Study Of Robustness In Interdependent Networks With Clustering And The Multilayer Networks

As the rapid development of information technology,computer science and other high technology,not only a network internal structure is complex,the relationships between other systems are more closely,these complex relations can be described by complex networks theory.Recently,network systems with clustering and multilayer network have been given much attention due to their widely occurrence in real world.For network with clustering,one focus of these studies has been on robustness of single clustered networks and interdependent clustered networks under random attack or hub-targeted attack.However,infrastructure networks could suffer from a damage that is localized,i.e.,a group of neighboring nodes be attacked or fail at the same time.In this paper,we analytically and numerically study the robustness of single Erd?s-Rényi(ER)clustered network and interdependent clustering network under localized attack(LA).For LA case,the system shows a second order phase transition behavior for single a clustered network,but for dependent networks,the system undergoes a change of percolation phase transition from a first order to a second order transition when the coupling strength decreases below a critical value.Besides,increasing clustering coefficient and decreasing average degree can make network with clustering more vulnerable for LA,This conclusion is contrast to random attacks(RA),the system robustness is almost unaffected by changing clustering coefficient.We also solved analytically the Network of Network(NON)under localized attack,which includes star-like NON of ER networks with clustering and random regular network(RR)of ER network with clustering.Star-like NON is formed of n clustering networks,one of which network depends on other n1-networks that are unrelated between any two.The external construction of RR NON is a random regular network with average degree m,which means any one of the ER network with clustering depends on other m networks.The results state clearly decreasing clustering coefficient and increasing the value of n both enhance the robustness of star-like NON.Moreover,increasing clustering coefficient and the value of m can make RR NON more vulnerable.For multilayer network,there putforward two kinds of random attack strategies to study its robustness according to the reality situation.One way is to randomly remove a fraction of nodes from each layer of network respectively,keep the giant component of each layer and then make the rest part of the each layer merge into one network.Another approach is to firstly map the several layers of networks into a network,move a fraction of nodes from the new network by random attack,and then remain only the largest connected component.On the other hand,there are three kinds of network structure: single layer network,two layer networks and n layer networks,each layer network is a ER network or a Scale Free network.Finally,we found that the robustness of all kinds of networks are increasing with the increase of the layers of network and decreasing the connection density can lead to the network more fragile and vulnerable.