A class of models for random hypergraphs

TL;DR

This paper introduces a flexible class of random hypergraph models, including Erdős-Rényi, preferential attachment, and spatial variants, with analytical results on phase transitions and new measures for geometric properties.

Contribution

It develops a unified framework for random hypergraphs with various features, providing analytical thresholds and new measures for spatial hypergraphs, serving as benchmarks for empirical data analysis.

Findings

Erdős-Rényi hypergraph transition at 1/√(EN)

Percolation transition in spatial hypergraphs at r_c*∼1/√E

Framework enables analysis of complex hypergraph properties

Abstract

Despite the recently exhibited importance of higher-order interactions for various processes, few flexible (null) models are available. In particular, most studies on hypergraphs focus on a small set of theoretical models. Here, we introduce a class of models for random hypergraphs which displays a similar level of flexibility of complex network models and where the main ingredient is the probability that a node belongs to a hyperedge. When this probability is a constant, we obtain a random hypergraph in the same spirit as the Erdos-Renyi graph. This framework also allows us to introduce different ingredients such as the preferential attachment for hypergraphs, or spatial random hypergraphs. In particular, we show that for the Erdos-Renyi case there is a transition threshold scaling as $1/\sqrt{EN}$ where $N$ is the number of nodes and $E$ the number of hyperedges. We also discuss a random geometric hypergraph which displays a percolation transition for a threshold distance scaling as $r_c^*\sim 1/\sqrt{E}$. For these various models, we provide results for the most interesting measures, and also introduce new ones in the spatial case for characterizing the geometrical properties of hyperedges. These different models might serve as benchmarks useful for analyzing empirical data.