Albert-Lszl Barabsi with Emma K. Towlson, Sebastian Ruf, Michael - PowerPoint PPT Presentation

Network Science Class 8: Network Robustness Albert-László Barabási with Emma K. Towlson, Sebastian Ruf, Michael Danziger, and Louis Shekhtman www.BarabasiLab.com

Section 8.5 Cascading failures: Empirical Results

Cascades: The Domino Effect Large events triggered by small initjal shocks Network Science: Robustness Cascades

Northeast Blackout of 2003 Origin A 3,500 MW power surge (towards Ontario) afgected the transmission grid at 4:10:39 p.m. EDT. (Aug-14-2003) Before the blackout Afuer the blackout Consequences More than 508 generatjng units at 265 power plants shut down during the outage. In the minutes before the event, the NYISO-managed power system was carrying 28,700 MW of load. At the height of the outage, the load had dropped to 5,716 MW, a loss of 80%. Network Science: Robustness Cascades

Section 8.5

Network Science: Robustness Cascades

Cascades Size Distribution of Blackouts Unserved energy/power magnitude ( S) distributjon P ( S ) ~ S − α , 1< α < 2 P ( S ) ~ S − α , 1< α < 2 Source Exponent Quantjty Probability of energy North America 2.0 Power unserved during North American blackouts Sweden 1.6 Energy 1984 to 1998. Norway 1.7 Power New Zealand 1.6 Energy China 1.8 Energy I. Dobson, B. A. Carreras, V. E. Lynch, D. E. Newman, CHAOS 17, 026103 (2007) Network Science: Robustness Cascades

Cascades Size Distribution of Earthquakes Earthquake size S distributjon Earthquakes during 1977–2000. P ( S ) ~ S − α , α ≈ 1.67 P ( S ) ~ S − α , α ≈ 1.67 Y. Y. Kagan, Phys. Earth Planet. Inter. 135 (2–3), 173–209 (2003) Network Science: Robustness Cascades

Information Cascades − α p ( s )∼ s

Section 8.5 Empirical Results U.S. aviation map showing congested air- ports as purple nodes, while those with nor- mal traffj c as green nodes. The lines corre- spond to the direct fmights between them on March 12, 2010. The clustering of the con- gested airports indicate that the dealys are not independent of each other, but cascade through the airport network. After [22].

Section 8.5 Empirical Results: Summary

Section 8.6 Modeling Cascading failures

Section 8.6

Section 8.6 Failure Propagation Model (a) =0.4 (b) f=1/2 E E !" A A !" D D f=0 f=1/2 Initjal Setup B B • Random graph with N nodes $" C C f=1/2 • Initjally each node is functjonal. f=1/3 f=2/3 Cascade (a,b) The development of a cascade in a small • Initjated by the failure of one node. network in which each node has the same breakdown threshold = 0.4. Initially all • f i : fractjon of failed neighbors of node i . Node i nodes are in state 0, shown as green circles. After node A changes its state to 1 (purple), fails if f i is greater than a global threshold φ . its neighbors B and E will have a fraction f = 1/ 2 > 0.4 of their neighbors in state 1. Consequently they also fail, changing their state to 1, as shown in (b). In the next time step C and D will also fail, as both have f > 0.4. Consequently the cascade sweeps the whole network, reaching a size s = 5. One can check that if we initially fmip node B , it will not induce an avalanche. D. Watus, PNAS 99, 5766-5771 (2002) Network Science: Robustness Cascades

Section 8.6 Failure Propagation Model (a) =0.4 (b) f=1/2 E E !" A A !" D D f= 0 f=1/2 B B C $" C f= 1/2 f= 1/3 f= 2/3 ) (c ) 10 0 16 LOW ER CRIT ICAL POIN T SUBCRIT ICAL U PPER CRIT ICAL POIN T 14 SU B CRIT ICAL 10 -1 SU PERCRIT ICAL 12 10 -2 10 P(s) k 8 10 -3 6 4 10 -4 SU PERCRITICAL 2 10 -5 0 10 0 10 1 10 2 10 3 10 4 s 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 Erdos-Renyi network Erdos-Renyi network P ( S ) ~ S −3/2 P ( S ) ~ S −3/2 D. Watus, PNAS 99, 5766-5771 (2002) Network Science: Robustness Cascades

Section 8.6 Branching Model (a) =0.4 (b) E D f=1/2 .$ -$ E E !" A A !" A *$ D D f=0 f=1/2 C B B +$ ,$ B $" C C f=1/2 f=1/3 f=2/3

Section 8.6 Branching Model (a) (b) p =0.5 E D .$ -$ A *$ C B +$ ,$ p =0.5 (c ) 3 x(t) 2 1 0 0 1 2 3 4 5 t s = t max +1=6

Section 8.6 Branching Model SUBCRITICAL SUPERCRITICAL CRITICAL (e) (f) (d)

Section 8.6 Branching Model

Section 8.6 Branching Model

Section 8.7 Building Robustness

Section 8.7 Building Robustness (a) (b) k =12/ 7 k =24/ 7 Can we maximize the robustness of a network to both random failures and targeted attacks without changing the cost?

Section 8.7 Building Robustness A network’s robustness against random failures is captured by its per- colation threshold f c , which is the fraction of the nodes we must remove for the network to fall apart. To enhance a network's robustness we ) f c depends only on k and k 2 . Conse- must increase f c . According to (8 .7 quently the degree distribution which maximizes f c needs to maximize k 2 if we wish to keep the cost k fjxed. This is achieved by a bimodal distribution, corresponding to a network with only two kinds of nodes, with degrees k m in and k m ax (Figure 8 .2 3 a ,b) . targ . tot rand f c f c f c

Section 8.7 Building Robustness targ . (c) tot rand f c f c f c 1.5 R A ND O M TARGETED TOTAL 1 k min ) + r ( k p k (1 r ) ( k k max ) , f c 0.5 k max = AN 2/3 . 0 0 5 10 15 20

targ . tot rand f c f c f c k min ) + r ( k p k (1 r ) ( k k max ) , k max = AN 2/3 .

Section 8.7 Halting Cascading Failures Simulations indicate that to limit the size of the cascades we must remove nodes with small loads and links with large excess load in the vicinity of the initial failure. The mechanism is similar to the method used by firefighters, who set a controlled fire in the fire- line to consume the fuel in the path of a wildfire.

Section 8.7 Lazarus Effect

Section 8.7 Case Study: Power Grid (a) (b) (c ) (d)

Section 8.7 Case Study: Power Grid (e) (f) 0 10 ˜ BREAKDOWN c K B ˜ K/ γ γ +1 e (14) 0,5 umulative distribution GRO UP 2 G ROUP 1 0,4 -1 it is straightforward to see that: 10 t arg f c P 0,3 k (ln p 1)p (15) ITALY c c -2 10 0,2 tK is large enough to ignore the U K A ND IR E LAND C us, an equiv alent network with CONNECTED 0,1 U C T E been built after arandom remov al -3 10 act that the absenceof correlations 0,0 0 5 k 10 15 re of links. In order to obtain the 1 1,5 2 k k Group 2: these networks are more robust to attacks than expected based on their degree distribution.

Section 8.8 Summary

Section 8.8 Summary

Section 8.8 Achilles’ Heel

The end Network Science: Evolving Network Models