Bifractal Networks
A network is fractal (with respect to shortest-path lengths) if the minimum number of subgraphs (boxes) of diameter l_{\textup{B}} required to cover the network scales as \begin{equation} N_{\textup{B}}(l_{\textup{B}}) \sim l_{\textup{B}}^{-D_{\textup{f}}} \end{equation} where D_{\textup{f}} is the fractal dimension of the network (Song, Havlin, and Makse 2005). In fractal networks, the average path lengths also scales as the power of the network size, \begin{equation} \langle l \rangle \sim N^{1/D_{\textup{f}}}. \end{equation} While many empirical networks exhibit the small-world property, where the average path lengths scales logarithmically with the number of nodes, the fractal property can still hold at length scales shorter than the network’s characteristic size. Detecting this scaling through shortest paths, however, is often challenging as most real networks do not have a sufficiently large diameter.
Fractal networks are of particular interest because their structure gives rise to distinctive features, such as long-range degree correlations manifested in hub repulsion (Rozenfeld et al. 2008).
When scale-free and fractal properties coexist, a broad class of networks exhibits bifractality, characterized by two distinct local fractal dimensions, d_{\textup{f}}^{\min} and d_{\textup{f}}^{\max} that depend on position within the network (Yamamoto and Yakubo 2023). In particular, consider a fractal scale-free network in which, under renormalization, the number of nodes \nu_{\textup{B}} in a supernode (subgraph) of diameter l_{\textup{B}} is proportional to the degree k_{\textup{B}} of that supernode (i.e., the number of adjacent subgraphs): \begin{equation} \nu_{\textup{B}} \propto k_{\textup{B}}. \end{equation} Under this condition, the network exhibits two local fractal dimensions (equivalently, Hölder exponents): \begin{equation} d_{\textup{f}}^{\max} = D_{\textup{f}}, \quad d_{\textup{f}}^{\min} = D_{\textup{f}} \left(\frac{\gamma-2}{\gamma-1}\right), \end{equation} where \gamma is the exponent of the degree distribution and D_{\textup{f}} is the global fractal dimension of the network.
Extensive analytical and numerical calculations for both deterministic and stochastic models suggest that this proportionality condition holds broadly across fractal scale-free networks. We therefore conjecture that bifractality is a general property of fractal scale-free networks.
For details, see (Yamamoto and Yakubo 2023).
Building on the bifractality conjecture, we studied the walk and spectral dimensions of random walks on fractal scale-free networks. We found that the walk dimension d_{\textup{w}}, which characterizes how the mean topological displacement of random walkers scales, remains constant across the network and does not depend on position.
In contrast, the spectral dimension, which governs the scaling of the return probability of a random walker, splits into two distinct values, d_{\textup{s}}^{\min} and d_{\textup{s}}^{\max}. This positional dependence arises directly from the network’s bifractal structure.
For details, see (Yakubo, Shimojo, and Yamamoto 2024).