Spectra and eigenvectors of scale-free networks

Kwang-Il Goh, B. Kahng, D. Kim

Research output: Contribution to journalArticle

153 Citations (Scopus)

Abstract

We study the spectra and eigenvectors of the adjacency matrices of scale-free networks when bidirectional interaction is allowed, so that the adjacency matrix is real and symmetric. The spectral density shows an exponential decay around the center, followed by power-law long tails at both spectrum edges. The largest eigenvalue [formula presented] depends on system size N as [formula presented] for large N, and the corresponding eigenfunction is strongly localized at the hub, the vertex with largest degree. The component of the normalized eigenfunction at the hub is of order unity. We also find that the mass gap scales as [formula presented]

Original languageEnglish
Number of pages1
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume64
Issue number5
DOIs
Publication statusPublished - 2001 Jan 1
Externally publishedYes

Fingerprint

Scale-free Networks
Eigenvector
eigenvectors
hubs
Adjacency Matrix
Eigenfunctions
Largest Eigenvalue
Spectral Density
matrices
Exponential Decay
unity
Tail
Power Law
apexes
eigenvalues
decay
Vertex of a graph
Interaction
interactions

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics

Cite this

Spectra and eigenvectors of scale-free networks. / Goh, Kwang-Il; Kahng, B.; Kim, D.

In: Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Vol. 64, No. 5, 01.01.2001.

Research output: Contribution to journalArticle

@article{f4a3217332d140bb9f40021c863f9936,
title = "Spectra and eigenvectors of scale-free networks",
abstract = "We study the spectra and eigenvectors of the adjacency matrices of scale-free networks when bidirectional interaction is allowed, so that the adjacency matrix is real and symmetric. The spectral density shows an exponential decay around the center, followed by power-law long tails at both spectrum edges. The largest eigenvalue [formula presented] depends on system size N as [formula presented] for large N, and the corresponding eigenfunction is strongly localized at the hub, the vertex with largest degree. The component of the normalized eigenfunction at the hub is of order unity. We also find that the mass gap scales as [formula presented]",
author = "Kwang-Il Goh and B. Kahng and D. Kim",
year = "2001",
month = "1",
day = "1",
doi = "10.1103/PhysRevE.64.051903",
language = "English",
volume = "64",
journal = "Physical Review E",
issn = "2470-0045",
publisher = "American Physical Society",
number = "5",

}

TY - JOUR

T1 - Spectra and eigenvectors of scale-free networks

AU - Goh, Kwang-Il

AU - Kahng, B.

AU - Kim, D.

PY - 2001/1/1

Y1 - 2001/1/1

N2 - We study the spectra and eigenvectors of the adjacency matrices of scale-free networks when bidirectional interaction is allowed, so that the adjacency matrix is real and symmetric. The spectral density shows an exponential decay around the center, followed by power-law long tails at both spectrum edges. The largest eigenvalue [formula presented] depends on system size N as [formula presented] for large N, and the corresponding eigenfunction is strongly localized at the hub, the vertex with largest degree. The component of the normalized eigenfunction at the hub is of order unity. We also find that the mass gap scales as [formula presented]

AB - We study the spectra and eigenvectors of the adjacency matrices of scale-free networks when bidirectional interaction is allowed, so that the adjacency matrix is real and symmetric. The spectral density shows an exponential decay around the center, followed by power-law long tails at both spectrum edges. The largest eigenvalue [formula presented] depends on system size N as [formula presented] for large N, and the corresponding eigenfunction is strongly localized at the hub, the vertex with largest degree. The component of the normalized eigenfunction at the hub is of order unity. We also find that the mass gap scales as [formula presented]

UR - http://www.scopus.com/inward/record.url?scp=33847624239&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847624239&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.64.051903

DO - 10.1103/PhysRevE.64.051903

M3 - Article

VL - 64

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 5

ER -