Multifractal analysis of eigenvectors of smallworld networks

TL;DR

This paper uses multifractal analysis to study how the eigenvectors of small-world network adjacency matrices transition between localized and delocalized states, revealing a sharp change at the transition point.

Contribution

It introduces a multifractal approach to analyze eigenvector localization in small-world networks and identifies a critical transition in their spectral properties.

Findings

Central spectrum shows strong multifractality

Tail spectrum approaches delocalization (Dq->1)

Sharp change in correlation dimension at transition

Abstract

Many real-world complex systems have small-world topology characterized by the high clustering of nodes and short path lengths.It is well-known that higher clustering drives localization while shorter path length supports delocalization of the eigenvectors of networks. Using multifractals technique, we investigate localization properties of the eigenvectors of the adjacency matrices of small-world networks constructed using Watts-Strogatz algorithm. We find that the central part of the eigenvalue spectrum is characterized by strong multifractality whereas the tail part of the spectrum have Dq->1. Before the onset of the small-world transition, an increase in the random connections leads to an enhancement in the eigenvectors localization, whereas just after the onset, the eigenvectors show a gradual decrease in the localization. We have verified an existence of sharp change in the correlation dimension at the localization-delocalization transition