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Connectivity of Random Geometric Hypergraphs.

Henry-Louis de Kergorlay1, Desmond J Higham1

  • 1School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, UK.

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Summary
This summary is machine-generated.

This study introduces a random geometric hypergraph model. A specific radius condition ensures connectivity in this model as nodes and hyperedges increase.

Keywords:
bipartiteradiusrandom graph

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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.