Zhang Hang, Zhou Jie, Zou Yong, Tang Ming, Xiao Gaoxi, Stanley H Eugene
School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China.
Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts, 02215, USA.
Phys Rev E. 2020 Feb;101(2-1):022314. doi: 10.1103/PhysRevE.101.022314.
In this paper, we study the robustness of interdependent networks with multiple-dependence (MD) relation which is defined that a node is interdependent on several nodes on another layer, and this node will fail if any of these dependent nodes are failed. We propose a two-layered asymmetric interdependent network (AIN) model to address this problem, where the asymmetric feature is that nodes in one layer may be dependent on more than one node in the other layer with MD relation, while nodes in the other layer are dependent on exactly one node in this layer. We show that in this model the layer where nodes are allowed to have MD relation exhibits different types of phase transitions (discontinuous and hybrid), while the other layer only presents discontinuous phase transition. A heuristic theory based on message-passing approach is developed to understand the structural feature of interdependent networks and an intuitive picture for the emergence of a tricritical point is provided. Moreover, we study the correlation between the intralayer degree and interlayer degree of the nodes and find that this correlation has prominent impact to the continuous phase transition but has feeble effect on the discontinuous phase transition. Furthermore, we extend the two-layered AIN model to general multilayered AIN, and the percolation behaviors and properties of relevant phase transitions are elaborated.
在本文中，我们研究具有多重依赖（MD）关系的相互依存网络的鲁棒性，该关系定义为一个节点依赖于另一层上的几个节点，并且如果这些依赖节点中的任何一个发生故障，该节点也将发生故障。我们提出了一种两层非对称相互依存网络（AIN）模型来解决此问题，其中非对称特征在于一层中的节点可能通过MD关系依赖于另一层中的多个节点，而另一层中的节点仅依赖于该层中的一个节点。我们表明，在该模型中，允许节点具有MD关系的层表现出不同类型的相变（不连续和混合），而另一层仅呈现不连续相变。基于消息传递方法发展了一种启发式理论，以理解相互依存网络的结构特征，并提供了一个关于三临界点出现的直观图景。此外，我们研究了节点的层内度和层间度之间的相关性，发现这种相关性对连续相变有显著影响，但对不连续相变影响微弱。此外，我们将两层AIN模型扩展到一般的多层AIN，并阐述了相关相变的渗流行为和性质。
