Mitigation strategies on scale-free networks against cascading failures Jianwei Wang

1. Mitigation strategies on scale-free networks against cascading failuresJianwei Wang Adviser: Frank,Yeong-Sung Lin Present by Chris Chang

2. Agenda • Introduction • The mitigation strategy • The analysis of four mitigation methods • Conclusion

3. Agenda • Introduction • The mitigation strategy • The analysis of four mitigation methods • Conclusion

4. Introduction • Cascading failures induced by random failure or intentional attacks are common phenomena and can occur in many of the real-world networks • By studying the dynamic evolving mechanism of cascading failure in real-world networks, a number of important aspects of cascading failures have been discussed and many valuable results have been obtained in the literature ： • the models for describing cascade phenomena • the cascade control and defense strategies • cascading failures in real networks • the attack strategies • the robustness of coupled networks

5. Introduction • Thus far, to obtain the optimal robustness of networks against cascading failures, there are few works about the mitigation mechanism on the overload node and selecting what type of and how many nodes to adopt the mitigation strategy.

6. Introduction • However, in many infrastructure networks, when the load on a node exceeds its capacity… • There exist some measures or replacement equipment to maintain its normal and efficient function to avoid the cascading propagation. • The choice of the mitigation strategy on some key nodes is very important. (under the limitation of the total protection cost in realistic networks)

7. Introduction • We introduce a simple efficient mitigation mechanism for an overload node and develop two new methods to select nodes to adopt the mitigation strategy. • By the normalized average avalanche size and the new unique robustness measure, we numerically investigate the efficiencies of the different mitigation strategies on enhancing the robustness of Barabási–Albert scale-free networks against cascading failures.

8. Agenda • Introduction • The mitigation strategy • The analysis of four mitigation methods • Conclusion

9. The mitigation strategy • In the real-world networks, to avoid the propagation of cascading-failure-induced disasters, due to some protection measures the overload nodes will be not removed from the network, and can redistribute some of the load on them to its neighboring nodes to maintain their normal and efficient functions. • Our aim is to examine the effectiveness of the mitigation strategy and to find a better method to suppress cascading failures.

10. The mitigation strategy • In the cascading model, the initial load Li of node i with the degree ki is defined as Li = (ki)α. • α(> 0) is a tunable parameter which controls the strength of the initial load of node i. • The capacity Ci of node i is proportional to its initial load and is defined as Ci=(1+β)Li • β is the tolerance parameter characterizing the tolerance of the network against cascading failures.

11. The mitigation strategy • After node i fails, the additional load Lji received by node j among its neighboring nodes is proportional to its initial load, where Γirepresents the set of all neighboring nodes of node i.

12. The mitigation strategy • Suppose that a node in a network at most adopt once mitigation strategy and mark the number that node i adopt the mitigation strategy by δi • for any node i in a network, δi≤ 1 • the node once again overloads may be removed from the network. • We will choose some of the nodes to protect them by the mitigation strategy, and set ξi = 1 if node i can adopt the mitigation, otherwise set ξi = 0.

13. The mitigation strategy • To avoid the malfunction of the node and consider the limitation of the mitigation strategy, we think that after adopting the mitigation strategy, the load on the node should be lower than its capacity and be higher than its initial load.

14. The mitigation strategy • At t + 1 time, we define the load L(i)t +1 on node i that adopt the mitigation strategy at t time as：L(i)t +1 = Li + p(Ci − Li )

15. The mitigation strategy • Therefore, at t time the redistributed load ∆Li,t by the node i adopted the mitigation strategy is defined as：∆Li,t = L(i)t − Li − p(Ci − Li)

16. The mitigation strategy • Under the above assumptions, node j among the neighboring nodes of node i can receive the extra load ∆Lji as follows:

17. The mitigation strategy • Here we only consider cascading failures triggered by removing a single node and calculate the number si of broken nodes induced by removing node i. • By the normalized average avalanche size S, we quantify the network robustness against cascading failures. S is defined as： • where N represents the number of all nodes in a network.

18. The mitigation strategy • Our aim in this paper is to discuss the optimal mitigation strategy with the lower price and higher robustness. • We propose four methods to select nodes and apply the mitigation strategy to these nodes.

19. The mitigation strategy • (1) The mitigation on the node with the highest load (MHL) : • Generally speaking, the nodes with the highest load can be more likely to trigger the cascading propagation. • According the value of the parameter α and the degree of the node, the bigger the degree of a node, the higher the load of this node. • Selecting the nodes in the descending order of their load

20. The mitigation strategy • (2) The mitigation on the node with the lowest load (MLL): • In previous studies, we discuss the effects of the removal of the nodes with the lowest load on the network robustness against cascading failures and find that these nodes also play vital roles in the cascading propagation. • Selecting the nodes in the ascending order of their load

21. The mitigation strategy • (3) The mitigation on the node that the breakdowns of its neighboring nodes can be more likely to trigger its malfunction (MHHC): • (4) The mitigation on the node that its removal can be more likely to trigger the breakdowns of its neighboring nodes (MHC):

22. The mitigation strategy we select the highest value of βi,i1 , βi,i2 , and βi,i3 to define βi,c on node i. we select the highest value of βi1,i, βi2,i, and βi3,i to define βi of node i.

23. The mitigation strategy • In the strategy MHHC, we select the nodes in the descending order of βi in the network to adopt the mitigation strategy. • In the MHC, we select the nodes in the descending order of βi,c in the network to adopt the mitigation strategy.

24. Agenda • Introduction • The mitigation strategy • The analysis of four mitigation methods • Conclusion

25. The analysis of four mitigation method • We investigate the efficiencies of four mitigation methods on improving the scale-free network robustness against cascading failures. • The scale-free networks are generated by the well-known Barabási–Albert model. In computer simulations, we fix the total network size as N = 1000 and set the average degree of the network to ⟨k⟩ = 4.

26. The analysis of four mitigation method • In the cascading model, the bigger the value α, the higher the heterogeneity of the load distribution. Therefore, in four cases of α = 0.4, α = 0.7, α = 1.0, and α = 1.3, we analyze four mitigation methods. • m(n) :representing the proportion between the number of mitigation nodes and the total number of nodes in a network; we choose the proportion of the mitigation nodes to m(n) = 0.1, m(n) = 0.3, m(n) = 0.5, and m(n) = 0.7, respectively.

27. The analysis of four mitigation method • First, we compare four mitigation methods and investigate the effects of four mitigation methods on improving the robustness of BA networks against cascading failures by two measures : • the normalized average avalanche size S • p(s) : represents the proportion between the number of protected nodes after adopting the mitigation methods and the number of broken nodes induced before adopting the mitigation strategy. • (p(s) can better reflect the significant improvement resilience of the BA network)

28. The analysis of four mitigation method • we compare the normalized average avalanche size S between before and after adopting the mitigation strategy and analyze the improved proportion p(s) of four mitigation methods. • when α = 0.4…

29. The analysis of four mitigation method • For the bigger normalized average avalanchesize  choose the MHL as the optimal protection • For the smaller normalized average avalanche size the MHHC is mostefficient. • Therefore, according to the different value of the capacity parameterβ, we can select the optimal strategy and thenumber of nodes adopted the mitigation method.

30. The analysis of four mitigation method • when α = 0.7…

31. The analysis of four mitigation method • when α = 1.0…

32. The analysis of four mitigation method • when α = 1.3…

33. The analysis of four mitigation method • we can observe that as the value α increases, for the smaller β the original order of the efficiency of four mitigation methods is gradually changing from MHL > MHHC > MHC > MLL toMHL > MHC > MHHC > MLL • This is mainly originated from that as the value α increases, the strength of the heterogeneity of the distribution of the load makes the nodes with the higher load more important.

34. The analysis of four mitigation method • However, for the bigger β, owing to the earlier protection on the easy-to-failure nodes among the neighboring nodes of a removal node, the MHHC is most efficient than others to suppress cascading failures.

35. The analysis of four mitigation method • Next, we analyze the effect of the value α on the mitigation strategy.

36. Agenda • Introduction • The mitigation strategy • The analysis of four mitigation methods • Conclusion

37. Conclusion • In this paper, according to the load on a node and the local characteristics of the dynamical evolving process of cascading failures, we propose four mitigation methods and address the optimization of protection strategies in BA scale-free networks.

38. Conclusion • We conclude that to avoid the smaller avalanche size (correspond to the bigger value β) the MHHC and the MHC generally are more efficient than the MHL and the MLL. • For the bigger avalanche size (correspond to the smaller value β), the MHL is most efficient.

39. Conclusion • From the choice of the mitigation methods and the number of nodes adopted the mitigation strategy, according to the parameters in the cascading model, we give the better strategy to protect BA scale-free networks.

40. Thank you for your attention