  The spectral dimension has been widely used to understand transport
properties on regular and fractal lattices. Nevertheless, it has been little
studied for complex networks such as scale-free and small world networks. Here
we study the spectral dimension and the return-to-origin probability of random
walks on hierarchical scale-free networks, which can be either fractals or
non-fractals depending on the weight of shortcuts. Applying the renormalization
group (RG) approach to the Gaussian model, we obtain the spectral dimension
exactly. While the spectral dimension varies between $1$ and $2$ for the
fractal case, it remains at $2$, independent of the variation of network
structure for the non-fractal case. The crossover behavior between the two
cases is studied through the RG flow analysis. The analytic results are
confirmed by simulation results and their implications for the architecture of
complex systems are discussed.
