Catastrophic Cascade of Failures in Interdependent Networks

Transcription

1 Catastrophic Cascade of Failures in Interdependent Networs Wor with: S. Buldyrev (NY) R. Parshani (BIU) G. Paul (BU) H. E. Stanley (BU) Nature 464, 1025 (2010 ) New results: Jia Shao (BU) R. Parshani (BIU) (PRL-in press) Electricity, Communication, Transport Social, Biological.. Shlomo Havlin Bar-Ilan University Israel Two types: Connectivity Dependency

2 Interacting Networs Until now studies focused on the case of a single networ which is isolated and does not interact or influenced by other systems. Such systems rarely occur in nature or in technology - analogous to non-interacting particles (molecules, spins). Results for interacting networs are surprisingly different from those of single networs.

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4 Blacout in Italy (28 September 2003) Cyber ttacs- CNN Simul. 02/10 Internet SCD Power grid CSCDE OF FILURES Railway networ, health care systems, financial services, communication systems

5 Blacout in Italy (28 September 2003) SCD Power grid

6 Blacout in Italy (28 September 2003) SCD Power grid

7 Blacout in Italy (28 September 2003) SCD Power grid

8 Further Examples of Interdependent Networs ppear in all aspects of life, nature and technology Economy: Networs of bans, insurance companies, and firms interact and depend on each other. Networ Networ B Physiology: The human body can be regarded as inter-dependent networs. For example, the cardio-vascular networ system, the respiratory system, the brain networ, and the nervous system all depend on each other. Biology: specific cellular function is performed by a networ of interacting proteins. These networs depend on each other through proteins that perform several functions. Further examples: social networs, transportation networs etc. Failure in networ causes failure in B further failure in..cscdes What are the critical percolation thresholds for such interdependent networs? Size of cascade failures? MUTUL PERCOLTION Buldyrev, Parshani, Paul, Stanley, SH, Nature, 464, 1025 (2010)

9 Robustness of a single networ: Percolation Remove randomly (or targeted) a fraction nodes 1 p (ER) (SF) P Size of the largest connected component (cluster) ORDER PRMETER 1 ER: p = 1/ c λ SF, p ( ) (2 λ 3) : p FOR RNDOM REMOVL c = 0 very robust 2 nd order Broader degree-more robust P 0 0 pc Single 2 nd order 1 st order p p c Coupled Cascades, Sudden breadown 1

10 P RESUTS: THEORY and SIMULTIONS: ER Networs after n-cascades of failures Removing 1-p nodes in P p = 2.45 / < pc ER networ Single realizations n Catastrophic cascades just below p c p = f f f = c ( 1)/2 1/ (2 (1 )); e f f f = , p = / ; P = p (1 f ) c FIRST ORDER TRNSITION c 2 For a single networ p = 1/ c

11 Generalization of ER nown result Solution for two interdependent ER with ; B for >> B, p c and = 1/ P = 0 as in single ER p c P ( p ) c If B is fully connected NO further damage can occur to!!! / B

12 PDF of number of cascades n at criticality for ER of size N ln[p(n)] +1/4 ln N n N 1/4 N=1000 N=2000 N=8000 N=8000 N=16000 N=32000 N= nn -1/4

13 IN CONTRST TO SINGLE NETWORKS, COUPLED NETWORKS RE MORE VULNERBLE WHEN DEGREE DIST. IS BRODER ll with = 4 Buldyrev et al, Nature 2010

14 GENERLIZTION: PRTIL DEPENDENCE: Theory and Simulations P Parshani, Buldyrev, S.H. PRL (2010, in press) arxiv: Wea q=0.1: 2 nd Order Strong q=0.8: 1 st Order

15 Strong Coupling Wea Coupling P P q=0.8 q=0.1

16 nalogous to critical point in liquid-gas transition: PRTIL DEPENDENCE

17 Summary and Conclusions First statistical physics approach --mutual percolation-- for Interdependent Networs cascading failures Generalization to Partial Dependence: Strong coupling: first order phase transition; Wea: second order Generalization to Networ of Networs: n interdependent networs Larger n --- more vulnerable Extremely vulnerable, broader degree distribution - more robust in single networ becomes less robust in interacting networs Rich problem: different types of networs and interconnections. Buldyrev et al, NTURE (2010) Parshani et al, (PRL, in press); arxiv: Networ Networ B

18 RNDOM REMOVL PERCOLTION FRMEWORK p nodes left p ( ) p p p1 giant cluster giant cluster p ( ) p p p p ( ) p p p 1 nodes left p ( B p ) 1 p p 2 p ( p ) 2 p p 3 giant cluster p ( ) B p p p p ( B p ) 1 p1 p2 giant cluster

19 Theoretical pproach: Two Networs and B General recursive relations: p2m = pb( p2 m 1) p2 m 1; p2 m = pb( p2 m 1) p on B p = p ( p ) p ; p = p ( p ) p on 2m+ 1 2m 2m 2m+ 1 2m t the final stationar y stage: p = p = p + 2m 1 2m 2m+ 1 define x p and y p x= p( y) p and y= pb( x) p or x= p ( p ( x) p) p B 2m 1 2m

20 nalytical Results: using generating functions x= p ( p ( x) p) p 2 dp ( p ( x) p) dp ( x) dx dx B B 1= p p ( p) = 1 G [1 p(1 f )] 0 f = G [1 p(1 f ) 1 where G ( u) P ( ) u degree generating function 0 B and G ( u) P ( ) u / P ( ) brunching GF 1 SME EQUTIONS FOR NETWORK B Newman, PRE 66, (2002) Shao et al, PRE 80, (2009)

21 Theoretical pproach: Two Networs and B General recursive relations: x = p( y) p and y = pb ( x) p or x= p ( p ( x) p) p transcendental equation B 1 nalytical Results: using generating functions for different types of networs and B Newman, PRE 66, (2002) Shao et al, PRE 80, (2009) pp (pp B (x)) p=0.70 p=0.752 p=0.80 y=x First order x=p 2 x=p 0 =p x

22 Case of two ER networs of same mean degree P FIRST ORDER PHSE- TRNSITION f 1 1 f = exp ; pc = = / 2f 2 f(1 f) nalytical solution for two random networs with any distribution of degrees