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

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

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

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

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

Blackout in Italy (28 September 2003) SCADA Power grid

Blackout in Italy (28 September 2003) SCADA Power grid

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

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

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

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 complete disintegration of both metworks

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

Probability of existence of mutual giant component for ER

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

IN CONTRAST TO SINGLE NETWORKS, COUPLED NETWORKS ARE MORE VULNERABLE WHEN DEGREE DIST. IS BROADER 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 Correspondenty coupled networks: Randomly 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 Parshani, Buldyrev, S.H. PRL, 105, 048701 (2010) arXiv:1004.3989 Weak q=0.1: 2nd Order Strong q=0.8: 1st Order q-fraction of dependency links

Strong Coupling Weak Coupling q=0.8 q=0.1

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

{ 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:1010.5829)

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

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 of networks 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

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).

Hysteresis

Switching Phenomena

Simulations of ST2 water near the liquid-liquid critical point