Research On Robustness Of Multilayer Interdependent Networks

In recent years,complex network theory has become an effective tool for modeling,analyzing and optimizing complex systems in the real world.With the continuous development of society,the interdependence between multi-layer networks in complex systems is increasing day by day.This interdependence improves the operating efficiency of the system,but also increases the vulnerability and risk of the system.This is mainly manifested in the fact that the internal node or edge failure of a single network will propagate and amplify with the interdependence between multilayer networks,eventually leading to the catastrophic collapse of the entire system.Therefore,the research on the robustness of multilayer interdependent networks has become a hot spot and frontier in the field of complex network structure security.At present,most of the research on the robustness of multilayer interdependent networks is based on the traditional percolation theory.This theory believes that only nodes and edges located in the huge connected components are functional,but ignores the phenomenon that some small functional components have a significant impact on the robustness of the network,resulting in the incompleteness of the current robustness analysis model.Based on this analysis,this paper focuses on the robustness of the reinforcement of the two basic units of nodes and dependent edges in multi-layer interdependent networks.This kind of reinforcement node or reinforcement dependent edge can support the function of the small component,even if the small component is separated from the huge connected component.In view of the fact that there are multiple interactions among nodes in multi-layer interdependent networks in real life,this paper focuses on the influence of functional small components under different reinforcement strategies on the robustness of the above three types of multi-layer interdependent networks from three levels:low-order,high-low-order and high-order networks,and further expands the application scope of percolation theory.The main research results are as follows:(1)In low-order network,the evaluation of the robustness of the existing multilayer weakly interdependent networks ignore the impact of reinforced finite components on the robustness of the networks.A novel generalized percolation model of multilayer interdependent networks with weak dependency links and reinforced nodes.Firstly,the theoretical calculations and simulations are performed on ER-ER and SF-SF networks,respectively,showing good agreement and verifying the accuracy of the theoretical analysis framework;Secondly,it is found that the system robustness can be significantly enhanced by increasing the weak dependency parameter a and the fraction ρ of reinforced nodes.Considering the expensive cost of deploying reinforced nodes,the model can achieve a balance between high robustness and cost efficiency by adjusting the weak dependence parameter;Furthermore,by adjusting the weak dependence parameter and the fraction of reinforced nodes,rich phase transition behaviours emerge:discontinuous,continuous,double,and mixed phase transitions,and theoretical methods are proposed to solve the corresponding phase transition points and shifts points of the phase transition types.More importantly,in symmetric multilayer networks,there exists a minimum weak dependence parameter αc*or a minimum reinforced fraction ρc*that can prevent catastrophic collapse of the system;In particular,in ER-ER networks,the upper bound αmax*of αc*is only related to the average degree of the network,and the upper bound ρmax*of ρc*is a constant value 0.1756;Finally,the validity of the proposed model is tested and further verified in empirical networks consisting of power grids and Internet autonomous systems.These findings provide insights for designing resilient multilayer interdependent networks.(2)In low-order networks,existing researches have shown that a certain fraction of reinforcing nodes can significantly improve the robustness of multilayer weakly interdependent networks and prevent the system from catastrophic collapse,ignoring the effects of reinforced dependent links and the numbers of layers of the network on the robustness of the system.A novel generalized percolation model of multilayer interdependent networks with weak dependency links and reinforced dependency links.Firstly,the theoretical calculations and simulations are performed on ER-ER and SF-SF networks,respectively,showing good agreement and verifying the accuracy of the theoretical analysis framework;Secondly,it is found that the system robustness can be significantly enhanced by increasing the weak dependency parameter a and the fraction p of reinforced dependency links.Furthermore,by adjusting the weak dependence parameter and the fraction of reinforced dependency links,rich phase transition behaviours emerge:discontinuous,continuous phase transitions,and theoretical methods are proposed to solve the corresponding phase transition points and shifts points of the phase transition types.More importantly,only a small proportion of reinforced dependency links in a symmetric two-layer interdependent network can prevent catastrophic collapse of the system.In particular,in the ER-ER network,no matter how the network average degree changes,the upper bound ρmax*of ρc*is always 0.088068,which is significantly smaller than the upper bound 0.1756 of the reinforced nodes;Finally,the model is extended to multilayer strongly interdependent networks.It is found that as the number of network layers increases,the system becomes more vulnerable,and the upper bound Pmax*of the minimum reinforcement ratio ρc*to prevent catastrophic collapse also increases.The exact analytical solution of ρmax*is derived in the multi-layer ER network,which is only related to the number of network layers.In addition,in this multilayer network,the reinforcement efficiency of the reinforcement dependency edge strategy is significantly better than that of the reinforcement node strategy.These findings provide theoretical guidance for designing more realistic and resilient multilayer interdependent networks.(3)In higher-low order networks,previous group percolation models in multilayer interdependent networks are always discontinuous phase transition.When this system is attacked,the system is difficult to resist catastrophic collapse.A generalized group percolation model in multilayer interdependent networks with reinforcement network layers is investigated numerically and analytically.Some backup devices that are equipped for a fraction of reinforced nodes constitute the reinforcement network layer.For each group,we assume that if all the nodes in a group fail on one network,a node on another network that depends on that group will fail.Firstly,the theoretical calculations and simulations are performed on ER-ER and SF-SF networks,respectively,showing good agreement and verifying the accuracy of the theoretical analysis framework;Secondly,it is found that the system robustness can be significantly enhanced by increasing group size S and the fraction p of reinforced nodes.Considering the expensive cost of deploying reinforced nodes,the model can achieve a balance between high robustness and cost efficiency by adjusting the group size;In addition,a variety of phase transition behaviours emerge:discontinuous,continuous,and hybrid phase transitions,and theoretical methods are proposed for solving the phase transition points of each type and the turning points of the phase transition types;Furthermore,a universal solution for the minimum reinforced fraction ρmin(or minimum group size Smim)and the corresponding phase transition points to prevent catastrophic collapse of the system is accurately solved,which is universal to RR-RR,ER-ER,SF-SF networks;Finally,when ρ>ρmin(or S>Smin),the system undergoes a continuous phase transition,and the general solution of the continuous phase transition is accurately solved,which is also universal for RR-RR,ER-ER,SF-SF networks.These findings provide a broad perspective for designing more resilient interdependent infrastructure networks.(4)In higher-low order networks,previous studies of group percolation models in interdependent networks with reinforced nodes have rarely addressed the effects of the degree of reinforced nodes and the heterogeneity of group size distribution.A percolation model in multilayer interdependent networks with reinforced crucial nodes and dependency groups is investigated numerically and analytically.Firstly,our theoretical results can well agree with numerical simulations.Secondly,we find that rich percolation transitions can be classifieded into three types:discontinuous,continuous,and hybrid phase transitions,which depend on the density of reinforced crucial nodes,the group size,and the heterogeneity of group size distribution.Importantly,our proposed crucial reinforced method has higher reinforcement efficiency than the random reinforced method.More significantly,we develop a general theoretical framework to calculate the percolation transition points and the shift point of percolation types.Finally,simulation results show that the robustness of interdependent networks can be improved by increasing the density of reinforced crucial nodes,the group size,and the heterogeneity of group size distribution.These findings might develop a new perspective for designing more resilient interdependent infrastructure networks.(5)In higher-order network,most of the existing percolation models of multilayer interdependent hypergraphs ignores the influence of the small component with reinforced nodes and the dependency and fault-tolerance mechanisms of hyperedges on internal nodes.A higher-order percolation model in partially interdependent hypergraphs with reinforced nodes is investigated numerically and analytically.Firstly,the theoretical calculations and simulations are performed on random and scale-free hypergraphs,respectively,showing good agreement and verifying the accuracy of the theoretical analysis framework;Secondly,the robustness of the network can be significantly improved by increasing the reinforced density,the proportion of autonomous nodes,and the tolerance of failed nodes in the hyperedges.Finally,a theoretical framework is proposed to calculate discontinuities,continuous phase transition points,and turning points of phase transition types in hypergraphs.These findings contribute to a more comprehensive understanding of the robustness of higher-order networks.