Spectral properties of complex networks

TL;DR

This review summarizes key findings on the spectral properties of complex networks, covering extremal eigenvalues, bulk spectrum, and degeneracies across various network models and real-world systems.

Contribution

It provides a comprehensive overview of spectral analysis in complex networks, highlighting the understanding of eigenvalue phenomena and their applications.

Findings

Spectral properties vary significantly across different network models.

Eigenvalue distributions reveal structural and dynamical features.

Spectral analysis aids in understanding natural processes on networks.

Abstract

This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part of the spectrum and (c) degenerate eigenvalues, based on the intrinsic properties of eigenvalues and the phenomena they capture. We have reviewed the works done for spectra of various popular model networks, such as the Erd\H{o}s-R\'enyi random networks, scale-free networks, 1-d lattice, small-world networks, and various different real-world networks. Additionally, potential applications of spectral properties for natural processes have been reviewed.