Multifractal properties of growing networks

We introduce a new family of models for growing networks. In these networks new edges are preferentially attached to vertices with a higher number of connections, and new vertices are created by already existing ones, partially inheriting (partially copying) connections of their parents. We show that the combination of these two features produces multifractal degree distributions. Here 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 form $\Pi \left( q\right) \sim q^{-\gamma }$ with $\gamma =\sqrt{2}$ in the infinite network limit. For finite networks, because of multifractality, any attempt to interpret the distribution as scale free will result in an ambiguous value of the exponent $\gamma $.