgraph states and are referred to as moments. Due to interaction between neighbors, each moment equationis a function of higher-order moments, such that an infinite hierarchy of equationsarises. Hence, the derivation requires truncation at a given order and an approximation of the highest-order moments in terms of lower-order ones, known as a closure formula. Recent systematic approximations have either restricted focus to closed moment equationsfor SIR epidemic spreading or to unclosed moment equationsfor arbitrary dynamics. In this paper, we develop a general procedure that automates both derivation and closure of arbitrary order moment equationsfor dynamics with nearest-neighbor interactions on undirected networks. Automation of the closure step was made possible by our generalized closure scheme, which systematically decomposes the largest subgraphs into their smaller components. We show that this decomposition is exact if these components form a tree, there is independence at distances beyond their graph diameter, and there is spatial homogeneity. Testing our method for SIS epidemic spreading on lattices and random networks confirms that biases are larger for networks with many short cycles in regimes with long-range dependence. A Mathematica package that automates the moment closure is available for download.