  Complex networks are pervasive in the real world, capturing dyadic
interactions between pairs of vertices, and a large corpus has emerged on their
mining and modeling. However, many phenomena are comprised of polyadic
interactions between more than two vertices. Such complex hypergraphs range
from emails among groups of individuals, scholarly collaboration, or joint
interactions of proteins in living cells.
  A key generative principle within social and other complex networks is
transitivity, where friends of friends are more likely friends. The previously
proposed Iterated Local Transitivity (ILT) model incorporated transitivity as
an evolutionary mechanism. The ILT model provably satisfies many observed
properties of social networks, such as densification, low average distances,
and high clustering coefficients.
  We propose a new, generative model for complex hypergraphs based on
transitivity, called the Iterated Local Transitivity Hypergraph (or ILTH)
model. In ILTH, we iteratively apply the principle of transitivity to form new
hypergraphs. The resulting model generates hypergraphs simulating properties
observed in real-world complex hypergraphs, such as densification and low
average distances. We consider properties unique to hypergraphs not captured by
their 2-section. We show that certain motifs, which are specified
subhypergraphs of small order, have faster growth rates in ILTH hypergraphs
than in random hypergraphs with the same order and expected average degree. We
show that the graphs admitting a homomorphism into the 2-section of the initial
hypergraph appear as induced subgraphs in the 2-section of ILTH hypergraphs. We
consider new and existing hypergraph clustering coefficients, and show that
these coefficients have larger values in ILTH hypergraphs than in comparable
random hypergraphs.
