The Study On Robustness Of Asymmetric Interdependent Networks With Multi-dependence Relation

During the past decades,there are increasing number of academic focusing on the research of complex network system,leading to the booming of overall researching context.The limitation of theory on single-layered networks seems apparent in the real networks,owing to the dependence among the real coupled systems,i.e,power communication system,the port network.For ensuring the function of coupled system,it is significant and necessary to strengthen the robustness of coupled system.In this article,we study the robustness of interdependent networks with multi-dependence 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 AIN model to aderess this problem.In this model the layer where nodes allow to have multi-dependence 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 intuitive picture for emergence of a tricritical point is provided.Moreover,we study the correlation between the intra-layer degree and inter-layer degree of the nodes and find that this correlation has prominent impact to the continuous phase transition but feeble effect on the discontinuous phase transition.Furthermore,we extend the two-layered AIN model to general multi-layered AIN,and the percolation behaviors and properties of phase transitions are elaborated.The research in this article can help to have further understanding on the robustness of asymmetric interdependent network(AIN),and provide a reference for the design of the real complex systems.