Robustness of complex networks with the local protection strategy against cascading failure

1. Robustness of complex networks with the local protection strategy against cascading failure 2013 Safety Science Presenter : Yen Fen Kao Advisor : Yeong Sung Lin

2. Agenda • Introduction • The mitigation strategy • Simulation analysis of mitigation strategy • Theoretical analysis of cascading model • Conclusions

3. Introduction • Cascading failures induced by targeted attacks and random failures can occur in many natural and man- made systems, and frequently trigger many catastrophic events.

4. Introduction • Earlier studies on cascading failures pay close attention to analyzing the evolving characteristics of cascading failures and mainly focus on the cascading modeling, the cascade control and defense, the attack strategies, the cascading phenomena in the diverse networks, and so on. • Later, many researchers find that catastrophic events induced by cascading failures can also occur in coupled networks

5. Introduction • Motivated by that fact, Buldyrev et al. (2010) developed a framework for understanding the robustness of interacting networks against cascading failures and present exact analytical solutions for the critical fraction of nodes. • After that, cascading failures on coupled networks have started to be studied actively and a number of important aspects of cascading failures in coupled networks have been discussed.

6. Introduction • Most works on cascading failures from the single network to coupled networks have focused only on the modeling of cascading failure, the cascade control and defense strategies, or a fundamental property of coupled networks without considering the load. • In many infrastructure networks, when a node overloads, its neighboring nodes can provide some protection resources to avoid its failure.

7. Introduction • This paper propose a new mitigation method. • Without changing the total protection capacities of the whole network, thispapernumerically investigate its effectiveness on improving the robustness of BA scale-free networks against cascading failures and find that the simple local protection method can dramatically optimize the resilience of BA networks.

8. Introduction • Adopting the mitigation strategy, this paper numerically discuss how BA networks can reach the strongest robust level against cascading failures and find, compared with previous results, the optimal value of the parameter decreases. We verify this result by the theoretical analysis.

9. Agenda • Introduction • The mitigation strategy • Simulation analysis of mitigation strategy • Theoretical analysis of cascading model • Conclusions

10. The mitigation strategy • In previous studies, the rule that the overload nodes are immediately removed from the network is widely adopted. However, few works discuss this problem: when the load on a node exceeds its capacity, whether there exist some strategies to maintain its normal and efficient functioning to avoid the cascading propagation or not? • Wang and Rong (2009)constructed a cascading model of an overload node with the breakdown probability. However, this mechanism with the breakdown probability increases the total price of the whole network. • Therefore, without changing the total price of the whole network, how to protect the overload nodes has become one of the key issues in the control and the defense of cascading failures. • Considering that the neighboring nodes of an overload node may provide some protection resources to this overload node to maintain its normal and efficient functioning, we propose a local protection method.

11. The mitigation strategy ：the set of the neighboring nodes of node i Cm：the capacity of node mto handle the load Lm： the initial loadof nodem The parameter p：a random number in 0 and 1, which decides to the strength of the protection resources provided by the neighboring nodes. p= 0, theneighboring nodes of the overload node cannot provide the extracapacity p= 1, the strongest protection is provided by the neighboring nodes

12. The mitigation strategy • For simplicity, this paper apply this strategy to a simple cascadingmodel. • Our purpose of this paper is to analyze to what extent this mitigation strategy can enhance the network robustness against cascading failures and what is the optimal strategy of the parameter selection after adopting this protection method.

13. The cascading model • The initial load Lion node iin a network is defined as ki：the degree of node I a：a tunable parameter • The capacity Ciof node iis defined as β：an adjustable parameter characterizing the tolerance of the network against cascading failures • F • After nodeifails, the additional loadLjreceived by nodejis proportional to its initial loadLj.

14. The cascading model • When the load on node I exceeds its capacity, the capacity Ci,tof node ican dynamically be adjusted to • The capacities of the neighboring nodes of node ican simultaneously be adjusted to

15. The mitigation strategy • This paper quantify the effectiveness of the mitigation strategy on improving the network robustness against cascading failures by two measures： 1. the avalanche size S , 2.theproportionP(S)

16. Agenda • Introduction • The mitigation strategy • Simulation analysis of mitigation strategy • Theoretical analysis of cascading model • Conclusions

17. Simulation analysis of mitigation strategy • The degree distribution of the scale-free networks satisfies a power lawform: k：the number of links of a randomly chosen node in the network ：the scaling exponent.

18. Simulation analysis of mitigation strategy

19. Simulation analysis of mitigation strategy

20. Simulation analysis of mitigation strategy • The role of the mitigation strategy on improving the network robustness when the diverse types of nodes are attacked. • This paper simply apply two attacks 1. attacking the nodes with the highest load (HL) 2.attacking the ones with the lowest load (LL) • Normalized average avalanche size,we quantify the networkrobustness after and before adopting the mitigation strategy A：the set of nodes attacked nA： the number of nodes attacked n： the number of all nodes in a network

21. Simulation analysis of mitigation strategy

22. Simulation analysis of mitigation strategy

23. Agenda • Introduction • The mitigation strategy • Simulation analysis of mitigation strategy • Theoretical analysis of cascading model • Conclusions

24. Theoretical analysis of cascading model • A • f

25. Theoretical analysis of cascading model • By the probability theory and the network structure, this paperfocus on the mathematical expectations of the expressions. and • Taking into account the characteristic of the no degree–degree correlation in BA networks, we have

26. Theoretical analysis of cascading model • Whereis the conditionalprobability that node i with the degree kihas a neighbor node withthe degree k’ . Similarly, we have • The above inequality (6) is simplified to

27. Theoretical analysis of cascading model • Analyzing the above inequality (9), we can obtain the critical threshold βcin three ranges of a < 1, a = 1, and a > 1 • kmax and kmin represent the minimum and maximum degree of nodes in a network, respectively. When the value p is equal to 0, the equation has been verified by Wang et al. (2008).

28. Theoretical analysis of cascading model • When a > 1, we have • According the Eq. (10) and the inequality (12) and the correctness of the Eq. (12) when p = 0, we can get βc(a > 1) > βc (a = 1). • In the case of a < 1, we can also obtain βc(a < 1) > βc(a = 1).

29. Theoretical analysis of cascading model • However, in Fig. 3, we observe that the optimal value a is about 0.8. • The smaller deviation between computer simulations and theoretical analysis mainly derives from the above approximation in the analytical predictions. i.e., two nodes i and j connected by other simultaneously have the minimum degree. • In fact, in BA networks generated by the BA model, according to the evolving mechanism of the BA model, two nodes with the lowest degree can be not connected by each other.

30. Theoretical analysis of cascading model • D • For a BA networkwith the finite size, its degree distribution is, wherem is equal to kmin (here). Thus, in a BA network canbe calculated by

31. Theoretical analysis of cascading model • In BA networks, kmax can be calculated by

32. Theoretical analysis of cascading model • In computer simulations, the parameter p, the average degree <k>, and the network size N is 0.5, 4, and 5000, respectively. So, only in the case of p = 0.5, we can obtain,whichshows that the network robustness can dramatically be improved. • This paper also find that the bigger the value p, the smaller the value βc, the stronger the network robustness.

33. Agenda • Introduction • The mitigation strategy • Simulation analysis of mitigation strategy • Theoretical analysis of cascading model • Conclusions

34. Conclusions • By the local dynamical adjustment of the capacity of an overload node, without changing the total price of the whole network, this paper propose a method to effectively protect the overload node to avoid its breakdown Characterized a fundamental property, by which the links that affect the total survivability level of the optimal routing paths belong to a typically small subset. • Applying a simple cascading model, this papernumerically investigate the effectiveness of the mitigation strategy on improving the robustness against cascading failures in BA scale-free networks.

35. Conclusions • According to the average avalanche size and the proportion between the protected nodes by the mitigation strategy and the failed nodes before adopting the mitigation strategy, this paperfind that the mitigation strategy can effectively enhance the network robustness and obtain the correlation between the network robustness and some parameters. • After applying the mitigation strategy, we obtain the optimal value a, at which BA networks can reach the strongest robust level against cascading failures

36. Thanks for your attention