Hypergraphs, a generalization of graphs, naturally represent groupwise relationships among multiple individuals or objects, which are common in many application areas, including web, bioinformatics, and social networks. The flexibility in the number of nodes in each hyperedge, which provides the expressiveness of hypergraphs, brings about structural differences between graphs and hypergraphs. Especially, the overlaps of hyperedges lead to complex high-order relations beyond pairwise relations, raising new questions that have not been considered in graphs: How do hyperedges overlap in real-world hypergraphs? Are there any pervasive characteristics? What underlying process can cause such patterns? In this work, we closely investigate thirteen real-world hypergraphs from various domains and share interesting observations of the overlaps of hyperedges. To this end, we define principled measures and statistically compare the overlaps of hyperedges in real-world hypergraphs and those in null models. Additionally, based on the observations, we propose HyperLap, a realistic hypergraph generative model. HyperLap is (a) Realistic: it accurately reproduces overlapping patterns of real-world hypergraphs, (b) Automatically Fittable: its parameters can be tuned automatically using HyperLap+ to generate hypergraphs particularly similar to a given target hypergraph, (c) Scalable: it generates and fits hypergraphs with 0.7 billion hyperedges within few hours.

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