Meanfield 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 lowdimensional evolution equations amenable to analysis. The variables in the evolution equations are mean counts of subgraph states and are referred to as moments. Due to interaction between neighbours, each moment equation is a function of higherorder moments, such that an infinite hierarchy of equations arises. Hence, the derivation requires truncation at a given order, and, an approximation of the highestorder moments in terms of lowerorder ones,known as a closure formula. Recent systematic approximations have either restricted focus to closed moment equations for SIR epidemic spreading or to unclosed moment equations for arbitrary dynamics. In this paper, we develop a general procedure that automates both derivation and closure of arbitrary order moment equations for dynamics with nearestneighbour interactions on undirected networks. Automation of the closure step was made possible by our generalised 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 loops in regimes with longrange dependence. A Mathematica package that automates the moment closure is available for download.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.07643
 Bibcode:
 2021arXiv211107643W
 Keywords:

 Mathematics  Dynamical Systems