Exploring the "Middle Earth" of Network Spectra via a Gaussian Matrix Function

TL;DR

This paper introduces the Gaussian Estrada index, a spectral measure capturing the significance of eigenvalues near zero in network spectra, revealing structural insights in both artificial and real-world networks.

Contribution

It provides bounds and formulas for the Gaussian Estrada index across various graph types and links this index to structural features like bicliques in real networks.

Findings

Maximum index for star and complete bipartite graphs

Formulas for Erdős-Rényi and Barabási-Albert graphs

Index correlates with structural patterns like bicliques

Abstract

We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. In particular, we study the Gaussian Estrada index---an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Here we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erd\H{o}s-R\'enyi random graphs as well as for the Barab\'asi-Albert graphs. We also show that in real-world networks this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs.