A framework for second order eigenvector centralities and clustering coefficients

Francesca Arrigo, Desmond Higham, Francesco Tudisco

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10 Citations (Scopus)
6 Downloads (Pure)


We propose and analyse a general tensor-based framework for incorporating second-order features into network measures. This approach allows us to combine traditional pairwise links with information that records whether triples of nodes are involved in wedges or triangles. Our treatment covers classical spectral methods and recently proposed cases from the literature, but we also identify many interesting extensions. In particular, we define a mutually reinforcing (spectral) version of the classical clustering coefficient. The underlying object of study is a constrained nonlinear eigenvalue problem associated with a cubic tensor. Using recent results from nonlinear Perron–Frobenius theory, we establish existence and uniqueness under appropriate conditions, and show that the new spectral measures can be computed efficiently with a nonlinear power method. To illustrate the added value of the new formulation, we analyse the measures on a class of synthetic networks. We also give computational results on centrality and link prediction for real-world networks.
Original languageEnglish
Number of pages21
JournalProceedings of the Royal Society A: Mathematical Physical and Engineering Sciences
Issue number2236
Publication statusPublished - 31 Mar 2020


  • link prediction
  • clustering coefficient
  • higher order network analysis
  • tensor
  • hypergraph
  • Perron–Frobenius theory


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