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

Cascading failures: Empirical Results

Section 8.5

Cascades: The Domino Effect

Large events triggered by small initjal shocks

Network Science: Robustness Cascades

Northeast Blackout of 2003

Consequences More than 508 generatjng units at 265 power plants shut down during the

• utage. 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

• f 80%.

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

Network Science: Robustness Cascades

Section 8.5

Network Science: Robustness Cascades

Cascades Size Distribution of Blackouts

Probability of energy unserved during North American blackouts 1984 to 1998.

Source Exponent Quantjty North America 2.0 Power Sweden 1.6 Energy Norway 1.7 Power New Zealand 1.6 Energy China 1.8 Energy

Unserved energy/power magnitude (S) distributjon

• I. Dobson, B. A. Carreras, V. E. Lynch, D. E. Newman, CHAOS 17, 026103 (2007)

P(S) ~ S −α, 1< α < 2 P(S) ~ S −α, 1< α < 2

Network Science: Robustness Cascades

Cascades Size Distribution of Earthquakes

P(S) ~ S −α,α ≈ 1.67 P(S) ~ S −α,α ≈ 1.67

Earthquake size S distributjon

• Y. Y. Kagan, Phys. Earth Planet. Inter. 135 (2–3), 173–209 (2003)

Earthquakes during 1977–2000.

Network Science: Robustness Cascades

Information Cascades

p(s)∼s

−α

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

Modeling Cascading failures

Section 8.6

Section 8.6

Section 8.6 Failure Propagation Model

• D. Watus, PNAS 99, 5766-5771 (2002)

Initjal Setup

• Random graph with N nodes
• Initjally each node is functjonal.

Cascade

• Initjated by the failure of one node.
• fi : fractjon of failed neighbors of node i. Node i

fails if fi is greater than a global threshold φ.

Network Science: Robustness Cascades

!" $"

f=1/2 f=0 f=1/3 f=1/2 =0.4

A D E C B

f=1/2 f=2/3

!"

A D E B C

(a) (b)

(a,b) The development of a cascade in a small network in which each node has the same breakdown threshold = 0.4. Initially all nodes are in state 0, shown as green circles. After node A changes its state to 1 (purple), 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.

Section 8.6 Failure Propagation Model

• D. Watus, PNAS 99, 5766-5771 (2002)

Erdos-Renyi network

P(S) ~ S −3/2

Erdos-Renyi network

P(S) ~ S −3/2

Network Science: Robustness Cascades

!" $"

f=1/2 f= f= 1/3 f= 1/2 =0.4

A D E C B

f=1/2 f= 2/3

!"

A D E B C

(a) (b) 2 4 6 8 10 12 14 16

SUBCRIT ICAL SU PERCRITICAL

k

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26

(c )

10 10 10

10

10

10

10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

LOW ER CRIT ICAL POIN T U PPER CRIT ICAL POIN T SU B CRIT ICAL SU PERCRIT ICAL

P(s)

s

)

Section 8.6 Branching Model !" $"

f=1/2 f=0 f=1/3 f=1/2 =0.4

A D E C B

f=1/2 f=2/3

!"

A D E B C

(a) (b)

*$ +$ ,$

• $

.$

A E D C B

Section 8.6 Branching Model

1 2 3 1 2 3 4 5

p =0.5 p =0.5

*$ +$ ,$

• $

.$

x(t)

t

s = tmax +1=6

A E D C B

(a) (c ) (b)

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

Section 8.6 Branching Model

Section 8.6 Branching Model

Building Robustness

Section 8.7

Section 8.7 Building Robustness

k =12/ 7

(a)

k =24/ 7

(b) Can we maximize the robustness of a network to both random failures and targeted attacks without changing the cost?

Section 8.7 Building Robustness

fc

tot

fc

rand

fc

targ . A network’s robustness against random failures is captured by its per- colation threshold fc, which is the fraction of the nodes we must remove for the network to fall apart. To enhance a network's robustness we must increase fc. According to (8 .7 ) fc depends only on k and k2 . Conse- quently the degree distribution which maximizes fc 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 km in and km ax (Figure 8 .2 3 a ,b).

Section 8.7 Building Robustness

fc

tot

fc

rand

fc

targ .

0.5 1 1.5 5 10 15 20 R A ND O M

TARGETED TOTAL

f

c

(c)

pk (1 r) (k kmin)+ r (k kmax) ,

kmax = AN 2/3.

fc

tot

fc

rand

fc

targ .

pk (1 r) (k kmin)+ r (k kmax) ,

kmax = 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) (c ) (b) (d)

Section 8.7 Case Study: Power Grid

1 1,5 2 0,0 0,1 0,2 0,3 0,4 0,5

B

BREAKDOWN

GRO UP 2 G ROUP 1

CONNECTED

˜ K

γ +1

e

˜ K/ γ

(14) it is straightforward to see that: (ln p

c

1)p

c

(15) tK is large enough to ignore the us, an equiv alent network with been built after arandom remov al act that the absenceof correlations re of links. In order to obtain the

5 10 15 k 10

• 3

10

• 2

10

• 1

10 C umulative distribution

U C T E U K A ND IR E LAND ITALY

c

k k P

k

fc

t arg

(e) (f)

Group 2: these networks are more robust to attacks than expected based

• n their degree distribution.

Section 8.8 Summary

Section 8.8 Summary

Section 8.8 Achilles’ Heel

The end

Network Science: Evolving Network Models