  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 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 a hypergraph with 0.7 billion
hyperedges within a few hours.
