Spectra and eigenvectors of scale-free networks

2001/03/16 by K. -I. Goh, K.-I. Goh, B. Kahng +1 · 2 citations
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech

paper · pdf · doi:10.1103/physreve.64.051903

published as Phys. Rev. E 64, 051903 (2001) · 6 pages, 10 figures, submitted to PRE

arxiv created 2001/03/16 · openalex publication_date 2001/10/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 lambda1 depends on system size N as lambda1 approximately N1/4 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 N(-0.68).

Citations

• Emergence of scaling in random networks
• Error and attack tolerance of complex networks
• The large-scale organization of metabolic networks

Cited by

• Spectra of complex networks
• Laplacian spectra as a diagnostic tool for network structure and dynamics