Multifractality of complex networks

TL;DR

This paper explores the multifractal nature of scale-free networks, showing that their structural measures exhibit multifractal scaling due to large fluctuations in local node density.

Contribution

It analytically and numerically demonstrates that scale-free networks can have multifractal structures, providing a general expression for their mass exponents.

Findings

Mass exponents $ au(q)$ are nonlinear functions of $q$.

Structural measures obey multifractal scaling.

Multifractality arises from large fluctuations in node density.

Abstract

We demonstrate analytically and numerically the possibility that the fractal property of a scale-free network cannot be characterized by a unique fractal dimension and the network takes a multifractal structure. It is found that the mass exponents $\tau(q)$ for several deterministic, stochastic, and real-world fractal scale-free networks are nonlinear functions of $q$, which implies that structural measures of these networks obey the multifractal scaling. In addition, we give a general expression of $\tau(q)$ for some class of fractal scale-free networks by a mean-field approximation. The multifractal property of network structures is a consequence of large fluctuations of local node density in scale-free networks.