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

## TL;DR

This paper shows that disease dynamics on hypergraphs with higher-order interactions can be effectively replicated by pairwise models with scaled parameters, simplifying complex contagion processes.

## Contribution

It introduces a unified agent-based model that bridges classical and hypergraph contagion models, demonstrating how lower-order interactions can mimic higher-order dynamics through parameter scaling.

## Key findings

- Pairwise models can replicate higher-order hypergraph dynamics with parameter adjustments.
- Dynamic variation of disease parameters improves the approximation of transient and steady-state behaviors.
- Heterogeneous hypergraph topologies weaken the accuracy of pairwise approximations.

## 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.