  Graphs have been utilized as a powerful tool to model pairwise relationships
between people or objects. Such structure is a special type of a broader
concept referred to as hypergraph, in which each hyperedge may consist of an
arbitrary number of nodes, rather than just two. A large number of real-world
datasets are of this form - for example, list of recipients of emails sent from
an organization, users participating in a discussion thread or subject labels
tagged in an online question. However, due to complex representations and lack
of adequate tools, little attention has been paid to exploring the underlying
patterns in these interactions.
  In this work, we empirically study a number of real-world hypergraph datasets
across various domains. In order to enable thorough investigations, we
introduce the multi-level decomposition method, which represents each
hypergraph by a set of pairwise graphs. Each pairwise graph, which we refer to
as a k-level decomposed graph, captures the interactions between pairs of
subsets of k nodes. We empirically find that at each decomposition level, the
investigated hypergraphs obey five structural properties. These properties
serve as criteria for evaluating how realistic a hypergraph is, and establish a
foundation for the hypergraph generation problem. We also propose a hypergraph
generator that is remarkably simple but capable of fulfilling these evaluation
metrics, which are hardly achieved by other baseline generator models.
