  Non-uniform hypergraphs appear in various domains of computer science as in
the satisfiability problems and in data analysis. We analyse a general model
where the probability for an edge of size $t$ to belong to the hypergraph
depends of a parameter $\omega_t$ of the model. It is a natural generalization
of the models of graphs presented in "The first cycles in an evolving graph"
[Flajolet, Knuth, Pittel, 1989] and in the "Birth of the giant component"
[Janson, Knuth, \L{}uczak, Pittel, 1993]. The present paper follows the same
general approach based on analytic combinatorics. We show that many analytic
tools developed for the analysis of graphs can be extended surprisingly well to
non-uniform hypergraphs. Specifically, we investigate random hypergraphs with a
large number of vertices $n$ and a complexity, defined as the "excess",
proportional to $n$. We analyze their typical structure before, near and after
the birth of the "complex" components, that are the connected components with
more than one cycle. Finally, we compute statistics of the model to link number
of edges and excess.
