1. Cascading failures of loads in interconnected networks under intentional attack Yongxiang Xia Department of Information Science and Electronic Engineering Zhejiang University Apr. 25, 2013 1

2. Contents Introduction Models Results Conclusions 2

5. Blackout in northeast America in 2003 Steps: 1. A generating plant in Eastlake, Ohio (a suburb of Cleveland) went offline amid high electrical demand 2. It strained high-voltage power lines (located in a distant rural setting) 3. Power lines went out of service when they came in contact with "overgrown trees" 4. More fails 5. … Final result: The blackout affected an estimated 10 million people in the Canadian province of Ontario and 45 million people in eight U.S. states.

6. Blackout in northeast America in 2003

7. Blackout: Can It Happen Again? 1959, 1961, 1965, 1977, 2003…

8. Modeling cascading failures Cascading failures in isolated networks ER: robust BA: robust-yet-fragile 8 Motter and Lai, PRE 66, 065102(R), 2002

9. Introduction Cascading failures in interdependent networks Buldyrev et al. Nature 464, 1025-1028 (2010) Modelling an iterative process of a cascade of failures.

10. Introduction Interconnected networks: yet another coupled networks The power grid of the continental United States, illustrating the three main regions or “interconnects”—Western, Eastern, and Texas—and new lines (in red) proposed by American Electric Power to transport wind power. Brummitt et al. PNAS 2012;109:E680-E689 10

11. Cascades of load based on Sandpile Dynamics Introduction Brummitt et al. PNAS 2012;109:E680-E689 A random three- and four-regular graph connected by Bernoulli-distributed coupling with interconnectivity parameter p = 0.1 [R(3)-B(0.1)-R(4)]. 11

12. Introduction Brummitt et al. PNAS 2012;109:E680-E689 Interconnectivity is locally stabilizing, but only up to a critical point. NOTE: Local dynamics 12

13. Different cascading failure model Motter and Lai, PRE 66, 065102(R), 2002 –Isolated network –Load Traffic Global dynamics Buldyrev et al. Nature 464, 1025-1028 (2010) –Interdependent network –Topology –No load Brummitt et al. PNAS 2012;109:E680-E689 –Interconnected network –Load Sandpile Local dynamics ?? –Load –Global dynamics

14. Network Model –Two BA scale-free networks Network size Average Degree –Coupling probability (How many interconnected links?) –Coupling preference (How to add interconnected links?) Assortative Disassortative Random Models : Number of interconnected links Each node has at most one interconnected link 14

15. Models Traffic Model Data-packet transport based on the shortest path routing Load : node betweenness Capacity Attack strategy Removal of one node with the highest load in the whole system initial load of node tolerance parameter Note: Global dynamics 15

16. Models The dynamical process of cascading failures of loads in interconnected networks. α=1.6 16 Note: Global dynamics

17. Results ------ vs  (a) Assortative (b) Disassortative (c) Random  Giant connected component Tolerance parameter Coupling probability  Assortative >Random >Disassortative 17

18. Results ------ VS Critical tolerance parameter at which G= 0.5 Coupling probability      , decreases sharply, increases slowly the coupling probability at the turning point 18

19. Results ------ Effect of the average degree Results: 1. Similar shape 2. higher average degree more robust to cascades  (a) Assortative (b) Disassortative (c) Random  → 19

20. Results ------ & Results: 1. similar shape 2. but turning point Po changes 20

21. Results ------ &  the coupling probability at the turning point  at the turning point Result: The coupling preference can largely affect, but has much less impact on. 21

22. How many –Enhancing the coupling probability can mitigate cascading failure for sparse coupling, but intensifies the cascades for dense coupling. –Such relationship between cascades and the coupling probability was found previously using the sandpile model. How –The coupling preference can largely affect the optimal coupling probability, but has much less impact on the minimal performance indicator. –Assortative coupling is more prone to reduce the malfunction caused by cascading failures than disassortative or random coupling in most cases. Conclusions 22

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