Phase Transition of Random Non-Uniform Hypergraphs

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 used by Flajolet, Knuth and Pittel [1989] and Janson, Knuth, \L{}uczak and 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. More specifically, we analyze their typical structure before and near the birth of the \emph{complex} components, that are the connected components with more than one cycle. We derive the asymptotic number of sparse connected hypergraphs as their complexity, defined as the \emph{excess}, increases. Although less natural than the number of edges, this parameter allows a precise description of the structure of hypergraphs. Finally, we compute some statistics of the model to link number of edges and excess.