Connectivity of Random Geometric Hypergraphs

Area of Science:

• Graph theory
• Network science
• Probability theory

Background:

• Real-world data often exhibits complex, higher-order relationships beyond simple pairwise connections.
• Existing network models may not fully capture these intricate structures.
• Hypergraphs offer a framework for representing multi-way relationships.

Purpose of the Study:

• To introduce and analyze a novel random geometric hypergraph model.
• To understand how this model captures higher-order connections.
• To investigate the conditions for network connectivity within this model.

Main Methods:

• Developing a random geometric hypergraph model based on an underlying bipartite graph.
• Sampling nodes and hyperedges uniformly within a domain.
• Assigning nodes to hyperedges based on proximity (radius).
• Analyzing connectivity properties in an asymptotic regime.

Main Results:

• The model effectively represents higher-order connections found in real datasets.
• A precise condition on the radius is established for guaranteeing network connectivity.
• Connectivity is analyzed in the asymptotic limit of growing nodes and hyperedges.

Conclusions:

• The proposed random geometric hypergraph model provides a valuable tool for studying complex networks.
• The derived radius condition offers a theoretical guarantee for network connectivity.
• This work contributes to the understanding of network formation and properties in higher-order structures.