Spectral properties, localization transition and multifractal eigenvectors of the Laplacian on heterogeneous networks

TL;DR

This paper investigates the spectral characteristics and eigenvector localization in heterogeneous networks with random and constant couplings, revealing transitions from delocalized to localized states with multifractal properties.

Contribution

It provides a detailed analysis of how spectral density and eigenvector localization depend on network heterogeneity and coupling randomness, highlighting a transition to multifractality.

Findings

Spectral density diverges with strong degree fluctuations.

Transition from non-ergodic delocalized to localized eigenvectors.

Eigenvectors exhibit multifractal scaling at the transition.

Abstract

We study the spectral properties and eigenvector statistics of the Laplacian on highly-connected networks with random coupling strengths and a gamma distribution of rescaled degrees. The spectral density, the distribution of the local density of states, the singularity spectrum and the multifractal exponents of this model exhibit a rich behaviour as a function of the first two moments of the coupling strengths and the variance of the rescaled degrees. In the case of random coupling strengths, the spectral density diverges within the bulk of the spectrum when degree fluctuations are strong enough. The emergence of this singular behaviour marks a transition from non-ergodic delocalized states to localized eigenvectors that exhibit pronounced multifractal scaling. For constant coupling strengths, the bulk of the spectrum is characterized by a regular spectral density. In this case, the corresponding eigenvectors display localization properties reminiscent of the critical point of the Anderson localization transition on random graphs.