  We propose a simple model for a hypergraph generalization of the stochastic
block model, using the clustering connection probability $P_{ij}$ between
communities $i$ and $j$, and integrating explicitly the hyperedge formation
process. Indeed, hyperedges are groups of nodes and we can expect that
different real-world networks correspond to different formation processes of
these groups and to different levels of homophily between nodes. We describe a
simple model where we can explicitly introduce the hyperedge formation process
and study its impact on the composition of hyperedges. We focus on the standard
case where $P_{ij}=p\delta_{ij}+q(1-\delta_{ij})$, and when $0\leq q\leq p$, we
show that the degree and hyperedge size distributions can be approximated by
binomials with effective parameters that depend on the number of communities
and on $q/p$. Also, the composition of hyperedges goes for $q=0$ from `pure'
hyperedges (comprising nodes belonging to the same community) to `mixed'
hyperedges that comprise nodes from different communities for $q=p$. We tested
various formation processes and our results suggest that when they depend on
the composition of the hyperedge, they tend to favor the dominant community and
lead to hyperedges with a smaller diversity. In contrast, for formation
processes that are independent from the hyperedge structure, we obtain
hyperedges comprising a larger diversity of communities. The advantages of the
model proposed here are its simplicity and flexibility that make it a good
candidate for testing community-related problems, from their detection to their
impact on various dynamics.
