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

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. A key component 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.Author summary: Higher-order structure in networks encompasses group-level interactions beyond simple pairwise links. These group structures can profoundly shape dynamics like epidemics and synchronization, often in counterintuitive ways. Studying these effects is challenging because even basic measures like the clustering coefficient have multiple, non-equivalent higher-order generalizations. We introduce a flexible hypergraph model that smoothly interpolates between purely pairwise and higher-order interactions while preserving network connectivity. The model incorporates geometric or feature-based node information from sources such as spatial data or embeddings, enabling realistic network constructions. We demonstrate its utility through case studies on clustering, higher-order PageRank diffusions, and epidemic spreading. Our model provides a simple and flexible method to better delineate the distinct roles of pairwise and higher-order structures in complex networks.