| Complex network is an important tool to describe and analyze the structure and dynamics of complex systems.In recent years,remarkable achievements have been made in the area of complex networks,in particular,the study on network attack,network recovery and structural controllability are very important hot research focuses.Current theoretical and practical research works on network attack are mostly focused on single and isolated networks,considering that many real-world complex systems are usually interdependent on each other,we extend the study on network attack from single networks to interdependent networks with emphasis on the influence of degree heterogeneity on the structure vulnerability of interdependent systems.Implementing effective repair strategies after attacks is one of the most important methods to assure the safety and robustness of complex systems,thus we proposed new network repair strategies under both random and intentional attack modes and analyzed their effectiveness in this paper.Network structural controllability refers to the ability to drive the network from any initial state to the desired final state within a limited time,and controllability optimization aims at reducing the input of control signals as much as possible.In our study,based on the optimization of networks to improve the attack robustness,the structural controllability of optimized networks is analyzed.The main contents and innovative results of the paper are summarized as follows:1)Study on the influence of degree heterogeneity on the structure vulnerability of interdependent Local-world networkFirstly,an interdependent system model composed of two network components is constructed.By adjusting an important parameter,the extent of degree heterogeneity of each network can be controlled.Then,numerical simulations are implemented to demonstrate the impact of degree heterogeneity.The research results verify that the vulnerability of both single and interdependent networks can be affected by degree heterogeneity,that is,degree heterogeneity can significantly increase the vulnerability of these networks.Additionally,the existence of interdependent links between two networks makes both networks much more vulnerable against targeted attacks.In particular,when coupling preference is taken into account,it is found that compared with disassortative coupling and random coupling,assortative coupling can lead to a more fragile interdependent system.2)Analysis of the effectiveness of network repair strategy under random and intentional attack modeFirstly,two edge attack modes are proposed.Random attack mode refers to random removal of edges,and intentional attack mode refers to removing the edges belonging to a node whose degree is above the average node degree.Secondly,by setting up the edge repair model,two recovery strategies are considered:random repair and targeted repair.Random repair is randomly selecting and reconnecting the redundant edges,and targeted repair prefers to repairing the edges of a node whose degree is greater than the average degree.Finally,taking random network,scale-free network and real network as research objects,the applicability of two different repair strategies under different attack modes is studied by numerical simulation method,and the most effective restoration strategy under various attack modes is given.3)Analysis of structural controllability of networks after optimizationFirstly,scale-free networks are studied on the relationships between the robustness measure R,the controllability parameter n_D and two important topological parameters-the scale index?and the average nodal degree?k?.Then,based on numerical simulation method,the changes of structure and controllability of scale-free networks and some real-world systems real network system are analyzed.The research results verified that after optimization networks'attack robustness and controllability are greatly improved,and optimized networks exhibit typical“onion-like”topologies. |
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