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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. 

We demonstrate a spatial hypergraph model that allows us to vary the amount of higher-order structure in the generated hypergraph. Specifically, we can vary from a model that is a pure pairwise graph into a model that is almost a pure hypergraph. We use this spatial hypergraph model to study higher-order effects in epidemic spread. We use a susceptible-infected-recovered-susceptible (SIRS) epidemic model designed to mimic the spread of an airborne pathogen. We study three types of airborne effects that emulate airborne dilution effects. For the scenario of linear dilution, which roughly corresponds to constant ventilation per person as required in many building codes, we see essentially no impact from introducing small hyperedges up to size 15 whereas we do see effects when the hyperedge set is dominated by large hyperedges. Specifically, we track the mean infections after the SIRS epidemic has run for a while so it is in a ?steady state? and find the mean is higher in the large hyperedge regime whereas it is unchanged from pairwise to small hyperedge regime.