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Area of Science:

  • Network Science
  • Graph Theory
  • Statistical Physics

Background:

  • Higher-order interactions are crucial in many complex systems.
  • Existing theoretical models for hypergraphs lack flexibility.
  • There is a need for adaptable models to study hypergraph properties.

Purpose of the Study:

  • Introduce a flexible class of random hypergraph models.
  • Provide a framework for incorporating features like preferential attachment and spatial properties.
  • Establish benchmarks for analyzing empirical hypergraph data.

Main Methods:

  • Developed a general framework for random hypergraphs based on node-hyperedge membership probability.
  • Analyzed Erdos-Renyi type random hypergraphs with constant node membership probability.
  • Introduced and analyzed spatial random hypergraphs with geometric properties.

Main Results:

  • Identified a phase transition in Erdos-Renyi random hypergraphs at a threshold scaling as 1/sqrt[EN].
  • Demonstrated a percolation transition in random geometric hypergraphs with a threshold distance scaling as r_{c}^{*}∼1/sqrt[E].
  • Introduced novel measures for characterizing geometrical properties of hyperedges in spatial models.

Conclusions:

  • The proposed hypergraph models offer significant flexibility, comparable to complex network models.
  • These models provide valuable benchmarks for the analysis of real-world network data with higher-order interactions.
  • The framework facilitates the study of diverse hypergraph structures and their properties.