  We view hyper-graphs as incidence graphs, i.e. bipartite graphs with a set of
nodes representing vertices and a set of nodes representing hyper-edges, with
two nodes being adjacent if the corresponding vertex belongs to the
corresponding hyper-edge. It defines a random hyper-multigraph specified by two
distributions, one for the degrees of the vertices, and one for the sizes of
the hyper-edges. We develop the logical analysis of this framework and first
prove a convergence law for first-order logic, then characterise the limit
first-order theories defined by a wide class of degree distributions.
Convergence laws of other models follow, and in particular for the classical
Erd\H{o}s-R\'enyi graphs and $k$-uniform hyper-graphs.
