Connectivity of random hypergraphs with a given hyperedge size distribution

TL;DR

This paper investigates the connectivity threshold of random hypergraphs with diverse hyperedge sizes, revealing that average hyperedge size primarily determines connectivity, largely unaffected by distribution shape or higher moments.

Contribution

It establishes a threshold criterion for connectivity in hypergraphs with heavy-tailed hyperedge size distributions, emphasizing the role of average hyperedge size.

Findings

Connectivity threshold depends mainly on average hyperedge size.

Heavy-tailed hyperedge distributions do not significantly alter the connectivity threshold.

Results extend to related random intersection graph models.

Abstract

This article discusses random hypergraphs with varying hyperedge sizes, admitting large hyperedges with size tending to infinity, and heavy-tailed limiting hyperedge size distributions. The main result describes a threshold for the random hypergraph to be connected with high probability, and shows that the average hyperedge size suffices to characterise connectivity under mild regularity assumptions. Especially, the connectivity threshold is in most cases insensitive to the shape and higher moments of the hyperedge size distribution. Similar results are also provided for related random intersection graph models.