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

TL;DR

The paper addresses modeling social contagion with higher-order, polyadic interactions on hypergraphs by extending the Watts threshold model to continuous time using approximate master equations (AMEs). It develops a three-variable, reduced AME framework for the double-threshold polyadic WTM and derives an exact reduction to three coupled ODEs with a cascade condition obtained from linearization; the cascade criterion is $\frac{\partial \dot{\theta}}{\partial \phi}|_{(0,0)} \cdot \frac{\partial \dot{\phi}}{\partial \theta}|_{(0,0)} > 1$. The authors validate their approach on empirical networks (a French school contact network and a DBLP coauthorship subhypergraph), showing good agreement with full AMEs and simulations, while noting limitations due to correlations and finite-size effects. They provide code for reproducibility and discuss future work to incorporate structural correlations for even more accurate modeling of real-world polyadic contagions.

Abstract

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.

Figures (10)

• Figure 1: A small hypergraph with active nodes shaded in blue and active hyperedges shaded in purple. We show the state of two hyperedges, where $n$ is the number of nodes in the hyperedge and $i$ is the number of active nodes in the hyperedge. We show hyperedges with $(n = 5, i = 3)$ and $(n = 4, i = 2)$. We also show the states of two nodes, where $k$ is the degree of the node and $m$ is the number of active groups (i.e., hyperdges) to which a node is attached. We show nodes with $(k = 2, m = 2)$ and $(k = 2, m = 1)$.
• Figure 1: The reduced AME system \ref{['eq:rho_dot']}--\ref{['eq:phi_dot']} accurately recovers the solutions of the full AME system \ref{['eq:fni']}--\ref{['eq:alpha']} for several choices of degree distributions, hyperedge-size distributions, node thresholds, and group thresholds in configuration-model hypergraphs. The black markers give the means of 500 simulations on 500 different 10,000-node networks, the solid blue curves give results for the full AME system, and the dotted green curves give results for the reduced AME system. (a) An example with degree distribution $g_k\sim\mathrm{Pois}(11)$, hyperedge-size distribution $p_n \sim \mathrm{Pois}(6)$, node threshold $\sigma_k = 0.2$, group threshold ${\tau_n} = 0.1$, and initially active node fraction $\rho_0 = 0.01$. (b) An example with degree distribution $g_k \sim \mathrm{Pois}(3)$, hyperedge-size distribution $p_n \sim \mathrm{Pois}(2)$, node threshold $\sigma_k = 0.2$, group threshold ${\tau_n} = 0.1$, and initially active node fraction $\rho_0 = 0.01$. (c) An example with degree distribution $g_k = k^{-2.2}/\sum_{j = 1}^{100}j^{-2.2}$, hyperedge-size distribution $p_n = n^{-2.2}/\sum_{m = 1}^{100}m^{-2.2}$, node threshold $\sigma_k = 0.1$, group threshold ${\tau_n} = 0.1$, and initially active node fraction $\rho_0 = 0.01$. (d) An example with degree distribution $g_k = k^{-2.2}/\sum_{j = 1}^{100}j^{-2.2}$, hyperedge-size distribution $p_n = n^{-2.5}/\sum_{m = 1}^{100}m^{-2.5}$, node threshold $\sigma_k = 0.05$, group threshold ${\tau_n} = 0.1$, and initially active node fraction $\rho_0 = 0.01$. For the heavy-tailed distributions in (c) and (d), we impose a maximum degree of $100$ and maximum group size of $100$.
• Figure 1: (a) The steady-state fraction $\rho^{*}$ of active nodes in the reduced AME system \ref{['eq:rho_dot']}--\ref{['eq:phi_dot']} for an initially active node fraction $\rho_0 = 10^{-3}$, degree distribution $g_k \sim \mathrm{Pois}(\langle k \rangle)$, hyperedge-size distribution $p_n\sim\mathrm{Pois}(3)$, node threshold $\sigma_k = 0.18$, and group threshold ${\tau_n} = 0.1$. The black markers indicate the mean steady-state densities of active nodes from 100 WTM simulations on a single 50,000-node configuration-model hypergraph. In our numerical calculations, we suppose that $\rho(t)$ has attained a steady state by time $t = 100$. (b) The steady-state fraction of active nodes in the reduced AME system for initially active node fractions $\rho_0 = 10^{-5}$ (solid blue curve), $\rho_0 = 10^{-4}$ (dashed orange curve), and $\rho_0 = 10^{-3}$ (dotted green curve). The black arrow points to the critical value of $\langle k \rangle$ that we calculate from the linearization of the reduced AME system \ref{['eq:rho_dot']}--\ref{['eq:phi_dot']}. For each of these values of $\rho_0$, the approximate critical degree from the linearization of the $(\dot{\theta}, \dot{\phi})$ system is $\langle k \rangle\approx 8.02$ (with different values in the third decimal place for the three values of $\rho_0$), as the linearization is most accurate near $(\rho(0),\theta(0),\phi(0)) = (0,0,0)$.
• Figure 1: The fraction $\rho(t)$ of active nodes in our continuous-time double-threshold hypergraph WTM on the French primary school face-to-face contact network stehle2011highstonge2022. This hypergraph has $N = 242$ nodes, 1188 hyperedges, mean degree $\langle k\rangle \approx 11.79$, mean hyperedge (i.e., group) size $\langle n\rangle \approx 2.4$, a maximum degree of 32, and a maximum group size of 5. We show the results of computations with (a) an initially active node fraction $\rho_0 = 0.05$, node threshold $\sigma_k = 0.25$, and group threshold ${\tau_n} = 0.3$ and (b) an initially active node fraction $\rho_0 = 0.05$, node threshold $\sigma_k = 0.15$, and group threshold ${\tau_n} = 0.2$. The solid gray curves are solutions of the reduced AME system \ref{['eq:rho_dot']}--\ref{['eq:phi_dot']}, and the purple circles are mean values of $\rho(t)$ from 500 simulations of the double-threshold hypergraph WTM on the original contact hypergraph. The blue crosses are means of simulations of the double-threshold hypergraph WTM on 500 different 242-node configuration-model hypergraphs that we generate with the same degree distribution $\{g_k\}$ and group-size distribution $\{p_{n}\}$ as in the original contact hypergraph, and the green triangles are the means of simulations of the double-threshold hypergraph WTM on 500 different 5000-node configuration-model hypergraphs with the same ${g_k}$ and ${p_n}$ as in the original contact hypergraph.
• Figure 1: The steady-state fraction $\rho^*$ of active nodes from the reduced AME system \ref{['eq:rho_dot']}--\ref{['eq:phi_dot']} (solid green curve) and from the discrete-time system \ref{['eq:chen_steady_state']}--\ref{['eq:f_u']} of Chen et al. chen2025simple (dashed orange curve) for initially active seed fraction $\rho_0 = 10^{-3}$. We show results of computations on configuration-model hypergraphs with (a) degree distribution $g_k\sim\mathrm{Pois}(\langle k \rangle$), hyperedge-size distribution $p_n\sim\mathrm{Pois}(3)$, node threshold $\sigma_k = 0.18$, and group threshold ${\tau_n} = 0.1$ and (b) degree distribution $g_k\sim\mathrm{Pois}(3)$, hyperedge-size distribution $p_n\sim\mathrm{Pois}(\langle n \rangle)$, node threshold $\sigma_k = 0.1$, and group threshold ${\tau_n} = 0.18$.