Localization and Universality of Eigenvectors in Directed Random Graphs

TY - JOUR

T1 - Localization and Universality of Eigenvectors in Directed Random Graphs

AU - Metz, Fernando Lucas

AU - Neri, Izaak

N1 - Funding Information: The authors thank Jacopo Grilli for interesting discussions. F. L. M. thanks the London Mathematical Laboratory and CNPq/Brazil for financial support. Publisher Copyright: © 2021 American Physical Society. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/1/29

Y1 - 2021/1/29

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85100231542&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.126.040604

DO - 10.1103/PhysRevLett.126.040604

M3 - Article

SN - 0031-9007

VL - 126

JO - Physical Review Letters

JF - Physical Review Letters

IS - 4

M1 - 040604

ER -