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

TL;DR

This paper investigates the spectral and structural properties of hierarchical random networks, revealing scale-free and poly-scale behaviors through numerical analysis of spectral densities and degree distributions.

Contribution

It introduces a novel analysis of block-hierarchical random matrices and networks, demonstrating scale-free spectral density and broad poly-scale fractal degree distributions.

Findings

Spectral density exhibits scale-free behavior.

Tail of spectral density follows a power-law for certain parameters.

Vertex degree distribution shows broad poly-scale fractal behavior.

Abstract

In this paper the question about statistical properties of block--hierarchical random matrices is raised for the first time in connection with structural characteristics of random hierarchical networks obtained by mipmapping procedure. In particular, we compute numerically the spectral density of large random adjacency matrices defined by a hierarchy of the Bernoulli distributions $\{q_1,q_2,...\}$ on matrix elements, where $q_{\gamma}$ depends on hierarchy level $\gamma$ as $q_{\gamma}=p^{-\mu \gamma}$ ($\mu>0$). For the spectral density we clearly see the free--scale behavior. We show also that for the Gaussian distributions on matrix elements with zero mean and variances $\sigma_{\gamma}=p^{-\nu \gamma}$, the tail of the spectral density, $\rho_G(\lambda)$, behaves as $\rho_G(\lambda) \sim |\lambda|^{-(2-\nu)/(1-\nu)}$ for $|\lambda|\to\infty$ and $0<\nu<1$, while for $\nu\ge 1$ 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 to a very broad range of scales.