  It has recently been shown that networks possessing the scale-free and
fractal properties may exhibit the bifractal nature, in which local structures
are described by two different fractal dimensions. In this study, we
investigate random walks on fractal scale-free networks (FSFNs) by examining
the walk dimension $d_{\text{w}}$ and the spectral dimension $d_{\text{s}}$, to
understand how the bifractality of FSFNs affects their dynamical properties.
The walk dimension is found to be unaffected by the difference in local
fractality of an FSFN and remains constant regardless of the starting node of a
random walk, whereas the spectral dimension takes two values,
$d_{\text{s}}^{\text{min}}$ and $d_{\text{s}}^{\text{max}}(>
d_{\text{s}}^{\text{min}})$, depending on the starting node. The dimension
$d_{\text{s}}^{\text{min}}$ characterizes the return probability of a random
walker starting from an infinite-degree hub node in the thermodynamic limit,
while $d_{\text{s}}^{\text{max}}$ describes that of a random walker starting
from a finite-degree non-hub node infinitely distant from hub nodes and is
equal to the global spectral dimension $D_{\text{s}}$. The existence of two
local spectral dimensions is a direct consequence of the bifractality of the
FSFN. Furthermore, analytical expressions of $d_{\text{w}}$,
$d_{\text{s}}^{\text{min}}$, and $d_{\text{s}}^{\text{max}}$ are presented for
FSFNs formed by the generator model and the giant components of critical
scale-free random graphs, and are numerically confirmed.
