Random walks on bifractal networks

It has recently been shown that networks possessing scale-free and fractal properties may exhibit a bifractal nature, in which local structures are described by two different fractal dimensions. In this study, we investigate random walks on such 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 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.