Anderson transition for Google matrix eigenstates

TL;DR

This paper introduces random matrix models for the Google matrix of directed networks, revealing an Anderson transition in eigenstates and spectral properties similar to real networks, with implications for PageRank decay.

Contribution

It presents new random matrix models that replicate key spectral and eigenstate properties of real directed networks, including Anderson transition phenomena.

Findings

Eigenstates exhibit Anderson transition from localized to delocalized states.

Spectrum lacks spectral gap and has broad eigenvalue distribution.

PageRank vector shows algebraic decay similar to real networks.

Abstract

We introduce a number of random matrix models describing the Google matrix G of directed networks. The properties of their spectra and eigenstates are analyzed by numerical matrix diagonalization. We show that for certain models it is possible to have an algebraic decay of PageRank vector with the exponent similar to real directed networks. At the same time the spectrum has no spectral gap and a broad distribution of eigenvalues in the complex plain. The eigenstates of G are characterized by the Anderson transition from localized to delocalized states and a mobility edge curve in the complex plane of eigenvalues.