The Fragility of interdependency: Coupled networks and switching phenomena

Sergey BuldyrevDepartment of Physics

Yeshiva UniversityCollaborators:

Gabriel Cwilich (YU), Nat Shere(YU),Shlomo Havlin (BIU), Roni Parshani (BIU), Antonio Majdandzic,

Jianxi Gao, Gerald Paul, Jia Shao, and

H. Eugene Stanley(BU)Thanks to DTRA

Electricity, Communication,TransportWater …..

Two types of links:ConnectivityDependency

For want of a nail the shoe was lost.For want of a shoe the horse was lost.For want of a horse the rider was lost.For want of a rider the battle was lost.For want of a battle the kingdom was lost.And all for the want of a horseshoe nail.

• Modern systems are coupled together and therefore should be modeled as interdependent networks.

• Node in A fails node in B fails

• Node in B fails node in C fails

• This leads to the cascade of failures

The modelof mutual percolation

Giant component of network A at the I stage of the cascade

Giant component of network B at the II stage of the cascade

Blackout in Italy (28 September 2003)

CASCADE OF FAILURES

Railway network, health care systems, financial services, communication systems

Power grid

InternetSCADA

Cyber Attacks-CNN Simul.02/10

Rosato et allInt. J. of Crit.Infrastruct. 4,63 (2008)

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

Robustness of a single network: PercolationRemove randomly (or targeted) a

fraction 1-p nodes

ER: 1/

SF, ( ) (2 3) :

0 very robust

c

c

p k

p k k

p

FOR RANDOM REMOVAL

ORDER PARAMETER:

μ∞(p) Size of the largest

connected component (cluster)

2nd order

(ER) (SF)

Broader degree-more robust

Breakdown threshold cp

μ∞(p) can be expressed in terms ofgenerating functions of the degreedistribution

In the infinite randomly connected networks, the probability of loops is negligible. These networks can be modeled as branching processes in which each open link of a growing aggregate is randomly attached to a node with k-1 outgoing links with a probability kP(k)/<k>, where P(k) is the degree distribution.For the branching processes the apparatus of generating functions is applicable.

Generating Functions

RANDOM REMOVAL – PERCOLATION FRAMEWORK

Equivalent to random removal

Nodes in the giantcomponent

Nodes in the giantcomponent

Equivalent to random removal

Nodes left afterinitial random removal

Nodes in the giantcomponent

Recursion Relations

=pgA(y)gB(x)

Graphical solution for SF networks

Our model predicts the existence of the all or nothing, first order phase transition, in the vicinity of which failure of a small fraction of nodes in one of the networks may lead to completedisintegration of both metworks

n

after n-cascades of failures

Catastrophic cascadesjust below cp

For a single network 1/cp k

ER networkSingle realizations

RESUTS: THEORY and SIMULATIONS: ER Networks

Removing 1-p nodes in A

( 1)/21/ (2 (1 )); e f fcp k f f f

20.28467, 2.4554 / ; (1 )c cf p k P p f

2.45 / cp k p

FIRST ORDER TRANSITION

min1, 2.4554k

μn(p)

Probability of existence of mutual giant component for ER

PDF of number of cascades n at criticality for ER of size N

1/4n N

IN CONTRAST TO SINGLE NETWORKS, COUPLED NETWORKS ARE MORE VULNERABLE WHEN DEGREE DIST. IS BROADER

All with 4k

Buldyrev, Parshany, Paul, Stanley, S.H. Nature 2010

Networks with correlated degrees

• Why coupled networks with broadrer degree distribution are more vulnerable?

• Because “hubs” in one network can depend on isolated nodes in the other.

• What will happen if the hubs are more likely to depend on hubs?

• This situtation is probably more realistic.

Identical degrees of mutually dependent nodes

Randomly coupled networks:

Correspondenty coupled networks:

PRE 83, 016112 (2011)

Indeed, for CCN the networks the robustness increases with the broadness of the degree distribution.

CCN are more robust than RCN with the same degree distribution

For CCN with 2<<3 pc=0 as

for single networks, and the transition becomes of the second order

For =3,

GENERALIZATION: PARTIAL DEPENDENCE: Theory and Simulations

P

Parshani, Buldyrev, S.H.PRL, 105, 048701 (2010)arXiv:1004.3989

Strong q=0.8:1st Order

Weak q=0.1:2nd Order

q-fraction of dependency links

Strong Coupling Weak Coupling

P P

q=0.8 q=0.1

Analogous to critical point in liquid-gas transition:

PARTIAL DEPENDENCE

Network of Networks

Jianxi Gao et al (arXiv:1010.5829)

n=5

For ER, , full coupling , ALL loopless topologies (chain, star, tree):

Vulnerability increases significantly with m

m=1 known ER- 2nd order

1/cp k

P

ik k

m=1

m=2

m=5

m

{

Multiple random support links

Removal of 1-pA from network A

Removal of 1-pB from network B

Degree distribution support links from A to B is PAB(k)

Degree distribution support links from B to A is PBA(k)

Fraction of autonomous nodes in B is 1-qAB

Fraction of autonomous nodes in A is 1-qBA

Most general case of network of networks

qji fraction of nodes in i which depends on j

Gji generation function of the Degree distribution of the support links

from j to i

Summary and Conclusions

• First statistical physics approach --mutual percolation-- for Interdependent Networks—cascading failures

• Generalization to Partial Dependence: Strong coupling: first order phase transition; Weak: second order

• Generalization to Network of Networks: 50ys of classical percolation is a limiting case. E.g., only m=1 is 2nd order; m>1 are 1st order

• Extremely vulnerable, broader degree distribution - more robust in single network becomes less robust in interacting networks

Network A

Network B

Rich problem: different types ofnetworks and interconnections. Buldyrev et al, NATURE (2010)Parshani et al, PRL (2010);arXiv:1004.3989

Conclusions• Interdependent networks are more vulnerable than

independent networks. They disintegrate via all-or-nothing, first order, phase transition.

• Among the interdependent networks with the same average degree the networks with broader degree distribution are the most vulnerable.

• Scale free interdependent networks with 2<λ<3 have non-zero pc

• Our model allows many generalizations: more than two interdependent networks; some nodes are independent, some nodes depend on more than one node, networks embedded in d dimensions, etc.

• Analytical solutions exists for the case of randomly connected uncorrelated networks with arbitrary degree distributions. They are important because they give us general phenomenology as van der Waals does.

Switching in the dynamic Watts opinion model

(1) Visit every node in the network. Set its internal state to ”0”, with probability p. This

part simulates the failures of internal integrity.

(2) Visit every node in the network for the second time. For each node found in internalstate ”0”, we check the time elapsed from its last internal failure. If that time is equal to

τ, the node’s internal state becomes ”1”. This part simulates recovery of internalintegrity from a failure after recovery time has elapsed.

(3) Now we check the neighborhood of each node i. If a node i has more then m neighbors with total state active, we set its external state to ”1”. In contrast, if a node has

less or equal to m active neighbors, with probability p2 the node i is set to have external

state ”0” and with probability 1 − p2 it is set to have external state ”1”.

(4) Determine new total states of nodes, using Table 1. These are used in the next iteration (t + 1).

Rules of the game

Hysteresis

Switching Phenomena

Simulations of ST2 water near the liquid-liquid critical point