Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes
Owen T Courtney1, Ginestra Bianconi1
1School of Mathematical Sciences, Queen Mary University of London, E1 4NS, London, United Kingdom.
Physical Review. E
|July 15, 2016
Summary
This study introduces generalized network structures called simplicial complexes and their configuration model. We developed algorithms to construct these complexes and analyzed their structural properties and correlations.
Area of Science:
- Network Science
- Complex Systems
- Topology
Background:
- Simplicial complexes generalize network structures to model interactions among more than two nodes.
- They are applicable to diverse systems like brain, social, and collaboration networks.
Purpose of the Study:
- To characterize simplicial complex structure using generalized degrees.
- To introduce and analyze the configuration model and canonical ensemble of simplicial complexes.
- To derive the structural cutoff and analyze correlations in these models.
Main Methods:
- Characterization using generalized degrees (encoding multi-node interactions).
- Introduction of the configuration model and canonical ensemble for simplicial complexes.
- Entropy evaluation and asymptotic expression for the number of simplicial complexes.
- Development of algorithms for constructing simplicial complexes within these models.
- Derivation of the structural cutoff and numerical analysis of correlations.
Main Results:
- Generalized degrees capture fundamental properties of multi-node interactions.
- Asymptotic expression for the number of simplicial complexes in the configuration model.
- The structural cutoff for simplicial complexes is derived, reducing to simple networks for d=1.
- Natural correlations in the configuration model without structural cutoff are numerically analyzed.
Conclusions:
- Simplicial complexes offer a powerful framework for studying complex interacting systems.
- The developed models and methods provide tools for analyzing and constructing these generalized networks.
- Understanding generalized degrees and structural cutoffs is key to characterizing complex network behavior.
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