Random hypergraphs and algorithmics

TL;DR

This paper provides exact and asymptotic enumeration results for hypergraphs using exponential generating functions, analyzing their structure and component sizes through combinatorial and complex analysis techniques.

Contribution

It introduces new enumeration formulas and asymptotic analysis methods for hypergraphs, extending Wright inequalities and exploring component structures.

Findings

Bounded hypergraph components with Wright inequalities

Asymptotic enumeration via complex analysis and saddle point method

Characterization of hypergraph components and cycle emergence

Abstract

Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of hypergraph component is bounded, as a generalisation of Wright inequalities for graphs: the proof is a combinatorial understanding of the structure by inclusion exclusion. Asymptotic results are obtained, thanks to generating functions proofs are at the end very easy to read, through complex analysis by saddle point method. By this way, we characterized: - the components with a given number of vertices and of hyperedges by the expected size of a random hypermatching in these structures. - the random hypergraphs (evolving hyperedge by hyperedge) according to the expected number of hyperedges when the first cycle appears in the evolving structure. This work is an open road to further works on random hypergraphs such as threshold phenomenon, tools used here seem to be sufficient at first sight.