Research On The Robustness Of Interdependent Complex Networks

Since the beginning of the new century,network science has attracted wide attention in the academia.Many physical phenomena of complex systems in the real world can be properly explained by network science.With the further research on complex networks,it is found that some real-world networks are interdependent,this kind of relationship decreases the robustness.Therefore,more and more researchers begin to pay attention to the robustness of interdependent complex networks.According to the characteristics of real-world networks,many models of interdependent networks have been proposed,which reflect the abilities of real-world networks to resist random or deliberate attacks.However,some of the phenomena of real-world interdependent networks have not been well explained,including:(1)A real node may depend on a dependency cluster,which consists of several nodes in another network,even some of nodes in the cluster fail,the original node can still work;(2)In interdependent networks with weak inter-layer links,the failure probability of the connection links of each node can be different due to heterogeneity;(3)The failure of an important node definitely leads to the failure of a minor node,but the failure of a minor node may not lead to the failure of an important node;(4)It is usually assumed that a node fails when it loses all its connections to the giant component,however,it may have alternative methods to reconnect with the giant component.For the above phenomena,we carry out the studies the robustness of interdependent complex networks,and analyze the influence of links and nodes with different properties on the robustness of networks.The main research contents and innovations are as follows:1.For the existence of conditional dependency clusters,we modeled interdependent systems as interdependent networks and study,both analytically and numerically,the percolation in interdependent networks with conditional dependency clusters.A node in our model survives until the number of failed nodes in its dependency cluster is greater than a threshold.Our exact solutions of giant component size are in good agreement with the simulation results.Though our model does not have second order phase transition,we still find ways to improve the robustness of interdependent networks.One way is to increase the dependency cluster failure threshold.A higher threshold means that more nodes in the dependency cluster can be removed without breaking down the node depending on the cluster.Another way is to increase the size of dependency clusters,the more nodes in the dependency cluster,the more failure combinations,which increases the survival probability of the node depending on.Our model offers a useful strategy to enhance the robustness of coupled systems and makes a good contribution to the study of interdependent networks with dependency clusters.2.For the phenomenon of heterogeneity in weak interdependent networks,we develop a framework of interdependent networks with heterogeneous weak inter-layer links.Each connectivity links of a node with weak inter-layer dependency is removed with a certain probability after the failure of its counterpart node.The probabilities for different nodes are various because of heterogeneity.At the end,nodes can survive as long as one of the remaining connectivity links reaches the giant component.We present an analytical solution for solving the giant component size and analyzing the crossing point of the phase transition of arbitrary interdependent random networks.For homogeneous symmetric Erd?s-Rényi networks,we solve the continuous transition point and the triple point.The simulation results are in good agreement with our exact solutions.Furthermore,we introduce two kinds of weak inter-layer dependency probability distributions to analysis the impact of heterogeneous weak inter-layer links on the robustness of interdependent networks.The results of both distributions show that with the increasing of heterogeneity,the transition point decreases and the networks are getting more robust.For the first simple probability distribution,we also find the percolation transition changes from discontinuous one to continuous one by improving the heterogeneity.For the second Gaussian probability distribution,a higher variance makes the inter-dependent networks more difficult to collapses.Our work explains the robustness of real-world interdependent networks from a new perspective,and offers a useful strategy to enhance the robustness by increasing the heterogeneity of weak inter-layer links of interdependent networks.3.For the more important nodes are more likely to survive,we propose a model of interdependent networks with asymmetric inter-layer dependency links.The importance of a node is measured by its degree.If two nodes are interdependent,the node with higher a degree is the important one,the other is the minor one.The asymmetric interdependent failure includes the following two conditions: the failure of the important node causes the failure of the minor node directly;if the minor node fails,the failure probability of the important node is related to the degree ratio of the minor node to the important node.We theoretically analyze the percolation process of our model and offer a method to solve the phase transition points for networks with arbitrary degree distributions.We find the exact solutions of critical points for randomly interdependent networks with the same distribution.The simulation results are in good agreement with our theoretical values.Furthermore,we find that the more dispersed the distribution is,the more robust the interdependent networks are.4.For the failed nodes may have other ways to reconnect with the giant component,we propose a model of interdependent networks with multi-model addressing was proposed.In the model,each node can be a multi-model addressing node with some probability according to its degree.Even the multi-model addressing node is not in the giant component,it can find other ways to reestablish the connection to the giant component.What's more,the nodes,which are in the same small component with the multi-model addressing node,can survive simultaneously.Secondly,based on the percolation theory,we analyze the giant component equation for networks with arbitrary degree distributions and find the formula that satisfies transition point.For the interdependent networks with same degree distributions,we calculate the discontinuous transition point and the triple point.Finally,the simulation results are in agreement with the theoretical analysis.With the increasing of the probability of multi-model addressing,the robustness of interdependent networks with same degree distributions improves.This work is supported by the National Natural Science Foundation of China(Grant No.61872382)and the Guangdong Provincial Research and Development Program in Key Areas of China(Grant No.2018B010113001).The research results will not only provide theoretical supports for the robust control of information communication networks and the construction of new network systems,but also offer some useful strategies to enhance the robustness of interdependent networks.