An Approximate-Master-Equation Formulation of the Watts Threshold Model on Hypergraphs

Abstract

In traditional models of behavioral and opinion dynamics on networks, researchers suppose that interactions occur between pairs of individuals. However, in reality, social interactions also occur in sets of three or more individuals. A common way to incorporate such polyadic interactions is to study dynamical processes on hypergraphs, in which interactions can occur between any number of the individuals. A major current effort is to generalize popular dynamical processes from graphs to hypergraphs. One important model is the Watts threshold model (WTM), which describes a simplistic social spreading process and was extended recently by Chen et al. from dyadic networks (i.e., ordinary graphs) to polyadic networks (i.e., hypergraphs). In the present paper, we extend their discrete-time model to continuous time by using approximate master equations (AMEs). By employing AMEs, we are able to very accurately model the mean behavior of the continuous-time system. We reduce the high-dimensional AME system to a system of three coupled ordinary differential equations without any detectable loss of accuracy. This much lower-dimensional system is computationally faster to solve numerically, and it is also easier to interpret. We linearize the reduced AME system and derive a cascade condition, which allows us to determine when a large spreading event occurs. We then examine our reduced AME system on a contact hypergraph of a French primary school and on a hypergraph of computer-science coauthorships. We find that our full and reduced AME systems give reasonably good descriptions of the mean time-dependent fraction of active nodes in the polyadic WTM dynamics on these empirical networks.

arXiv comment25 pages, 10 figures