Spectra and eigenvectors of scale-free networks

K.-I. Goh, B. Kahng, D. Kim
2001 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
We study the spectra and eigenvectors of the adjacency matrices of scale-free networks when bi-directional 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 \lambda_1 depends on system size N as \lambda_1 \sim N^{1/4} for large N, and the corresponding eigenfunction is strongly localized at the hub, the vertex with largest
more » ... rtex 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}.
doi:10.1103/physreve.64.051903 pmid:11735964 fatcat:nnwxfncsl5dcfh7eiiunrykwka