  The presence of large-scale real-world networks with various architectures
has motivated an active research towards a unified understanding of diverse
topologies of networks. Such studies have revealed that many networks with the
scale-free and fractal properties exhibit the structural multifractality, some
of which are actually bifractal. Bifractality is a particular case of the
multifractal property, where only two local fractal dimensions
$d_{\text{f}}^{\text{min}}$ and $d_{\text{f}}^{\text{max}}
(>d_{\text{f}}^{\text{min}})$ suffice to explain the structural inhomogeneity
of a network. In this work, we investigate analytically and numerically the
multifractal property of a wide range of fractal scale-free networks (FSFNs)
including deterministic hierarchical, stochastic hierarchical,
non-hierarchical, and real-world FSFNs. Then we demonstrate how commonly FSFNs
exhibit the bifractal property. The results show that all these networks
possess the bifractal nature. We conjecture from our findings that any FSFN is
bifractal. Furthermore, we find that in the thermodynamic limit the lower local
fractal dimension $d_{\text{f}}^{\text{min}}$ describes substructures around
infinitely high-degree hub nodes and finite-degree nodes at finite distances
from these hub nodes, whereas $d_{\text{f}}^{\text{max}}$ characterizes local
fractality around finite-degree nodes infinitely far from the infinite-degree
hub nodes. Since the bifractal nature of FSFNs may strongly influence
time-dependent phenomena on FSFNs, our results will be useful for understanding
dynamics such as information diffusion and synchronization on FSFNs from a
unified perspective.
