Robustness Of Multilayer Interdependent Networks Under Cascading Failures

Recently,complex theory has become effective tools for modeling and analyzing complex systems in the real world.Due to the rapid development of social economy and science technology,the coupling strength and complexity between various infrastructures in real life,such as traffic networks,electric power information networks and internet networks,are increasing.Therefore,it is more difficult to study cascade failure behavior of infrastructure.Scholars modeled the coupling relationship in network as the interdependency links between nodes/edges and nodes/edges in the network.It is found that the dependency between networks increases the risk of cascading failure occasionally,and the complexity of networks is getting stronger and stronger.Studying the robustness of this type of network has certain practical significance.The percolation threshold based on the percolation theory and the phase transition behavior of the giant component is usually used to study the robustness of the interdependent networks.Based on the complex network,this paper adopts the self-consistent probability theoretical framework to analyze the multi-layer network and its robustness.The main research contents of this paper are as follows:1)The cascade failure and percolation theory analysis of complex network are introduced in detail.Firstly,the commonly used parameters describing the structural characteristics of complex networks are introduced,this paper studies the complex network model and its generation method.Then the network percolation model is modeled and solved based on percolation theory.It is mainly obtained by the method of generating function and self-consistent probability equations.According to the different coupling modes and coupling strength,the common interdependent network models are classified,and the cascading failure process of interdependent networks under attack is analyzed based on the percolation theory.2)The coupling relationship in infrastructure network as the interdependency links between nodes and nodes in the network.On the basis of previous studies,the interdependence relation of partial node-coupling between two networks was deeply considered,and the parameter q represents the coupling strength of the two networks.Under the given coupling strength,the phase transformation behavior and percolation threshold of the interdependent network under random failure were studied.We develop the theoretical analysis framework of this system based on the self-consistent probability method,which can be used to analyze the phase transition behavior of any PEIN as long as the degree distribution function of the network is known.We use this theoretical ananlysis framework to carry out theoretical ananlysis and simulation ananlysis on the Erd?s–Rényi（ER）and Random Regular（RR）networks to prove the correctness of the theoretical framework.We find,analytically and via simulations,that as the coupling strengthincreases,the phase transition behaviors also change from second-order phase transition to first-order phase transition.As coupling strengthdecreases from 1 to 0,the percolation threshold of partially node-coupled interdependent networks decreases,which means that the smaller the coupling strength in partially node-coupled interdependent networks,the better the robustness of the interdependent network.3)This paper presents a partial edge-coupled interdependence model.We also use the self-consistent probability method to develop the theoretical framework for this model to analyze the phase transition behavior and percolation threshold of partially edge-coupled interdependent networks（PEIN）under random failures.Theoretical analysis and simulation analysis are carried out on partially edge-coupled interdependent networks composed of the Random Regular（RR）network and the Erd?s–Rényi（ER）network,coupling strengthis referred as to the degree of interdependence of the two networks.We provide a detailed theoretical analysis framework to study the phase transition behavior,percolation threshold,critical phase transition point,and the giant component of the network.As the coupling strengthdecreases from 1 to 0,partially edge-coupled interdependent networks show a phase transition change from a first-order phase transition to a second-order phase transition under random failures.When=0,the system degenerates into two single networks;when=1,the two networks are completely edge-coupled interdependent networks,which shows a firstorder phase transition.At the critical coupling strength₍（8）,the system transforms from the first-order phase transition to the second-order phase transition.This phenomenon is the same as the corresponding node-coupled interdependent networks,and the critical value of coupling strength₍（8） is the same.However,for any value of,the percolation threshold of the partially edge-coupled interdependent network is always smaller than partially node-coupled interdependent networks.