Origin of the hub spectral dimension in scale-free networks

The return-to-origin probability and the first passage time distribution are essential quantities for understanding transport phenomena in diverse systems. The behaviors of these quantities typically depend on the spectral dimension $d_s$. However, it was recently revealed that in scale-free networks these quantities show a crossover between two power-law regimes characterized by $ d_s $ and the so-called hub spectral dimension $d_s^{\textrm{(hub)}}$ due to the heterogeneity of connectivities of each node. To understand the origin of $d_s^{\textrm{(hub)}}$ from a theoretical perspective, we study a random walk problem on hierarchical scale-free networks by using the renormalization group (RG) approach. Under the RG transformation, not only the system size but also the degree of each node changes due to the scale-free nature of the degree distribution. We show that the anomalous behavior of random walks involving the hub spectral dimension $d_s^{\textrm{(hub)}}$ is induced by the conservation of the power-law degree distribution under the RG transformation.