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Hypergraph assortativity: A dynamical systems perspective.

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The expansion eigenvalue of hypergraphs predicts dynamical processes. Reducing hypergraph dynamical assortativity can help extinguish epidemics, as shown by our new models and data validation.

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Area of Science:

  • Network science
  • Graph theory
  • Mathematical biology

Background:

  • The largest eigenvalue of a network's contact matrix is crucial for understanding dynamical processes.
  • This concept is extended to hypergraphs, where an analogous 'expansion eigenvalue' is vital for hypergraph dynamics.

Purpose of the Study:

  • To derive approximations for the expansion eigenvalue in both uncorrelated and assortative hypergraphs.
  • To introduce 'dynamical assortativity' for uniform hypergraphs.
  • To demonstrate how reducing dynamical assortativity can mitigate epidemic spread.

Main Methods:

  • Utilized a mean-field approach to approximate the expansion eigenvalue for uncorrelated hypergraphs based on degree sequences.
  • Developed a generative model for hypergraphs incorporating degree assortativity.
  • Employed a perturbation approach to approximate the expansion eigenvalue for assortative hypergraphs.
  • Defined and analyzed 'dynamical assortativity' for uniform hypergraphs.

Main Results:

  • Derived approximations for the expansion eigenvalue in terms of degree sequences for uncorrelated hypergraphs.
  • Obtained approximations for the expansion eigenvalue in assortative hypergraphs using a generative model and perturbation theory.
  • Introduced and quantified dynamical assortativity.
  • Showed that reducing dynamical assortativity via preferential rewiring can lead to epidemic extinction.

Conclusions:

  • The expansion eigenvalue is a key metric for hypergraph dynamical processes.
  • Dynamical assortativity provides a novel perspective on network structure's impact on dynamics.
  • Strategies for reducing dynamical assortativity offer potential for epidemic control.