Random Recursive Hypergraphs

TL;DR

This paper introduces a parameter-free model for random recursive hypergraphs, analyzing their properties and deriving explicit formulas using advanced combinatorial numbers, with extensions to deformed models.

Contribution

It presents a novel, parameter-free recursive hypergraph model and provides explicit mathematical characterizations of its properties, including extensions to deformed variants.

Findings

Explicit formulas involving harmonic, Bernoulli, Eulerian, and Stirling numbers.

The model's characteristics are analytically tractable and mathematically elegant.

Extensions lead to new models of growing random hypergraphs.

Abstract

Random recursive hypergraphs grow by adding, at each step, a vertex and an edge formed by joining the new vertex to a randomly chosen existing edge. The model is parameter-free, and several characteristics of emerging hypergraphs admit neat expressions via harmonic numbers, Bernoulli numbers, Eulerian numbers, and Stirling numbers of the first kind. Natural deformations of random recursive hypergraphs give rise to fascinating models of growing random hypergraphs.