Stochastic block hypergraph model.

Area of Science:

• Network Science
• Statistical Physics
• Data Mining

Background:

• The stochastic block model (SBM) is a standard for generating graphs with community structure.
• Existing models lack simple generalizations for hypergraphs, where hyperedges connect multiple nodes.
• There is a need for intuitive hypergraph models to study community phenomena.

Purpose of the Study:

• To propose a simple and flexible generalization of the SBM for hypergraphs.
• To introduce a hypergraph model with an explicit, modulable hyperedge formation process.
• To analyze the model's properties concerning degree and hyperedge size distributions.

Main Methods:

• Developed a hypergraph model based on clustering connection probability P_ij.
• Focused on the standard case P_ij = pδ_ij + q(1-δ_ij) with 0 ≤ q ≤ p.
• Analyzed degree and hyperedge size distributions, approximating them with binomial distributions.

Main Results:

• The model's degree and hyperedge size distributions approximate binomial distributions.
• Hyperedge composition transitions from 'pure' (intra-community) to 'mixed' (inter-community) as q/p increases.
• Formation processes influence hyperedge diversity: composition-dependent processes favor dominant communities, while structure-independent processes yield greater diversity.

Conclusions:

• The proposed hypergraph model is simple, flexible, and intuitive.
• It effectively captures community structure and allows for studying various formation processes.
• The model is suitable for investigating community detection, dynamics, and visualization in complex networks.