On scale-free and poly-scale behaviors of random hierarchical networks

Abstract

In this paper the question of statistical properties of block-hierarchical randommatrices is raised for the first time in connection with structural characteristicsof random hierarchical networks obtained by a ‘mipmapping’ procedure. Inparticular, we compute numerically the spectral density of large randomadjacency matrices defined via a hierarchy of the Bernoulli distributions{q1,q2,...} on matrix elements,where qγ depends onthe hierarchy level γ as qγ = p−μγ (μ>0). For the spectral density we clearly see scale-free behavior. We show also that forthe Gaussian distributions on matrix elements with zero mean and variancesσγ = p−νγ, the tail of thespectral density, ρG(λ), behaves as ρG(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 ofsuch hierarchical networks has a poly-scale fractal behavior extended over a very broadrange of scales.