Fund Project:Project supported by the National Key R&D Program of China (Grant Nos. 2017YFB0803200, 2018YFB0804002), and the National NaturalScience Foundationof China (Grant Nos. 61802429, 61872382).
Received Date:24 December 2018
Accepted Date:31 January 2019
Available Online:23 March 2019
Published Online:05 April 2019
Abstract:Modern systems are always coupled. Previous studies indicate that coupled systems are more fragile than single systems. In a single system, when a fraction of 1-p nodes are removed, the percolation process is often of the second order. In a coupled system, due to the lack of resilience, the phase transition is always of the first order when removing a fraction of nodes. Most of previous studies on coupled systems focus on one-to-one dependency relation. This kind of relationship is called a no-feedback condition. Existing studies suppose that coupled systems are much more fragile without a no-feedback condition. That is to say, if a node depends on more than one node, the coupled system will breakdown even when a small fraction of nodes are removed from the coupled system. By observing the real world system, real nodes are often dependent on a dependency cluster, which consists of more than one other node. For example, in an industry chain, an electronic equipment factory may need several raw material factories to supply production components. Despite part of the raw material factories being bankrupt, the electronic equipment factory can carry out productionnormally because the remaining raw material factories still supply the necessary production components. But theoretical analysis shows that the robustness of such a coupled system is worse than that of one-to-one dependency system. Actually, the coupled system in real world does not usually disintegrate into pieces after some nodes have become invalid. To explain this phenomenon, we model a coupled system 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. Other way is to increase the size of dependency clusters, the more the nodes in the dependency cluster, the more the failure combinations are, which increases the survival probability of the node depending on cluster. Our model offers a useful strategy to enhance the robustness of coupled system and makes a good contribution to the study of interdependent networks with dependency clusters. Keywords:interdependent networks/ cascading failures/ percolation/ phase transition
对于任意$\tilde P(\tilde k)$分布求解逾渗方程较为复杂, 本文只考虑依赖群规模固定的情况. 图3是不同$\gamma $取值的RR-RR相依网络逾渗的仿真结果. 图 3 不同$\gamma $取值的RR-RR相依网络逾渗, 每个网络节点数$N = 200000$, 平均度$\left\langle k \right\rangle = 6$, $\tilde P(10) = 1$, 空心标记为仿真值, 实线为逾渗方程数值解 (a) 巨分量${\mu _\infty }$与$p$对应关系; (b) 级联失效迭代次数(number of iterations, NOI) Figure3. The results of the percolation of RR-RR interdependent networks for different $\gamma $, each network has 200000 nodes, average degree is 6, $\tilde P(10) = 1$. The symbols represent the simulation results, and the solid linesshow the corresponding analytical predictions. (a) The size ofthe giant component ${\mu _\infty }$versus $p$; (b) number of iterations.
随着$p$值发生变化, $D(f)$会与横轴相切, 相切时$p$的取值${p_{\rm{c}}}$为相变点. 图4为同分布ER-ER相依网络相变点数值解, 图5为不同$\gamma $取值的ER-ER相依网络逾渗的仿真结果. 图 4 不同$\gamma $值对应的ER-ER相依网络相变点, 网络平均度$\left\langle k \right\rangle = 6$, $\tilde P(10) = 1$ Figure4. Graphical solutions of ER-ER interdependent networks transition for different $\gamma $. The average degree is 6, $\tilde P(10) = 1$.
图 5 不同$\gamma $取值的ER-ER相依网络逾渗, 每个网络节点数$N = 200000$, 平均度$\left\langle k \right\rangle = 6$, $\tilde P(10) = 1$, 空心标记为仿真值, 实线为逾渗方程数值解 (a) 巨分量${\mu _\infty }$与$p$对应关系; (b) 级联失效迭代次数 Figure5. The results of the percolation of ER-ER interdependent networks for different $\gamma $, each network has 200000 nodes, average degree is 6, $\tilde P(10) = 1$. The symbols represent the simulation results, and the solid linesshow the corresponding analytical predictions. (a) The size ofthe giant component ${\mu _\infty }$ versus $p$; (b) number of iterations.
图6、图7分别对比了ER-ER相依网络不同${\tilde P_i}(\tilde k)$分布的逾渗结果和不同$\gamma $取值的相变点. 图 6 不同依赖度分布的ER-ER相依网络逾渗, $\tilde P(10) = 1$, $\tilde P(5) = 1$分别用实心和空心标记表示, 每个网络节点数$N = 200000$, 平均度$\left\langle k \right\rangle = 6$ (a) 巨分量${\mu _\infty }$与$p$对应关系; (b) 级联失效迭代次数 Figure6. The results of the percolation of ER-ER interdependent networks for different dependency degrees, each network has 200000 nodes, average degree is 6. The dependency degrees are $\tilde P(10) = 1$ (solid symbols) and $\tilde P(5) = 1$ (empty symbols). (a) The size ofthe giant component ${\mu _\infty }$ versus $p$; (b) number of iterations.
图 7 不同$\gamma $取值的ER-ER相依网络相变点, 空心标记为仿真结果, 实线是理论值 Figure7. The critical point ${p_{\rm{c}}}$ versus $\gamma $ of ER-ER interdependent networks, The symbols represent the simulation results, and the solid linesshow the corresponding analytical predictions.
将两个生成函数代入(7)式可求出巨分量. 图8给出了不同$\gamma $取值的SF-SF相依网络逾渗仿真结果. 图 8 不同$\gamma $取值的SF-SF相依网络逾渗, 每个网络节点数$N = 200000$, 平均度$\left\langle k \right\rangle = 6$, $\lambda = 2.7$, $\tilde P(10) = 1$, 空心标记为仿真值, 实线为逾渗方程数值解 (a) 巨分量${\mu _\infty }$与$p$对应关系; (b) 级联失效迭代次数 Figure8. The results of the percolation of SF-SF interdependent networks for different $\gamma $, each network has 200000 nodes, average degree is 6, $\lambda = 2.7$, $\tilde P(10) = 1$. The symbols represent the simulation results, and the solid linesshow the corresponding analytical predictions. (a) The size ofthe giant component ${\mu _\infty }$ versus $p$; (b) number of iterations.