A class of models for random hypergraphs

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 $r_c^{\ast }\sim 1/\sqrt{E}$. 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.

11 pages, 11 figures (final version with additional results)

References in corpus (6)

• Hyperbolic Geometry of Complex Networks
• The physics of higher-order interactions in complex systems
• Random hypergraphs and their applications
• Random walks on hypergraphs
• Continuum Percolation Thresholds in Two Dimensions
• Latent geometry of bipartite networks

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• The theory of percolation on hypergraphs
• Percolation and Topological Properties of Temporal Higher-order Networks
• The simpliciality of higher-order networks
• The nature of hypergraph $k$-core percolation problems
• A framework to generate hypergraphs with community structure
• Random Recursive Hypergraphs
• Entropy-based models to randomize real-world hypergraphs
• Impact of directionality on the emergence of Turing patterns on m-directed higher-order structures
• Growing Hypergraphs with Preferential Linking
• Higher-order null models as a lens for social systems
• Inference and Visualization of Community Structure in Attributed Hypergraphs Using Mixed-Membership Stochastic Block Models