  Fix a positive integer $n$, a real number $p\in (0,1]$, and a (perhaps
random) hypergraph $\mathcal{H}$ on $[n]$. We introduce and investigate the
following random multigraph model, which we denote $\mathbb{G}(n,p\, ;
\,\mathcal{H})$: begin with an empty graph on $n$ vertices, which are labelled
by the set $[n]$. For every $H\in \mathcal{H}$ choose, independently from
previous choices, a doubleton from $H$, say $D = \{i,j\} \subset H$, uniformly
at random and then introduce an edge between the vertices $i$ and $j$ in the
graph with probability $p$, where each edge is introduced independently of all
other edges.
