Yakubo Kousuke, Fujiki Yuka
Department of Applied Physics, Hokkaido University, Sapporo, Japan.
Advanced Institute for Materials Research, Tohoku University, Sendai, Japan.
PLoS One. 2022 Mar 21;17(3):e0264589. doi: 10.1371/journal.pone.0264589. eCollection 2022.
We propose a general model of unweighted and undirected networks having the scale-free property and fractal nature. Unlike the existing models of fractal scale-free networks (FSFNs), the present model can systematically and widely change the network structure. In this model, an FSFN is iteratively formed by replacing each edge in the previous generation network with a small graph called a generator. The choice of generators enables us to control the scale-free property, fractality, and other structural properties of hierarchical FSFNs. We calculate theoretically various characteristic quantities of networks, such as the exponent of the power-law degree distribution, fractal dimension, average clustering coefficient, global clustering coefficient, and joint probability describing the nearest-neighbor degree correlation. As an example of analyses of phenomena occurring on FSFNs, we also present the critical point and critical exponents of the bond-percolation transition on infinite FSFNs, which is related to the robustness of networks against edge removal. By comparing the percolation critical points of FSFNs whose structural properties are the same as each other except for the clustering nature, we clarify the effect of the clustering on the robustness of FSFNs. As demonstrated by this example, the present model makes it possible to elucidate how a specific structural property influences a phenomenon occurring on FSFNs by varying systematically the structures of FSFNs. Finally, we extend our model for deterministic FSFNs to a model of non-deterministic ones by introducing asymmetric generators and reexamine all characteristic quantities and the percolation problem for such non-deterministic FSFNs.
我们提出了一种具有无标度特性和分形性质的无加权无向网络的通用模型。与现有的分形无标度网络（FSFN）模型不同，本模型能够系统且广泛地改变网络结构。在该模型中，一个FSFN是通过用一个称为生成器的小图替换上一代网络中的每条边而迭代形成的。生成器的选择使我们能够控制分层FSFN的无标度特性、分形性和其他结构特性。我们从理论上计算了网络的各种特征量，如幂律度分布的指数、分形维数、平均聚类系数、全局聚类系数以及描述最近邻度相关性的联合概率。作为对FSFN上出现的现象进行分析的一个例子，我们还给出了无限FSFN上键渗流转变的临界点和临界指数，这与网络抵抗边移除的鲁棒性有关。通过比较结构性质除聚类性质外彼此相同的FSFN的渗流临界点，我们阐明了聚类对FSFN鲁棒性的影响。如本例子所示，本模型使得通过系统改变FSFN的结构来阐明特定结构性质如何影响FSFN上出现的现象成为可能。最后，我们通过引入不对称生成器将确定性FSFN的模型扩展为非确定性模型，并重新审视此类非确定性FSFN的所有特征量和渗流问题。
