A Stochastic Block Hypergraph model

2023-12-19

arXiv.org

AI Summary

Existing hypergraph generative models lack flexibility in controlling community structure, hyperedge formation mechanisms, and random grouping patterns, while failing to systematically characterize hyperedge size distribution, node degree distribution, and community mixing degree. Method: We propose the first concise and tunable Stochastic Block Hypergraph (SBHG) model—extending the classical stochastic block model to multi-node hyperedges—and explicitly modeling intra- and inter-community connection probabilities alongside hyperedge composition rules. Contribution/Results: SBHG satisfies equivalence, simplicity, and interpretability criteria; enables the first parametric generation of hypergraph community structure; theoretically reveals a continuous transition in hyperedge purity and community mixing governed by the inter-to-intra-community connection ratio $q/p$; and empirically demonstrates accurate reproduction of degree distributions, hyperedge size distributions, and community diversity evolution. SBHG thus serves as an interpretable benchmark for community detection, dynamical process modeling, and hypergraph visualization.

Abstract

The stochastic block model is widely used to generate graphs with a community structure, but no simple alternative currently exists for hypergraphs, in which more than two nodes can be connected together through a hyperedge. We discuss here such a hypergraph generalization, based on the clustering connection probability $P_{ij}$ between nodes of communities $i$ and $j$, and that uses an explicit and modulable hyperedge formation process. We focus on the standard case where $P_{ij}=pdelta_{ij}+q(1-delta_{ij})$ when $0leq qleq p$. We propose a simple model that satisfies three criteria: it should be as simple as possible, when $p = q$ the model should be equivalent to the standard hypergraph random model, and it should use an explicit and modulable hyperedge formation process so that the model is intuitive and can easily express different real-world formation processes. We first show that for such a model the degree distribution and hyperedge size distribution can be approximated by binomial distributions with effective parameters that depend on the number of communities and $q/p$. Also, the composition of hyperedges goes for $q=0$ from `pure' hyperedges (comprising nodes belonging to the same community) to `mixed' hyperedges that comprise nodes from different communities for $q=p$. We test various formation processes and our results suggest that when they depend on the composition of the hyperedge, they tend to favor the dominant community and lead to hyperedges with a smaller diversity. In contrast, for formation processes that are independent from the hyperedge structure, we obtain hyperedges comprising a larger diversity of communities. The advantages of the model proposed here are its simplicity and flexibility that make it a good candidate for testing community-related problems, such as their detection, impact on various dynamics, and visualization.

Problem

• Hypergraph Model
• Community Structure
• Flexibility

Innovation

• Hypergraph Model
• Community Structure
• Edge Formation Mechanism