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Shlomo Havlin
Israel
Interdependent NetworksFrom Single Networks to Network of Networks
Bar-Ilan University
Electric grid, CommunicationTransportation Services …..
Two types of links:1. Connectivity2. Dependency
Cascading disaster-abrupt transition
20002010
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Extensive Studies Since 2000 -- 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
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Complex Single Networks- Since 2000 Poisson distribution(ER - 1959)
Erdős-Rényi Network Scale-free Network
-λ
m K
Scale-free distribution(Barabasi -1999)
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Internet Network
Faloutsos et. al., SIGCOMM ’99
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WWW-Network
Barabasi et al (1999)
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Jeong, Tombor, Albert, Barabasi, Nature (2000)
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Many real networks are non-Poissonian
P k e
k
kk
k
( )!
( )
0
c k m k KP k
o t h e r w i s e
Classical Erdos-Renyi (1960) Barabasi-Albert (1999)
Homogeneous, similar to lattices Heterogeneous-translational symmetry breaks!New universality class-many anomalous laws
~ log log. ., 0; cd Ne g p ~ logd N -- Small world
Ultra Small worlds (Cohen and SH, PRL (2003))1 1/c cp q k
Breakthrough in understanding many problems! [1 exp( )]P p k P
SF more robust!!
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Many real networks are non-Poissonian
P k e
k
kk
k
( )!
( )
0
c k m k KP k
o t h e r w i s e
Classical Erdos-Renyi (1960) Barabasi-Albert (1999)
Homogeneous, similar to lattices Heterogeneous-translational symmetry breaks!New universality class-many anomalous laws
~ log log. ., 0; cd Ne g p ~ logd N -- Small world
Ultra Small worlds (Cohen and SH, PRL (2003))1 1/c cp q k
Breakthrough in understanding many problems! [1 exp( )]P p k P
SF more robust!!
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Many real networks are non-Poissonian
Classical Erdos-Renyi (1960) Barabasi-Albert (1999)
Homogeneous, similar to lattices Heterogeneous-translational symmetry breaks!New universality class-many anomalous laws
~ log log. ., 0; cd Ne g p ~ logd N -- Small world
Ultra Small worlds (Cohen and SH, PRL (2003))1 1/c cp q k
Breakthrough in understanding many problems! [1 exp( )]P p k P
1
p
SF ER
SF more robust!!
cp
P
0 1
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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:
1 1 1/c cq p k
1c cq p
WHAT IS DIFFERENT?
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Scale Free networks --immunization strategies
targeted
Random
Acquaintance order 1
robust
vulnerable
critical fraction of removed or immunized nodescq
order 2
Efficient immunization
Poor immunization
cq
Efficient Immunization Strategy:
Acquaintance Immunization
Cohen et al, Phys. Rev. Lett. 91 , 168701 (2003)
( ) ~p k k
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A. Vespignani and V. Collizza, PNAS (2009)
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Global spreading of epidemics, rumors, opinions, and innovations Brockmann and Helbing Nature 342, 1337 (2013)
Transportation Network (GAN)
Simulated Epidemics (HK)
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Network Motifs-System Biology
Uri Alon,, Science (2002)
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Detecting overlapping communities A Lancichinetti, S Fortunato, J Kertész
New Journal of Physics 11 (3), 033015 (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
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Bashan et al, Nature Communication [2012] Structure and Function
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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
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EL-NINO BECOMES AUTONOMOUS: ONLY INFUENCE-NOT INFLUENCED
Gozolchiani et al PRL (2011)
VERY EARLY PREDICTION!!! Ludescher et al, PNAS (2014)
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Mitigation of malicious attacks on networksPower Grid Internet
Schneider, Moreira, Andrade, SH and Herrmann PNAS (2011)
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Collaboration:Amir Bashan, BIU
Sergey Buldyrev, NYDong Zhou, BIUYang Wang, BIU
J. Gao: NortheasternYehiel Berezin, BIU
Michael Danziger, BIURoni Parshani: BIU
Christian Schneider, MITH. 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, 195701 (2011)Gao et al, Nature Phys.,8, 40 (2012)Bashan et al, Nature Com., 3, 702, (2012)Wei Li et al, PRL, 108, 228702 (2012)Bashan et al, Nature Phys. 9, 667 (2013)
From Single Network to Network of Networks2000 2010
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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 alInt. J. of Crit.Infrastruct. 4,63 (2008)
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Rinaldi S, et al IEEE (2001)
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BRAIN
HUMAN BODY: NETWORK OF NETWORKS
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• 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
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Comparing single and coupled networks: Robustness
P
1
01
Continuous abrupt
cp
Remove randomly (or targeted) a fraction nodes1 p
P Size of the largest connected component (cluster)
p
Single networks:Continuous transition
0 cp
(ER) (SF)
Coupled networks: New paradigm-Abrupt transitionCascading Failures
Single ER
Coupled
Cascades,Suddenbreakdown
Breakdown threshold cp
[1 exp( )]P p k P
Message: our world is extremely unsafe!
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RANDOM REMOVAL – PERCOLATION FRAMEWORK
A
B
nodes leftp
1( ) Ap p p p giant cluster
1( ) Ap p p p nodes left1 2( )Bp p p p
1 1 2( ) Bp p p p giant cluster
2 2 3 ( )Agiant cluster p p p p
2 3( )Ap p p p
3 3 4 ( )Bgiant cluster p p p p
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P
n
after τ-cascades of failuresP
Catastrophic cascadesjust below cp
For a single network 1/cp k
ER networkSingle realizations
RESULTS: THEORY and SIMULATIONS: ER Networks
Removing 1-p nodes in A
A
B
2.4554 /cp k
2.45 / cp k p
ABRUPT TRANSITION (1st order)
mmin in for single network2 455 1. 4 kk
τ
Dong Zhou et al (2013) 1/3N
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Simultaneous first and second order percolation transitions
Dong et al, arXiv:1211.2330 (2013)
A
B
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Partial Interdependent Networks Determining in simulations:cp
Theory and simulations
P
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GENERALIZATION: PARTIAL DEPENDENCE: Theory and Simulations
A
B P
Parshani, Buldyrev, S.H.PRL, 105, 048701 (2010) Strong q=0.8:
1st OrderWeak q=0.1:2nd Order
q-fraction of dependency nodes
0.2 for random coupling
0.9 for optimal robustnessc
c
q
q
Schneider et al arXiv:1106.3234Scientific Reports (2013)
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Designing Robust Coupled Networks:Italy 2003 blackout
Random interdependencies Nearly optimal interdependencies
Schneider, Araujo, Havlin , Herrmann, Designing Robust Coupled Networks, Scientific Reports (2013)
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IN CONTRAST TO SINGLE NETWORKS, COUPLED NETWORKS ARE MORE VULNERABLE WHEN DEGREE DIST. IS BROADER
All with 4k
Buldyrev, Parshani, Paul, Stanley, S.H. Nature (2010)
0cp
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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- 2nd order
1/cp k
[1 exp( )]nP p kP P
ik k
n=1
n=2
n=5
m
[1 exp( )]P p kP
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Random Regular Network of ER networks
2(1 )[1 (1 ) 4 ]2
k P mm
pP e q q qP
1
(1 )c mp
k q
2 2
c
k m m k mq
k
For 0 OR 0
the single network
is obtained!
m q
RR, m=3
ER = 2.2k
Surprisingly Independent on n![1 exp( )]P p k P
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Interdependent Spatially Embedded Networks
0.9q
0.1q
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
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Spatial embedded compared to random coupled networks when q changes:
q=0.2
q=0
Bashan et alhttp://arxiv.org/abs/1206.2062 Nature Physics, (2013)
0cq
P
NOI
~ ( )
5 / 36 1 for d=2
For ER and d=6, =1
cP p p
0.9q
0.1q
0q 0q
EXTREMELY VULNERABLE!!
1 ( )c c cp q P p
0.5cq q
Message: our world is extremely unsafe!-no safe zone!
( )cP p
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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)
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Interdependent Spatially Embedded Networks
Many networks are spatially embedded:Internet, Power grid, Transportation etc
Wei et al, PRL, 108, 228702 (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 r
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Interdependent Spatially Embedded Networks
Many networks are spatially embedded:Internet, Power grid, Transportation etc
Wei et al, PRL, 108, 228702 (2012)
Bashan et al, http://arxiv.org/abs/1206.2062
1st order
2nd order
cr
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New percolation-localized attacks
Y. Berezin et al, . arXiv:1310.0996
Localized attacks on spatially embedded systems with dependencies: critical size attack
Theory
Gradient percolation, Sapoval et al (1985)
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Simulation Results Analytical Results
New percolation-localized attacks
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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 randomRich problem: different types ofnetworks 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)
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PARTIAL DEPENDENCE:critical point
A
B
Analogous to critical point in liquid-gas transition:
Parshani et alPRL, 105, 048701 (2010)
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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)
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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)
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Interdependent Spatially Embedded Networks
Many networks are spatially embedded:Internet, Power grid, Transportation etc
Wei et al, PRL, 108, 228702 (2012)
Bashan et al, http://arxiv.org/abs/1206.2062
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The extreme vulnerability of spatial embedded coupled networks
q=0.2
q=0
Bashan et alhttp://arxiv.org/abs/1206.2062
0cq
P
NOI
EXTREMELY VULNERABLE!!
P
NOI
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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 ofnetworks and interconnections. Buldyrev et al., NATURE (2010)Parshani et al., PRL (2010)Gao et al., (preprint)
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CITATIONS PER YEAR –DIFFERENT FIELDS
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SCADA
Power grid
Blackout in Italy (28 September 2003)
SCADA=Supervisory Control And Data Acquisition
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Blackout in Italy (28 September 2003)
Power grid
SCADA
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Blackout in Italy (28 September 2003)
Power grid
SCADA
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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 2nd order; n>1 are 1st 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 ofnetworks 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)