Spectra and eigenvectors of scale-free networks

K.-I. Goh, B. Kahng, and D. Kim
Phys. Rev. E 64, 051903 – Published 15 October 2001
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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 λ1 depends on system size N as λ1N1/4 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 N0.68.

  • Received 16 March 2001

DOI:https://doi.org/10.1103/PhysRevE.64.051903

©2001 American Physical Society

Authors & Affiliations

K.-I. Goh, B. Kahng, and D. Kim

  • School of Physics, Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea

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Issue

Vol. 64, Iss. 5 — November 2001

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