2008/11/30 by В. А. Аветисов, V. A. Avetisov, Alexander V. Chertovich +4 · 1 citation
Mathematics · Physics and Astronomy · #Adjacency matrix #Combinatorics #Complex Network Analysis Techniques #Complex network #Degree distribution #Gaussian #Graph #Hierarchy #Mathematics #Matrix (chemical analysis) #Physics #Quantum mechanics #Random Matrices and Applications #Random matrix #Scale (ratio) #Spectral density #Spectral power distribution #Statistical physics #Statistics #advanced mathematical theories #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1088/1742-5468/2009/07/p07008
11 pages, 6 figures (paper is substantially revised)
arxiv created 2009/04/10 · openalex publication_date 2009/07/01 · arxiv updated 2015/05/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In this paper the question of statistical properties of block-hierarchical random matrices is raised for the first time in connection with structural characteristics of random hierarchical networks obtained by a ‘mipmapping’ procedure. In particular, we compute numerically the spectral density of large random adjacency matrices defined via a hierarchy of the Bernoulli distributions q 1 , q 2 ,... on matrix elements, where q γ depends on the hierarchy level γ as q γ = p −μγ (μ>0). For the spectral density we clearly see scale-free behavior. We show also that for the Gaussian distributions on matrix elements with zero mean and variances σ γ = p −νγ , the tail of the spectral density, ρ G (λ), behaves as ρ G (λ)∼|λ| −(2−ν)/(1−ν) for and 0<ν<1, while for ν≥1 the power-law behavior is terminated. We also find that the vertex degree distribution of such hierarchical networks has a poly-scale fractal behavior extended over a very broad range of scales.