Yamamoto Jun, Yakubo Kousuke
Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan.
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom.
Phys Rev E. 2023 Aug;108(2-1):024302. doi: 10.1103/PhysRevE.108.024302.
The presence of large-scale real-world networks with various architectures has motivated active research towards a unified understanding of diverse topologies of networks. Such studies have revealed that many networks with scale-free and fractal properties exhibit the structural multifractality, some of which are actually bifractal. Bifractality is a particular case of the multifractal property, where only two local fractal dimensions d_{f}^{min} and d_{f}^{max}(>d_{f}^{min}) suffice to explain the structural inhomogeneity of a network. In this work we investigate analytically and numerically the multifractal property of a wide range of fractal scale-free networks (FSFNs) including deterministic hierarchical, stochastic hierarchical, nonhierarchical, and real-world FSFNs. Then we demonstrate how commonly FSFNs exhibit the bifractal property. The results show that all these networks possess the bifractal nature. We conjecture from our findings that any FSFN is bifractal. Furthermore, we find that in the thermodynamic limit the lower local fractal dimension d_{f}^{min} describes substructures around infinitely high-degree hub nodes and finite-degree nodes at finite distances from these hub nodes, whereas d_{f}^{max} characterizes local fractality around finite-degree nodes infinitely far from the infinite-degree hub nodes. Since the bifractal nature of FSFNs may strongly influence time-dependent phenomena on FSFNs, our results will be useful for understanding dynamics such as information diffusion and synchronization on FSFNs from a unified perspective.
具有各种架构的大规模真实世界网络的存在，激发了人们对网络不同拓扑结构进行统一理解的积极研究。此类研究表明，许多具有无标度和分形特性的网络呈现出结构多重分形性，其中一些实际上是双分形的。双分形性是多重分形特性的一种特殊情况，其中仅两个局部分形维数(d_{f}^{min})和(d_{f}^{max})（(>d_{f}^{min})）就足以解释网络的结构不均匀性。在这项工作中，我们通过解析和数值方法研究了广泛的分形无标度网络（FSFNs）的多重分形特性，包括确定性分层、随机分层、非分层和真实世界的FSFNs。然后我们展示了FSFNs表现出双分形特性的普遍程度。结果表明，所有这些网络都具有双分形性质。我们从研究结果推测，任何FSFN都是双分形的。此外，我们发现，在热力学极限下，较低的局部分形维数(d_{f}^{min})描述了无限高连接度枢纽节点周围以及距这些枢纽节点有限距离处的有限连接度节点周围的子结构，而(d_{f}^{max})则表征了距无限高连接度枢纽节点无限远处的有限连接度节点周围的局部分形性。由于FSFNs的双分形性质可能会强烈影响FSFNs上的时间相关现象，我们的结果将有助于从统一的角度理解FSFNs上的信息扩散和同步等动力学。
