1. Sergey Buldyrev Department of Physics Yeshiva University
The Fragility of interdependency: Coupled networks and switching phenomena
Sergey Buldyrev
Department of Physics
Yeshiva University
Collaborators:
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

3. For want of a nail the shoe was lost
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
Electricity,
Communication,
Transport
Water …..
Two types of links:
Connectivity
Dependency

4. The model of 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

5. Blackout in Italy (28 September 2003)
Cyber
Attacks-
CNN
Simul.
02/10
Internet
SCADA
Rosato et all
Int. J. of Crit.
Infrastruct. 4,
63 (2008)
Power grid
CASCADE OF FAILURES
Railway network, health care systems, financial services, communication systems

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

7. Blackout in Italy (28 September 2003)
SCADA
Power grid

8. Blackout in Italy (28 September 2003)
SCADA
Power grid

9. Robustness of a single network: Percolation
(SF)
Remove randomly (or targeted) a
fraction 1-p nodes
ORDER PARAMETER:
μ∞(p) Size of the largest
connected component (cluster)
μ∞(p) can be expressed in terms of
generating functions of the degree
distribution
FOR RANDOM REMOVAL
2nd order
Broader degree-more robust

10. 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.

11. Generating Functions

13. A B RANDOM REMOVAL – PERCOLATION FRAMEWORK Nodes left after
initial random removal
Nodes in the giant
component
Nodes in the giant
component A
Equivalent to
random removal
Equivalent to
random removal
Nodes in the giant
component B

14. Recursion Relations 
=pgA(y)gB(x)

15. 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 complete
disintegration of both metworks

16. RESUTS: THEORY and SIMULATIONS: ER Networks
after n-cascades of failures
Removing 1-p nodes in A
ER network
Single realizations
μn(p)
Catastrophic cascades
just below
FIRST ORDER TRANSITION

17. Probability of existence of mutual giant component for ER

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

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

20. 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.

21. Identical degrees of mutually dependent nodes
Correspondenty coupled networks:
Randomly coupled networks:
PRE 83, (2011)

22. 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,

23. GENERALIZATION: PARTIAL DEPENDENCE: Theory and Simulations
Parshani, Buldyrev, S.H.
PRL, 105, (2010)
arXiv:
Weak q=0.1:
2nd Order
Strong q=0.8:
1st Order
q-fraction of dependency links

24. Strong Coupling
Weak Coupling
q=0.8
q=0.1

25. PARTIAL DEPENDENCE Analogous to critical point
in liquid-gas transition:

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

27. Multiple random support links
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
Removal of 1-pA from network A
Removal of 1-pB from network B

28. 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

29. 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 of
networks and interconnections.
Buldyrev et al, NATURE (2010)
Parshani et al, PRL (2010);
arXiv:

30. 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.

31. Switching in the dynamic Watts opinion model

32. Rules of the game
(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 internal
state ”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 internal
integrity 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).

33. Hysteresis

34. Switching Phenomena

35. Simulations of ST2 water near the liquid-liquid critical point