A framework for second order eigenvector centralities and clustering coefficients

TL;DR

This paper introduces a tensor-based framework for second order network measures, integrating triples of nodes into centrality and clustering coefficients, with theoretical analysis and applications to synthetic and real-world networks.

Contribution

It develops a novel spectral approach for second order network features using nonlinear eigenvalue problems, extending classical methods and enabling efficient computation.

Findings

New spectral measures for second order network features

Efficient nonlinear power method for computation

Improved link prediction and centrality analysis

Abstract

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.