Simplicity of random hypergraphs

TL;DR

This paper extends classical graph results to random hypergraphs, deriving explicit formulas for the expected number of non-simple features and showing their vanishing proportion in large hypergraphs.

Contribution

It provides the first explicit formulas for expected self-loops, multi-hyperedges, and degenerate hyperedges in random hypergraphs generated by the configuration model.

Findings

Expected number of non-simple hypergraph features derived explicitly.

Expected fraction of such features vanishes as the number of vertices grows.

Results extend classical graph theory to hypergraph models.

Abstract

Random hypergraphs extend the classical notion of random graphs by allowing hyperedges to join more than two vertices, making them well-suited for modeling higher-order interactions in complex systems. Despite their broad applicability, many structural properties of random hypergraphs remain less understood than in the graph setting. One such property is simplicity: the absence of self-loops, multi-hyperedges, and, in the hypergraph context, degenerate hyperedges where hyperedges contain a copy of the same vertex at least twice. While the behaviour of the number of such self-loops and multi-hyperedges is well understood for random graphs through the configuration model, analogous results for hypergraphs are comparatively sparse. In this work, we study both undirected and directed hypergraphs generated by the configuration model with prescribed vertex and hyperedge degrees. We derive exact, explicit expressions for the expected number of self-loops, multi-hyperedges and degenerate hyperedges, extending classical results from the graph setting. In addition, an asymptotical analysis shows that, under mild moment conditions on the degree distribution, the expected fraction of self-loops, multi-hyperedges and degenerate hyperedges vanishes as the number of vertices grows. Our results provide a systematic understanding of simplicity in directed and undirected hypergraph models.