Multifractality of complex networks

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.