Yakubo Kousuke, Shimojo Gentaro, Yamamoto Jun
Hokkaido University, Department of Applied Physics, Sapporo 060-8628, Japan.
Central European University, Department of Network and Data Science, A-1100 Wien, Austria.
Phys Rev E. 2024 Dec;110(6-1):064318. doi: 10.1103/PhysRevE.110.064318.
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_{w} and the spectral dimension d_{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_{s}^{min} and d_{s}^{max}(>d_{s}^{min}), depending on the starting node. The dimension d_{s}^{min} characterizes the return probability of a random walker starting from an infinite-degree hub node in the thermodynamic limit, while d_{s}^{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_{s}. The existence of two local spectral dimensions is a direct consequence of the bifractality of the FSFN. Furthermore, analytical expressions of d_{w}, d_{s}^{min}, and d_{s}^{max} are presented for FSFNs formed by the generator model and the giant components of critical scale-free random graphs, and are numerically confirmed.
最近有研究表明，具有无标度和分形特性的网络可能呈现双分形性质，即局部结构由两个不同的分形维数来描述。在本研究中，我们通过考察游走维数(d_{w})和谱维数(d_{s})，研究此类分形无标度网络（FSFNs）上的随机游走，以了解双分形性如何影响其动力学性质。我们发现游走维数不受FSFN局部分形性差异的影响，且无论随机游走的起始节点如何，游走维数都保持不变，而谱维数则根据起始节点取两个值，(d_{s}^{min})和(d_{s}^{max})（(>d_{s}^{min})）。维数(d_{s}^{min})表征了在热力学极限下从无限度中心节点出发的随机游走者的返回概率，而(d_{s}^{max})描述了从离中心节点无限远的有限度非中心节点出发的随机游走者的返回概率，且等于全局谱维数(D_{s})。两个局部谱维数的存在是FSFN双分形性的直接结果。此外，我们给出了由生成器模型和临界无标度随机图的巨分支形成的FSFN的(d_{w})、(d_{s}^{min})和(d_{s}^{max})的解析表达式，并通过数值方法进行了验证。
