Percolation In Multilayer Networks With Connected And Dependent Topologies

With the development of technology,the interaction among the complex networks in real life more and more complex,so once one of the network fails,the failure will spread to the entire network system through the interaction among the various network.People constantly adjust the network model in order to more realistic simulation process of failure propagation in real network system,so as to find ways to alleviate even solve the dangers of cascading failure.In the past studies,the coupled complex system is often described by using the dependent network model,in which the failure of a node in one network will lead to the failure of the dependent node in another network.In addition,the multilayer network can also describe the cascading failure process.The multi-layer network model can describe the complex system formed by the same set of nodes with different links,and these links with different properties represent different interactions between nodes.In this paper,we study the percolation dynamics in double-layer networks with two topological layers composed of connectivity links and interdependency links,and propose a failure mechanism of weak dependence with adjustable dependency intensity,that the failure of a node may cause damage to its neighbors in the connectivity layer through the neighbor nodes in the dependency layer,so that its remaining connectivity links are deleted with a certain probability,but not directly delete the nodes.In this paper,we analyze the size of the giant component when the network system reaches the steady state,the phase transition point and the phase transition boundary of different types of phase transition from the point of view of simulation and theory.Finally,our model is verified in the double-layer ER random network system,double-layer network system with scale-free distribution and the double-layer network system composed of real power grid and Internet autonomous system.The results show that the network system will undergo a first order discontinuity,a second order continuity and a double phase transition with a proportion,p,initial retained nodes by adjusting the dependence intensity α and the density of the dependency links β.In particular,in a double-layer network system with scale-free distribution,it can be found that the system will undergo two stages of collapse: most nodes will suddenly fail firstly,and then the remaining nodes will gradually fail,which is consistent with the cascading failure phenomenon in the critical infrastructure system.Our work not only provides possible qualitative explanations for mass destruction in real-world infrastructure systems,but also provides illuminating implications for how systems are designed or controlled to achieve ideal resilience levels.