Spectrum of complex networks

TL;DR

This paper analyzes the spectral properties of complex networks, especially preferential attachment graphs, revealing the limiting eigenvalue distribution and localization of eigenvectors, with implications for understanding network structure.

Contribution

It provides the first rigorous determination of the spectral distribution and eigenvector localization for preferential attachment networks, extending to other models.

Findings

Limiting spectral distribution of large networks determined

Leading eigenvectors are strongly localized

Analysis applicable to various network models

Abstract

The study of complex networks has been one of the most active fields in science in recent decades. Spectral properties of networks (or graphs that represent them) are of fundamental importance. Researchers have been investigating these properties for many years, and, based on numerical data, have raised a number of questions about the distribution of the eigenvalues and eigenvectors. In this paper, we give the solution to some of these questions. In particular, we determine the limiting distribution of (the bulk of) the spectrum as the size of the network grows to infinity and show that the leading eigenvectors are strongly localized. We focus on the preferential attachment graph, which is the most popular mathematical model for growing complex networks. Our analysis is, on the other hand, general and can be applied to other models.