Multifractal properties of growing networks

Abstract

We introduce a new family of models for growing networks. In these networks new edges are attached preferentially to vertices with higher number of connections, and new vertices are created by already existing ones, inheriting part of their parent's connections. We show that combination of these two features produces multifractal degree distributions, where degree is the number of connections of a vertex. An exact multifractal distribution is found for a nontrivial model of this class. The distribution tends to a power-law one, $Π(q) \sim q^{-γ}$, $γ=\sqrt{2}$ in the infinite network limit. Nevertheless, for finite networks's sizes, because of multifractality, attempts to interpret the distribution as a scale-free would result in an ambiguous value of the exponent $γ$.