A spatial hypergraph model to smoothly interpolate between pairwise graphs and hypergraphs to study higher-order structures

TL;DR

This paper presents a spatial hypergraph model that smoothly transitions between pairwise graphs and hypergraphs, enabling the study of higher-order network structures and their effects on clustering, diffusion, and epidemic dynamics.

Contribution

The authors introduce a novel spatial hypergraph model with a resolution parameter that interpolates between pairwise and hyperedge connections, invariant to hyperedge choice.

Findings

Model allows analysis of higher-order structure effects

Enables study of clustering coefficients and diffusion

Facilitates understanding of epidemic spread in networks

Abstract

We introduce a spatial graph and hypergraph model that smoothly interpolates between a graph with purely pairwise edges and a graph where all connections are in large hyperedges. The key component is a spatial clustering resolution parameter that varies between assigning all the vertices in a spatial region to individual clusters, resulting in the pairwise case, to assigning all the vertices in a spatial region to a single cluster, which results in the large hyperedge case. An important outcome of this model is that the spatial structure is invariant to the choice of hyperedges. Consequently, this model enables us to study clustering coefficients, graph diffusion, and epidemic spread and how their behavior changes as a function of the higher-order structure in the network with a fixed spatial substrate. We hope that our model will find future uses to distill or explain other behaviors in higher-order networks.