Do triangles matter? Replicating hypergraph disease dynamics with lower-order interactions

TL;DR

The paper investigates whether higher-order interactions (e.g., triangles) fundamentally alter disease dynamics or can be captured by lower-order (pairwise) interactions. It introduces an edge-based agent-based framework that unifies SIS/SIR with simplicial contagion across arbitrary hyperedge orders and develops network-activity normalisation, plus adaptive, time-varying infection rates to match higher-order trajectories. The key finding is a duality: steady-state behavior can be reproduced by normalised pairwise dynamics, while transient dynamics require adaptive parameters and are sensitive to topology; robustness holds under misspecification but macro-scale heterogeneities can limit accuracy. Practically, this work suggests that higher-order features may, in many cases, be approximated by carefully tuned lower-order dynamics, informing modeling choices and potential adaptive control strategies in complex networks.

Abstract

Disease spreading models such as the ubiquitous SIS compartmental model and its numerous variants are widely used to understand and predict the behaviour of a given epidemic or information diffusion process. A common approach to imbue more realism to the spreading process is to constrain simulations to a network structure, where connected nodes update their disease state based on pairwise interactions along the edges of their local neighbourhood. Simplicial contagion models (SCM) extend this to hypergraphs such that groups of three nodes are able to interact and propagate the disease along higher-order hyperedges (triangles). Though more flexible, it is not clear the extent to which the inclusion of these higher-order interactions result in dynamics that are characteristically different to those attained from simpler pairwise interactions. Here, we propose an agent-based model that unifies the classical SIS/SIR compartmental model and SCM, and extends it to allow for interactions along hyperedges of arbitrary order. Using this model, we demonstrate how the steady-state dynamics of pairwise interactions can be made to replicate those of simulations that include higher-order topologies by linearly scaling disease parameters based on a proposed measure of network activity. By allowing disease parameters to dynamically vary over time, lower-order pairwise interactions can be made to closely replicate both the transient and steady-state dynamics of higher-order simulations. We demonstrate that this relationship is robust to misspecification in the assumed higher-order interaction model, and applies to non-clique complex hypergraphs with non-trivial heterogeneous topology. For the latter case, it is found that heterogeneities in hypergraph topology result in weakened approximations of higher-order dynamics by pairwise interactions.

Figures (9)

• Figure 1: Epidemic trajectories from ODE model assuming homogeneous mixing, and 20 simulations of the numerical approximation using a $N=500$ node fully connected network. Mean and 90% quantiles shown. Parameters are $\beta_{ODE}=0.0003$, $\mu_{ODE}=0.5N\beta_{ODE}$, $\alpha_{ODE} = 0.04N\beta_{ODE}$.
• Figure 2: SI epidemic trajectories between the pairwise and higher-order simulations. Accompanying activity ratios are shown right revealing a saturation to 1 as the epidemic reaches the endemic steady state. Dotted trajectory in the unnormalised case corresponds to enabling of higher-order interactions in pairwise simulation.
• Figure 3: Simulations with dynamic disease parameters with $K=1$ for triangular lattice and random networks. Shaded regions correspond to $90\%$ CI. Left to right: target trajectories from the higher order simulation, comparison trajectories between pairwise and higher order case, four randomly chosen trajectories of $\beta_1(t)$ with adjustment events indicated by vertical lines, and the mean drift ratio $\bar{\xi}(t)$, with the $\rho$ threshold envelope in red.
• Figure 4: Phaseplots of the peak time, and peak and final proportions for simulated SIR model with dynamic disease parameters. Top and bottoms rows correspond to the pairwise $K=1$ and higher-order $K=4$ simulations for $N=500$ node ER random network. Dynamic adjustment shows good replication of phaseplots and epidemic outbreak threshold.
• Figure 5: Schematic of scaling function $f(p)$ and tested misspecified forms $\hat{f}_1(p)$ and $\hat{f}_2(p)$.