Delocalization transition for the Google matrix

TL;DR

This paper investigates the localization and delocalization phenomena of eigenvectors of the Google matrix, revealing a phase transition that impacts the efficiency of PageRank-based information retrieval.

Contribution

It introduces the concept of a delocalization transition in the Google matrix eigenvectors, linking network parameters to changes in PageRank localization.

Findings

Delocalization transition occurs when network parameters change.

Localized PageRank eigenvectors become delocalized in the complex eigenvalue plane.

Delocalization affects the efficiency of Google search algorithms.

Abstract

We study the localization properties of eigenvectors of the Google matrix, generated both from the World Wide Web and from the Albert-Barabasi model of networks. We establish the emergence of a delocalization phase for the PageRank vector when network parameters are changed. In the phase of localized PageRank, a delocalization takes place in the complex plane of eigenvalues of the matrix, leading to delocalized relaxation modes. We argue that the efficiency of information retrieval by Google-type search is strongly affected in the phase of delocalized PageRank.