  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.
