Phase Transition of Random Non-Uniform Hypergraphs

TL;DR

This paper extends analytic combinatorics methods to analyze phase transitions in random non-uniform hypergraphs, revealing structural changes and component statistics as the hypergraph evolves.

Contribution

It generalizes graph models to non-uniform hypergraphs and applies analytic combinatorics to study their phase transitions and component structures.

Findings

Identification of the phase transition point for hypergraph connectivity.

Characterization of the structure of complex components near the phase transition.

Quantitative relationships between edges, excess, and component complexity.

Abstract

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.