Localization and Universality of Eigenvectors in Directed Random Graphs

Fernando Lucas Metz, Izaak Neri

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)
93 Downloads (Pure)

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 distribution. For delocalized eigenvectors, we recover in this limit 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.
Original languageEnglish
Article number040604
JournalPhysical Review Letters
Volume126
Issue number4
DOIs
Publication statusPublished - 29 Jan 2021

Fingerprint

Dive into the research topics of 'Localization and Universality of Eigenvectors in Directed Random Graphs'. Together they form a unique fingerprint.

Cite this