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

Francesca Arrigo,Francesco Tudisco,Desmond J Higham

doi:10.1098/rspa.2019.0724

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.

Highlights

Introduction and motivationThe classical paradigm in network science is to analyse a complex system by focusing on pairwise interactions; 2020 The Authors
There is a sense in which many algorithms in network science indirectly go beyond pairwise interactions by considering traversals around the network
Our aim here is to develop and analyse a general framework for incorporating second-order features; see definition 3.1. This takes the form of a constrained nonlinear eigenvalue problem associated with a nonlinear mapping defined in terms of a square matrix and a cubic tensor

Summary

Introduction and motivation

The classical paradigm in network science is to analyse a complex system by focusing on pairwise interactions; 2020 The Authors. Our aim here is to develop and analyse a general framework for incorporating second-order features; see definition 3.1 This takes the form of a constrained nonlinear eigenvalue problem associated with a nonlinear mapping defined in terms of a square matrix and a cubic tensor. The classic PageRank algorithm [12] is perhaps the best known example of such a measure Within this setting, in definition 3.3, we define for the first time a mutually reinforcing version of the classical Watts–Strogatz clustering coefficient [13]; here, we give extra weight to nodes that form triangles with nodes that are themselves involved in important triangles. In definition 3.3 below, we show how a mutually reinforcing clustering coefficient can be defined

General eigenvector model

ELEGANS

Example network with theoretical comparison

Conclusion