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Hypergraph animals
1School of BioSciences, <a href="https://ror.org/01ej9dk98">University of Melbourne</a> School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3052, Australia.
We introduce hypergraph animals, simple structures for analyzing complex hypergraphs. These structures reveal the importance of high-cardinality edges and degree-hyperedge connections in random hypergraphs.
Area of Science:
- Graph theory
- Network science
- Combinatorics
Background:
- Complex systems rely on intricate network structures.
- Hypergraphs offer a more general framework than traditional graphs.
- Understanding local structures within hypergraphs is crucial.
Purpose of the Study:
- Introduce "hypergraph animals" as novel structures for hypergraph analysis.
- Explore combinatorial properties and relationships to existing concepts like lattice animals and network motifs.
- Analyze the abundance of hypergraph animals in random hypergraph models.
Main Methods:
- Define hypergraph animals to characterize local node neighborhoods.
- Leverage the connection between hypergraph animals and partition numbers for mathematical analysis.
- Investigate abundances in sparse, uncorrelated, and Erdös-Renyí-inspired random hypergraphs.
Main Results:
- Established relationships between hypergraph animals, lattice animals, and network motifs.
- Demonstrated the significance of high-cardinality edges in random hypergraph ensembles.
- Revealed a profound connection between node degree and hyperedge cardinality influencing animal abundances.
Conclusions:
- Hypergraph animals provide a powerful framework for analyzing complex hypergraphs.
- Findings highlight the critical role of edge cardinality and degree in random hypergraph structure.
- Suggests the need for advanced random hypergraph models to capture real-world dependencies.
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