Israel Interdependent Networks From Single Networks to Network of Networks Bar-Ilan University Electric grid, Communication Transportation Services ….. Two types of links: 1. Connectivity 2.Dependency Cascading disaster-abrupt transition

Extensive Studies Since Single Networks A Network is a structure of N nodes and M edges (or 2M links ) Called usually graph – in Mathematics Complex systems can be described and understood using networks Internet: nodes represent computers links the connecting cables Biological systems: nodes represent proteins links their relations Climate system: nodes represent locations links similar climate Successful research: Efficient immunization strategies, identifying key players, robustness, climate (KURTHS), physiology, protein networks- function ….. Wang et al (Science 2009) Brown-same operating system-now Percolation-Immunization

Complex Single Networks- Since 2000 Poisson distribution (ER ) Erdős-Rényi Network Scale-free Network -λ-λ m K Scale-free distribution (Barabasi -1999)

Internet Network Faloutsos et. al., SIGCOMM ’99

WWW-Network Barabasi et al (1999)

Jeong, Tombor, Albert, Barabasi, Nature (2000)

Classical Erdos-Renyi (1960)Barabasi-Albert (1999) Homogeneous, similar to lattices Heterogeneous-translational symmetry breaks! New universality class-many anomalous laws -- Small world Ultra Small worlds ( Cohen and SH, PRL (2003)) Breakthrough in understanding many problems! SF more robust!!

Classical Erdos-Renyi (1960)Barabasi-Albert (1999) Homogeneous, similar to lattices Heterogeneous-translational symmetry breaks! New universality class-many anomalous laws -- Small world Ultra Small worlds ( Cohen and SH, PRL (2003)) Breakthrough in understanding many problems! SF more robust!!

Classical Erdos-Renyi (1960)Barabasi-Albert (1999) Homogeneous, similar to lattices Heterogeneous-translational symmetry breaks! New universality class-many anomalous laws -- Small world Ultra Small worlds ( Cohen and SH, PRL (2003)) Breakthrough in understanding many problems! SFER SF more robust!! 1

Infectious disease Critical Threshold Malaria 99% Measles 90-95% Whooping cough 90-95% Fifths disease 90-95% Chicken pox 85-90% Internet more than 99% Known values of immunization thresholds: This puzzle is solved due to the broad degree distribution (HUBS) of social networks which does not occur in random graphs! Such immunization thresholds were not understood since they were well above the expected value of percolation in classical random networks: WHAT IS DIFFERENT?

Scale Free networks --immunization strategies targeted Random Acquaintance order 1 robust vulnerable order 2 Efficient immunization Poor immunization Efficient Immunization Strategy: Acquaintance Immunization Cohen et al, Phys. Rev. Lett. 91, (2003)

A. Vespignani and V. Collizza, PNAS (2009)

Global spreading of epidemics, rumors, opinions, and innovations Brockmann and Helbing Nature 342, 1337 (2013) Transportation Network (GAN) Simulated Epidemics (HK)

Network Motifs-System Biology Uri Alon,, Science (2002)

Detecting overlapping communities A Lancichinetti, S Fortunato, J Kertész New Journal of Physics 11 (3), (2009) Structure and Function: Useful for unveiling protein function: Irene Sendin˜a–Nadal et al, PlosOne (2011) Michele Tumminello, Roasario Mantegna, Janos Kertesz Networks in economy and finance

Bashan et al, Nature Communication [2012] Structure and Function

Climate networks are very sensitive to El Nino Sea surface temperature network 5km height temperature network Yamasaki, Gozolchiani, SH (PRL 2008, 2011) Challenge: Predicting El-Nino and other extreme events

EL-NINO BECOMES AUTONOMOUS: ONLY INFUENCE-NOT INFLUENCED Gozolchiani et al PRL (2011) VERY EARLY PREDICTION!!! Ludescher et al, PNAS (2014)

Mitigation of malicious attacks on networks Power Grid Internet Schneider, Moreira, Andrade, SH and Herrmann PNAS (2011)

Collaboration: Amir Bashan, BIU Sergey Buldyrev, NY Dong Zhou, BIU Yang Wang, BIU J. Gao: Northeastern Yehiel Berezin, BIU Michael Danziger, BIU Roni Parshani: BIU Christian Schneider, MIT H. E. Stanley, Boston PARTIAL LIST Buldyrev et al, Nature, 464, 1025 (2010 ) Parshani et al, PRL,105, 0484 (2010) Parshani et al, PNAS, 108, 1007 (2011) Gao et al, PRL, 107, (2011) Gao et al, Nature Phys.,8, 40 (2012) Bashan et al, Nature Com., 3, 702, (2012) Wei Li et al, PRL, 108, (2012 ) Bashan et al, Nature Phys. 9, 667 (2013) From Single Network to Network of Networks

Blackout in Italy (28 September 2003) CASCADE OF FAILURES Railway network, health care systems, financial services, communication systems Power grid Communication SCADA Cyber Attacks- CNN Simulation (2010) Rosato et al Int. J. of Crit. Infrastruct. 4, 63 (2008)

Rinaldi S, et al IEEE (2001)

BRAIN HUMAN BODY: NETWORK OF NETWORKS

Until 2010 studies focused on a single network which is isolated AND does not interact or influenced by other systems. Isolated systems rarely occur in nature or in technology -- analogous to non-interacting particles (molecules, spins). Results for interacting networks are strikingly different from those of single networks. Interdependent Networks

Comparing single and coupled networks: Robustness Continuous abrupt Remove randomly (or targeted) a fraction nodes Size of the largest connected component (cluster) Single networks: Continuous transition ( ER ) ( SF ) Coupled networks: New paradigm-Abrupt transition Cascading Failures Single ER Coupled Cascades, Sudden breakdown Message: our world is extremely unsafe!

RANDOM REMOVAL – PERCOLATION FRAMEWORK

after τ-cascades of failures Catastrophic cascades just below ER network Single realizations RESULTS: THEORY and SIMULATIONS: ER Networks Removing 1-p nodes in A ABRUPT TRANSITION (1 st order) τ Dong Zhou et al (2013)

Simultaneous first and second order percolation transitions Dong et al, arXiv: (2013)

Partial Interdependent Networks Determining in simulations: Theory and simulations

GENERALIZATION: PARTIAL DEPENDENCE: Theory and Simulations Parshani, Buldyrev, S.H. PRL, 105, (2010) Strong q=0.8: 1 st Order Weak q=0.1: 2 nd Order q-fraction of dependency nodes Schneider et al arXiv: arXiv: Scientific Reports (2013)

Designing Robust Coupled Networks: Italy 2003 blackout Random interdependenciesNearly optimal interdependencies Schneider, Araujo, Havlin, Herrmann, Designing Robust Coupled Networks, Scientific Reports (2013)

IN CONTRAST TO SINGLE NETWORKS, COUPLED NETWORKS ARE MORE VULNERABLE WHEN DEGREE DIST. IS BROADER Buldyrev, Parshani, Paul, Stanley, S.H. Nature (2010)

Network of Networks (tree) Gao et al PRL (2011) n=5 For ER,, full coupling, ALL loopless topologies (chain, star, tree): Vulnerability increases significantly with n n=1 known ER- 2 nd order n=1 n=2 n=5 m

Random Regular Network of ER networks RR, m=3 ER = 2.2 Surprisingly Independent on n!

Interdependent Spatially Embedded Networks Theory (based on critical exponent): NO continuous transition for any q>0-extreme vulnerability!! Bashan et al, Nature Physics ( 2013) Many networks are spatially embedded: Internet, Power grid, Transportation etc r=2

Spatial embedded compared to random coupled networks when q changes: q=0.2 q=0 Bashan et al Nature Physics, (2013) EXTREMELY VULNERABLE!! Message: our world is extremely unsafe!-no safe zone!

Interdependent European Communication Network and Power Grid Experimental test on real spatial embedded coupled networks Results for European and US Interdependent Communication Networks and Power Grids Bashan et al, Nature Physics (2013)

Interdependent Spatially Embedded Networks Many networks are spatially embedded: Internet, Power grid, Transportation etc Wei et al, PRL, 108, (2012) Bashan et al, Nature Physics (2013) When connectivity links are limited in their length---same universality class as lattices! THREE DIFFERENT BEHAVIORS DEPENDING ON

Interdependent Spatially Embedded Networks Many networks are spatially embedded: Internet, Power grid, Transportation etc Wei et al, PRL, 108, (2012) Bashan et al, 1 st order 2 nd order

New percolation-localized attacks Y. Berezin et al,. arXiv: Localized attacks on spatially embedded systems with dependencies: critical size attack Theory Gradient percolation, Sapoval et al (1985)

Simulation ResultsAnalytical Results New percolation-localized attacks

Summary and Conclusions First statistical physics approach for robustness of Networks of Interdependent Networks—cascading failures New paradigm: abrupt collapse compared to continuous in single network Generalization to “Network of Networks”: n interdependent networks- 50y of graph theory and percolation is only a limited case! Larger n is more vulnerable–spatial embedding-extremely unsafe: New percolation: Localized Attacks-more vulnerable compared to random Rich problem: different types of networks and interconnections. Buldyrev et al., NATURE (2010) Parshani et al., PRL (2010) Gao et al, PRL (2011) Parshani et al, PNAS (2011) Wei et al, PRL (2012) Gao et al., Nature Phys. (2012) Bashan et al, Nature Phys. (2013)

PARTIAL DEPENDENCE: critical point Analogous to critical point in liquid-gas transition: Parshani et al PRL, 105, (2010)

Unveiling Protein Functions by Synchronization in the Interaction Network Irene Sendin˜a–Nadal, Yanay Ofran, Juan A. Almendral1, Javier M. Buldu, Inmaculada Leyva, Daqing Li, Shlomo Havlin, Stefano Boccaletti, Plos One (2011)

Unveiling Protein Functions by Synchronization in the Interaction Network Irene Sendin˜a–Nadal, Yanay Ofran, Juan A. Almendral, Javier M. Buldu, Inmaculada Leyva, Daqing Li, Shlomo Havlin, Stefano Boccaletti, Plos One (2011)

Interdependent Spatially Embedded Networks Many networks are spatially embedded: Internet, Power grid, Transportation etc Wei et al, PRL, 108, (2012) Bashan et al,

The extreme vulnerability of spatial embedded coupled networks q=0.2 q=0 Bashan et al EXTREMELY VULNERABLE!!

Summary and Conclusions First statistical physics approach for robustness of interdependent networks Broader degree distribution is more vulnerable in interacting networks Generalization to “Network of Networks”: n interdependent networks-50y of graph theory and percolation is only a limited case! Larger n is more vulnerable Network A Network B Rich problem: different types of networks and interconnections. Buldyrev et al., NATURE (2010) Parshani et al., PRL (2010) Gao et al., (preprint)

CITATIONS PER YEAR –DIFFERENT FIELDS

SCADA Power grid Blackout in Italy (28 September 2003) SCADA=Supervisory Control And Data Acquisition

Blackout in Italy (28 September 2003) Power grid SCADA

Blackout in Italy (28 September 2003) Power grid SCADA

Summary and Conclusions First statistical physics approach --mutual percolation-- for a Network of Interdependent Networks—cascading failures Generalization to Network of Networks: 50ys of classical percolation is a limiting case. E.g., only n=1 is 2 nd order; n>1 are 1 st order Partial Dependence: Strong coupling: first order phase transition; Weak: second order Extremely vulnerable: broader degree distribution in single networks more robust ---- in interacting networks -less robust Network A Network B Rich problem: different types of networks and interconnections. Real data. Buldyrev et al, NATURE (2010) Parshani et al, PRL (2010); Gao et al, PRL (2011) Parshani et al, EPL (2010) Parshani et al, PNAS (2011) Bashan et al, PRE (2011)