An Approximate-Master-Equation Formulation of the Watts Threshold Model on Hypergraphs (arXiv 2503.04020. 2025)

Leah A. Keating, Kwang-Il Goh and Mason A. Porter

In traditional models of behavioral or opinion dynamics on social networks, researchers suppose that all interactions occur between pairs of individuals. However, in reality, social interactions also occur in groups of three or more individuals. A common way to incorporate such polyadic interactions is to study dynamical processes on hypergraphs. In a hypergraph, interactions can occur between any number of the individuals in a network. The Watts threshold model (WTM) is a well-known model of a simplistic social spreading process. Very recently, Chen et al. extended the WTM from dyadic networks (i.e., graphs) to polyadic networks (i.e., hypergraphs). In the present paper, we extend their discrete-time model to continuous time using approximate master equations (AMEs). By using AMEs, we are able to model the system with very high accuracy. We then reduce the high-dimensional AME system to a system of three coupled differential equations without any detectable loss of accuracy. This much lower-dimensional system is more computationally efficient to solve numerically and is also easier to interpret. We linearize the reduced AME system and calculate a cascade condition, which allows us to determine when a large spreading event occurs. We then apply our model to a social contact network of a French primary school and to a hypergraph of computer-science coauthorships. We find that the AME system is accurate in modelling the polyadic WTM on these empirical networks; however, we expect that future work that incorporates structural correlations between nearby nodes and groups into the model for the dynamics will lead to more accurate theory for real-world networks.

Multitype branching process method for modelling complex contagion on clustered networks (2022)
Leah A. Keating, James P. Gleeson and David J.P. O'Sullivan

This paper was an Editors' suggestion in Physical Review E.

Complex contagion adoption dynamics are characterised by a node being more likely to adopt after multiple network neighbours have adopted. We show how to construct multi-type branching processes to approximate complex contagion adoption dynamics on networks with clique-based clustering.  This involves tracking the evolution of a cascade via different classes of clique motifs that account for the different numbers of active, inactive and removed nodes. This discrete-time model assumes that active nodes become immediately and certainly removed in the next time step. This description allows for extensive Monte Carlo simulations (which are faster than network-based simulations), accurate analytical calculation of cascade sizes, determination of critical behaviour and other quantities of interest.

A generating-function approach to modelling complex contagion on clustered networks with multi-type branching processes  (Journal of Complex Networks, 2023)
Leah A. Keating, James P. Gleeson and David J.P. O'Sullivan

Understanding cascading processes on complex network topologies is paramount for modelling how diseases, information, fake news and other media spread. In this paper, we extend the multi-type branching process method developed in Keating et al., 2022, which relies on homogenous network properties, to a more general class of clustered networks. Using a model of socially-inspired complex contagion we obtain results, not just for the average behaviour of the cascades but for full distributions of the cascade properties. We introduce a new method for the inversion of probability generating functions to recover their underlying probability distributions; this derivation naturally extends to higher dimensions. This inversion technique is used along with the multi-type branching process to obtain univariate and bivariate distributions of cascade properties. Finally, using clique cover methods, we apply the methodology to synthetic and real-world networks and compare the theoretical distribution of cascade sizes with the results of extensive numerical simulations.