Graph Energies of Egocentric Networks and Their Correlation with Vertex Centrality Measures

TL;DR

This paper explores the use of graph energies in large networks, demonstrating strong correlations with vertex centrality measures and suggesting potential for efficient local algorithms to estimate these measures.

Contribution

It introduces the application of graph energies to egocentric networks and shows their strong correlation with centrality measures, enabling local estimation methods.

Findings

Graph energies correlate strongly with vertex centrality measures.

In some network models, energies align with betweenness and eigencentrality.

Potential for local algorithms to estimate centrality based on energy measures.

Abstract

Graph energy is the energy of the matrix representation of the graph, where the energy of a matrix is the sum of singular values of the matrix. Depending on the definition of a matrix, one can contemplate graph energy, Randi\'c energy, Laplacian energy, distance energy, and many others. Although theoretical properties of various graph energies have been investigated in the past in the areas of mathematics, chemistry, physics, or graph theory, these explorations have been limited to relatively small graphs representing chemical compounds or theoretical graph classes with strictly defined properties. In this paper we investigate the usefulness of the concept of graph energy in the context of large, complex networks. We show that when graph energies are applied to local egocentric networks, the values of these energies correlate strongly with vertex centrality measures. In particular, for some generative network models graph energies tend to correlate strongly with the betweenness and the eigencentrality of vertices. As the exact computation of these centrality measures is expensive and requires global processing of a network, our research opens the possibility of devising efficient algorithms for the estimation of these centrality measures based only on local information.