Localization and universality of eigenvectors in directed random graphs

TL;DR

This paper develops a comprehensive theory for the statistics of right eigenvectors in directed random graphs, revealing localization phenomena and universality properties depending on degree distribution and connectivity.

Contribution

It provides exact analytic expressions for eigenvector localization and shows how degree distribution influences eigenvector behavior in directed random graphs.

Findings

Right eigenvectors are localized in graphs with small average degree.

Localization transition occurs at a critical mean degree independent of degree fluctuations if the fourth moment is finite.

In high connectivity, eigenvector distribution depends only on degree fluctuations.

Abstract

Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, if the fourth moment of the degree distribution is finite, then the critical mean degree of the localization transition is independent of the degree fluctuations, which is different from localization in undirected graphs that is governed by degree fluctuations. We also show that in the high connectivity limit the distribution of the right eigenvector components is solely determined by the degree fluctuations. For delocalized eigenvectors, we recover the universal results from standard random matrix theory that are independent of the degree distribution, while for localized eigenvectors the eigenvector distribution depends on the degree distribution.