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

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 {q1,q2,...} on matrix elements, where qγ depends on the hierarchy level γ as qγ = 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 σγ = p−νγ, the tail of the spectral 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 of such hierarchical networks has a poly-scale fractal behavior extended over a very broad range of scales.