Localizations on Complex Networks

TL;DR

This paper investigates the localization properties of complex networks using eigenvector analysis, revealing multifractal behaviors and proposing new measures to characterize network structure.

Contribution

It introduces a novel approach using eigenvector components and probability distributions to analyze localization in complex networks, applicable to various network types.

Findings

Eigenvector components exhibit multifractal distributions.

Localization properties depend on network topology.

Proposed measures effectively characterize network structure.

Abstract

We study the structural characteristics of complex networks using the representative eigenvectors of the adjacent matrix. The probability distribution function of the components of the representative eigenvectors are proposed to describe the localization on networks where the Euclidean distance is invalid. Several quantities are used to describe the localization properties of the representative states, such as the participation ratio, the structural entropy, and the probability distribution function of the nearest neighbor level spacings for spectra of complex networks. Whole-cell networks in the real world and the Watts-Strogatz small-world and Barabasi-Albert scale-free networks are considered. The networks have nontrivial localization properties due to the nontrivial topological structures. It is found that the ascending-order-ranked series of the occurrence probabilities at the nodes behave generally multifractally. This characteristic can be used as a structural measure of complex networks.