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A framework for second-order eigenvector centralities and clustering coefficients.

Francesca Arrigo1, Desmond J Higham2, Francesco Tudisco3

  • 1Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|May 14, 2020
PubMed
Summary
This summary is machine-generated.

We introduce a novel tensor-based framework to enhance network analysis by including higher-order node interactions, like triangles, alongside traditional links. This method improves network measures and offers new insights into network structure and dynamics.

Keywords:
Perron–Frobenius theoryclustering coefficienthigher-order network analysishypergraphlink predictiontensor

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Area of Science:

  • Network science
  • Data analysis
  • Computational mathematics

Background:

  • Traditional network analysis often relies on pairwise relationships (links).
  • Higher-order structures like wedges and triangles provide richer network information.
  • Existing methods may not fully capture these complex interactions.

Purpose of the Study:

  • To develop a general tensor-based framework for incorporating second-order features into network measures.
  • To extend classical spectral methods and define new network metrics.
  • To analyze the utility of this framework for network analysis tasks.

Main Methods:

  • A tensor-based framework is proposed to integrate pairwise and higher-order network features.
  • A constrained nonlinear eigenvalue problem associated with a cubic tensor is formulated.
  • Nonlinear Perron-Frobenius theory is used to establish existence and uniqueness.
  • A nonlinear power method is employed for efficient computation of spectral measures.

Main Results:

  • The framework successfully incorporates information from wedges and triangles into network measures.
  • A novel spectral clustering coefficient is defined.
  • The existence and uniqueness of the proposed spectral measures are proven.
  • Efficient computational methods are demonstrated.

Conclusions:

  • The tensor-based framework offers a powerful and flexible approach to network analysis.
  • The new spectral measures provide enhanced insights into network structure.
  • The method shows practical value in centrality and link prediction on real-world networks.