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.

中文翻译：

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具有多依赖关系的非对称相互依赖网络

在本文中，我们研究具有多个依赖关系（MD）的相互依赖网络的鲁棒性，该关系定义一个节点与另一层上的多个节点相互依赖，如果这些依赖节点中的任何一个失败，则该节点也会失败。我们提出了两层不对称相互依赖网络（AIN）模型来解决此问题，其中不对称特征是一层中的节点可能依赖于另一层具有MD关系的节点，而另一层中的节点依赖于MD关系。完全取决于这一层中的一个节点。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），而另一层仅表现出不连续的相变。提出了一种基于消息传递方法的启发式理论，以理解相互依赖的网络的结构特征，并为三临界点的出现提供了直观的图像。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具有显着影响，但对不连续相变具有微弱的影响。此外，我们将两层AIN模型扩展到一般的多层AIN，并详细阐述了相关相变的渗透行为和性质。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具有显着影响，但对不连续相变具有微弱的影响。此外，我们将两层AIN模型扩展到一般的多层AIN，并详细阐述了相关相变的渗透行为和性质。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具有显着影响，但对不连续相变具有微弱的影响。此外，我们将两层AIN模型扩展到一般的多层AIN，并详细阐述了相关相变的渗透行为和性质。