Shao Shuai, Huang Xuqing, Stanley H Eugene, Havlin Shlomo
Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA and Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Mar;89(3):032812. doi: 10.1103/PhysRevE.89.032812. Epub 2014 Mar 25.
Clustering, or transitivity, a behavior observed in real-world networks, affects network structure and function. This property has been studied extensively, but most of this research has been limited to clustering in single networks. The effect of clustering on the robustness of coupled networks, on the other hand, has received much less attention. Only the case of a pair of fully coupled networks with clustering has recently received study. Here we generalize the study of clustering of a fully coupled pair of networks and apply it to a partially interdependent network of networks with clustering within the network components. We show, both analytically and numerically, how clustering within networks affects the percolation properties of interdependent networks, including the percolation threshold, the size of the giant component, and the critical coupling point at which the first-order phase transition changes to a second-order phase transition as the coupling between the networks is reduced. We study two types of clustering, one proposed by Newman [Phys. Rev. Lett. 103, 058701 (2009)] in which the average degree is kept constant while the clustering is changed, and the other by Hackett et al. [Phys. Rev. E 83, 056107 (2011)] in which the degree distribution is kept constant. The first type of clustering is studied both analytically and numerically, and the second is studied numerically.
聚类，即传递性，是现实世界网络中观察到的一种行为，它会影响网络结构和功能。这一特性已得到广泛研究，但大多数此类研究仅限于单个网络中的聚类。另一方面，聚类对耦合网络鲁棒性的影响受到的关注要少得多。直到最近才有关于一对具有聚类的完全耦合网络情况的研究。在这里，我们将一对完全耦合网络的聚类研究进行推广，并将其应用于网络组件内部具有聚类的部分相互依存的网络。我们通过解析和数值两种方法表明，网络内部的聚类如何影响相互依存网络的渗流特性，包括渗流阈值、巨分量的大小，以及随着网络间耦合的降低，一阶相变转变为二阶相变时的临界耦合点。我们研究了两种类型的聚类，一种是由纽曼[《物理评论快报》103, 058701 (2009)]提出的，其中平均度保持不变而聚类发生变化；另一种是由哈克特等人[《物理评论E》83, 056107 (2011)]提出的，其中度分布保持不变。第一种类型的聚类通过解析和数值两种方法进行研究，第二种则通过数值方法进行研究。
