Class 10: Robustness Cascades
Prof. Albert-László Barabási
Dr. Baruch Barzel, Dr. Mauro Martino
Network Science: Robustness Cascades March 23, 2011
A SIMPLE STORY (3):
Thex
Network Science: Introduction 2012
ROBUSTNESS IN COMPLEX SYSTEMS
Complex systems maintain their basic functions even under errors and failures
cell  mutations
There are uncountable number of mutations and other errors in our cells, yet, we do not notice their
consequences.
Internet  router breakdowns
At any moment hundreds of routers on the internet are broken, yet, the internet as a whole does not
loose its functionality.
Where does robustness come from?
There are feedback loops in most complex systems that keep tab on the
component’s and the system’s ‘health’.
Could the network structure affect a system’s robustness?
Network Science: Robustness Cascades March 23, 2011
ROBUSTNESS
Could the network structure affect a system’s robustness?
node failure
How do we describe in quantitave terms the breakdown of a
network under node or link removal?
~percolation theory~
Network Science: Robustness Cascades March 23, 2011
PERCOLATION THEORY
p= the probability that a node is occupied
Increasing p
Critical point pc: above pc we have a spanning cluster.
Network Science: Robustness Cascades March 23, 2011
PERCOLATION: CRITICAL EXPONENTS
p – play the role of T in thermal phase transitions
Order parameter:
P∞~(p-pc)β
Correlation length:
ξ~|p-pc|-ν
P∞
probability that a node (or link) belongs to the  cluster
mean distance between two sites on the same cluster
Cluster size:
S~|p-pc|-γ
Average size of finite clusters
•β, ν, γ: critical exponents—characterize the behavior near the phase transition
•The exponents are universal (independent of the lattice)
•pc depends on details (lattice)
•ν and γ are the same for p>pc and p<pc
•For ξ and S take into account all finite clusters
Network Science: Robustness Cascades March 23, 2011
I:
Subcritical
<k> < 1
II:
Critical
<k> = 1
III:
Supercritical
<k> > 1
IV:
Connected
<k> > ln N
N=100
<k>
<k>=0.5
<k>=1
<k>=3
<k>=5
ROBUSTNESS
Could the network structure contribute to robustness?
node failure
How do we describe in quantitave terms the breakdown of a
network under node removal?
~percolation theory~
Network Science: Robustness Cascades March 23, 2011
ROBUSTNESS: INVERSE PERCOLATION TRANSITION
Remove
nodes
Unperturbed
network
Remove
nodes
Gian Component
Persists
P
1
Network
Collapses
P∞:probability that a node
belongs to the giant
component
f: fraction of removed nodes.
0
fc
f
Network Science: Robustness Cascades March 23, 2011
Damage is modeled as an inverse percolation process
f= fraction of removed nodes
Graph
Component
structure
S
fc
f
(Inverse Percolation phase transition)
Network Science: Robustness Cascades March 23, 2011
BOTTOM LINE: ROBUSTNESS OF REGULAR NETWORKS IS WELL UNDERSTOOD
f=0: all nodes are
part of the giant
component, i.e.
S=N, P∞=1
P
1
0
fc
f
0<f<fc: the network is
fragmented into many clusters
with average size S~|p-pc|-γ
f>fc: the network
collapses, falling into
many small clusters;
there is a giant component;
the probability that a node
belongs to it: P∞~(p-pc)β
giant component
disappears
Network Science: Robustness Cascades March 23, 2011
ROBUSTNESS: OF SCALE-FREE NETWORKS
The interest in the robustness problem has three origins:
Robustness of complex systems is an important problem in many areas
Many real networks are not regular, but have a scale-free topology
In scale-free networks the scenario described above is not valid
Albert, Jeong, Barabási, Nature 406 378 (2000)
Network Science: Robustness Cascades March 23, 2011
ROBUSTNESS OF SCALE-FREE NETWORKS
Scale-free networks do not appear to
break apart under random failures.
Reason: the hubs.
The likelihood of removing a hub is small.
1
S
0
1
f
Albert, Jeong, Barabási, Nature 406 378 (2000)
Network Science: Robustness Cascades March 23, 2011
MALLOY-REED CRITERIA: THE EXISTENCE OF A GIANT COMPONENT
A giant cluster exists if each node is connected to at least two other nodes.
The average degree of a node i linked to the GC, must be 2, i.e.
Bayes’ theorem
P(ki|i <-> j): joint probability that a node has degree ki and is connected to nodes i and j.
For a randomly connected network (does NOT mean random network!) with P(k):
i can choose between N-1 nodes to
link to, each with probability 1/(N-1).
I can try ki times.
< k2 >
kº
=2
<k>
κ>2: a giant cluster exists
κ<2: many disconnected clusters
Network Science: Robustness Cascades March 23, 2011
Malloy, Reed, Random Structures and Algorithms (1995); Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
Probability Distribution Function
(PDF)
Apply the Malloy-Reed Criteria to an Erdos-Renyi Network
Discrete Formulation
Continuum Formulation
-binomial distribution-
-Poisson distribution-
æ N -1ö k
(N -1)-k
P(k) = ç
÷ p (1- p)
è k ø
P(k) = e
-<k>
< k >k
k!
< k >= (N -1) p
< k >=< k >
< k 2 >= p(1- p)(N -1) + p2 (N -1)2
< k 2 >=< k > (1+ < k >)
s k = (< k 2 > - < k >2 )1/ 2 = [ p(1- p)(N -1)]1/ 2
s k = (< k 2 > - < k >2 )1/ 2 =< k >1/ 2
Network Science: Robustness Cascades March 23, 2011
Apply the Malloy-Reed Criteria to an Erdos-Renyi Network
A giant cluster exists if each node is connected to at least two other nodes.
< k2 >
kº
=2
<k>
κ>2: a giant cluster exists;
κ<2: many disconnected clusters;
< k >=< k >
< k 2 >=< k > (1+ < k >)
s k = (< k 2 > - < k >2 )1/ 2 =< k >1/ 2
< k 2 > < k > (1+ < k >)
kº
=
= 1+ < k >= 2
<k >
<k >
< k >=1
Malloy-Reed; Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
Network Science: Robustness Cascades March 23, 2011
< k2 >
kº
=2
<k>
S
Graph
Component
structure
RANDOM NETWORK: DAMAGE IS MODELED AS AN INVERSE OERCOLATION PROCESS
f
fc
f= fraction of removed nodes
kc : <k>=1
<k>
Network Science: Robustness Cascades
(Inverse percolation phase transition)
March 23, 2011
BREAKDOWN THRESHOLD FOR ARBITRARY P(k)
Problem: What are the consequences of removing a fraction f of all nodes?
At what threshold fc will the network fall apart (no giant component)?
Random node removal changes
the degree of individual nodes [k  k’ ≤k]
the degree distribution [P(k)  P’(k’)]
A node with degree k will loose some links and become a node with degree k’ with probability:
æ k ö k -k'
(1 - f ) k'
ç ÷f
è k'ø
Remove k-k’
links, each with
probability f
k'£ k
Leave k’ links
untouched, each
with probability 1-f
The prob. that we had a k
degree node was P(k), so
the probability that we will
have a new node with
degree k’ :
ækö
P'(k') = å P(k)ç ÷ f k -k' (1 - f ) k'
è k'ø
k =k'
¥
Let us asume that we know <k> and <k2> for the original degree distribution P(k)
 calculate <k’> , <k’2> for the new degree distribution P’(k’).
Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
Network Science: Robustness Cascades March 23, 2011
BREAKDOWN THRESHOLD FOR ARBITRARY P(K)
ækö
P'(k') = å P(k)ç ÷ f k -k' (1 - f ) k'
è k'ø
k =k'
¥
¥
¥
Degree distribution after we removed f fraction of nodes.
¥
¥
k!
k -k'
k'
< k' > f = å k' P'(k') = å k' å P(k)
f
(1 - f ) = å
k'!(k
k')!
k' =0
k' =0 k =k'
k' =0
The sum is done over
the triangel shown in
the right, so we can
replace it with
¥
< k' > f = å
k' =0
k=[k’, ∞)
¥
¥
k(k -1)!
å P(k) (k' -1)!(k - k')! f
k -k'
(1 - f ) k'-1(1 - f )
k =k'
¥
åå
k' =0 k =k'
¥
=å
k =0
k
å
k' =0
k’
¥
¥
k
k(k -1)!
(k -1)!
k -k'
k'-1
å P(k) (k' -1)!(k - k')! f (1 - f ) (1 - f ) =å(1 - f )kP(k) å (k' -1)!(k - k')! f k -k' (1 - f )k' -1 =
k =k'
k =0
k' =0
¥
æ k -1ö k -k'
k'-1
åå(1 - f )kP(k) åçè k' -1÷ø f (1 - f ) =å(1 - f )kP(k) = (1 - f ) < k >
k' =0 k =0
k' =0
k =0
k
¥
k
Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
Network Science: Robustness Cascades March 23, 2011
BREAKDOWN THRESHOLD FOR ARBITRARY P(K)
ækö
P'(k') = å P(k)ç ÷ f k -k' (1 - f ) k'
è k'ø
k =k'
¥
Degree distribution after we removed f fraction of nodes.
¥
< k' > f =< k'(k' -1) - k' > f = å k'(k' -1)P'(k')- < k' > f
2
k' =0
k=[k’, ∞)
The sum is done over
the triangel shown in
the right, i.e. we can
replace it with
¥
< k'(1 - k') > f = å
k' =0
¥
¥
åå
k' =0 k =k'
¥
=å
k =0
k
å
k' =0
k’
¥
¥
k
k(k -1)(k - 2)! k -k'
(k - 2)!
k -2'
2
2
å P(k) (k' -2)!(k - k')! f (1 - f ) (1 - f ) =å(1 - f ) k(k -1)P(k) å (k' -2)!(k - k')! f k -k' (1 - f )k' -2 =
k =k'
k =0
k' =0
¥
æ k - 2ö k -k'
k' -2
åå(1 - f ) k(k -1)P(k) åçè k' -2÷ø f (1 - f ) =å(1 - f )2 k(k -1)P(k) = (1 - f )2 < k(k -1) >
k' =0 k =0
k' =0
k =0
k
¥
k
2
< k'2 > f =< k'(k' -1) - k' > f = (1 - f )2 (< k 2 > - < k >) - (1 - f ) < k >= (1 - f )2 < k 2 > + f (1 - f ) < k >
Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
Network Science: Robustness Cascades March 23, 2011
BREAKDOWN THRESHOLD FOR ARBITRARY P(K)
Robustness: we remove a fraction f of the nodes.
At what threshold fc will the network fall apart (no giant component)?
Random node removal changes
the degree of individuals nodes [k  k’ ≤k)
the degree distribution [P(k)  P’(k’)]
< k' > f = (1 - f ) < k >
kº
< k'2 > f = (1 - f )2 < k 2 > + f (1 - f ) < k >
Breakdown threshold:
fc = 1 -
< k' 2 > f
< k' > f
=2
κ>2: a giant cluster exists
κ<2: many disconnected clusters
1
k2
1
-1
k
f<fc: the network is still connected (there is a giant cluster)
0
f>fc: the network becomes disconnected (giant cluster vanishes)
Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
S
fc
f
Network Science: Robustness Cascades March 23, 2011
ROBUSTNESS OF SCALE-FREE NETWORKS
Scale-free networks do not appear to
break apart under random failures.
Reason: the hubs.
The likelihood of removing a hub is small.
1
S
0
1
f
Albert, Jeong, Barabási, Nature 406 378 (2000)
Network Science: Robustness Cascades March 23, 2011
ROBUSTNESS OF SCALE-FREE NETWORKS
fc = 1 -
Scale-free networks do not appear to break apart
under random failures. Why is that?
<k
m
>= (g -1)K min
g -1
K max
òk
K min
< k m >=
m -g
[
k2
k
K max
(g -1)
-1
m -g +1
dk =
K gmin
k
[
] K min
(m - g +1)
(g -1)
-1
K gmin
K max m -g +1 - K min m -g +1
(m - g +1)
1
-1
K max = K min N
1
g -1
]
(2 - g ) K max 3-g - K min 3-g
2 -g
g > 3: k =
=
K
(3 - g ) K max 2-g - K min 2-g
3 - g min
< k 2 > (2 - g ) K max 3-g - K min 3-g
=
<k>
(3 - g ) K max 2-g - K min 2-g
(2 - g ) K max 3-g - K min 3-g
2 -g
-2
3 >g > 2: k =
=
K max 3-g Kgmin
2-g
2-g
(3 - g ) K max
- K min
3 -g
(2 - g ) K max 3-g - K min 3-g
2 -g
2 > g > 1: k =
K
2-g
2-g =
(3 - g ) K max
- K min
3 - g max
Network Science: Robustness Cascades March 23, 2011
ROBUSTNESS OF SCALE-FREE NETWORKS
1
fc = 1 k -1
ì
K min
ï
<k >
2 -g
-2
íK max 3-g K gmin
k=
=
<k>
3 -g ï
K max
î
g >3
3 >g >2
2 > g >1
2
K max = K min N
1
g -1
γ>3: κ is finite, so the network will break apart at a finite fc that depens on Kmin
γ<3: κ diverges in the N ∞ limit, so fc  1 !!!
for an infinite system one needs to remove all the nodes to break the system.
For a finite system, there is a finite but large fc that scales with the system size as:
Internet: Router level map, N=228,263; γ=2.1±0.1;
AS level map, N= 11,164; γ=2.1±0.1;

κ=28
κ=264

k @1 - CN
-
3-g
g -1
fc=0.962
fc=0.996
Network Science: Robustness Cascades March 23, 2011
NUMERICAL EVIDENCE
Scale-free random graph with
P ( k )  Ak  , with k  m ,...K
1
if   3
 2
m 1
 3
1
fc  1 
if 2    3
  2   2 3 
m K
1
3
  2 .5
fc  1 
K  400
  3 .5
K  25
Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
Infinite scale-free networks with   3 do not break down under
random node failures.
Network Science: Robustness Cascades March 23, 2011
SIZE OF THE GIANT COMPONENT DURING RANDOM DAMAGE –WITHOUT PROOF-
S: size of the giant component.
(i) γ>4: S≈f-fc (similar to that of a random graph)
(i) 3>γ>4: S≈(f-fc)1/(γ-3)
(i) γ<3: fc =0 and S≈f1+1/(3-γ)
R. Cohen, D. ben-Avraham, S. Havlin,
Percolation critical exponents in scale-free networks
Phys. Rev. E 66, 036113 (2002);
See also: Dorogovtsev S, Lectures on Complex Networks, Oxford, pg44
Network Science: Robustness Cascades March 23, 2011
INTERNET’S ROBUSTNESS TO RANDOM FAILURES
failure
attack
Internet
1
fc = 1 k -1
R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)
Internet: Router level map, N=228,263; γ=2.1±0.1;
AS level map, N= 11,164; γ=2.1±0.1;
κ=28

fc=0.962
κ=264

fc=0.996
Internet parameters: Pastor-Satorras & Vespignani, Evolution and Structure of the Internet: Table 4.1 & 4.4
Network Science: Robustness Cascades March 23, 2011
Achilles’ Heel of scale-free networks
1
Attacks
Failures
  3 : fc=1
S
(R. Cohen et al PRL, 2000)
0
fc
Albert, Jeong, Barabási, Nature 406 378 (2000)
f
1
Network Science: Robustness Cascades March 23, 2011
Attack threshold for arbitrary P(k)
Attack problem: we remove a fraction f of the hubs.
At what threshold fc will the network fall apart (no giant component)?
Hub removal changes
the maximum degree of the network [Kmax  K’max ≤Kmax)
the degree distribution [P(k)  P’(k’)]
A node with degree k will loose some links because some of its neighbors will vanish.
Claim: once we correct for the changes in Kmax and P(k),we are back to the robustness problem.
That is, attack is nothing but a robusiness of the network with a new Kmax and P(k).
fc
Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
f
Network Science: Robustness Cascades March 23, 2011
Attack threshold for arbitrary P(k)
Attack problem: we remove a fraction f of the hubs.
the maximum degree of the network [Kmax  K’max ≤Kmax) `
If we remove an f fraction of hubs, the maximum degree changes:
K max
ò P(k)dk =
f
'
K max
K max
ò P(k)dk = (g -1)K
'
K max
æ K min ög -1
ç
÷ = f
è K'max ø
g -1
min
g -1 g -1 1-g
g
k dk =
K min (K max - K'1max )
1-g
K max
ò
-g
'
K max
K' max = K min f
Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
1
1-g
As K’max ≤Kmax
we can ignore
the Kmax term
 The new maximum degree after
removing f fraction of the hubs.
Network Science: Robustness Cascades March 23, 2011
Attack threshold for arbitrary P(k)
Attack problem: we remove a fraction f of the hubs.
the degree distribution changes [P(k)  P’(k’)]
A node with degree k will loose some links because some of its neighbors will vanish.
Let us calculate the fraction of links removed ‘randomly’ , f’, as a consequence of we removing f
K
fraction of hubs.
ò kP(k)dk
max
0
K max
ò kP(k)dk
f '=
'
K max
<k>
=
1
1 g -1 g -1 2-g
-1
(g -1)K gmin
k1-g dk =
K (K
ò
<k>
< k > 2 - g min max
'
K max
1 g -1 g -1 2-g
f '= K K f
< k > 2 - g min min
< k m >= -
(g -1)
m
K min
(m - g + 1)
K max
2-g
1-g
1 g -1
=K f
< k > 2 - g min
f '= f
2-g
1-g
(g -1)
K
(2 - g ) min
Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
< k >= -
as K’max ≤Kmax
1 g -1 g -1 2-g
g
- K'2K K'
max ) = < k > 2 - g min max
2-g
1-g
For γ2, f’1, which means that even
the removal of a tiny fraction of hubs will
destroy the network. The reason is that
for γ=2 hubs dominate the network
Network Science: Robustness Cascades March 23, 2011
Attack threshold for arbitrary P(k)
Attack problem: we remove a fraction f of the hubs.
At what threshold fc will the network fall apart (no giant component)?
Hub removal changes
1
1
the maximum degree of the network [Kmax  K’max ≤Kmax) K'
max  K min f
2
the degree distribution [P(k)  P’(k’)]
A node with degree k will loose some links because some of its neighbors will vanish. f '  f 1
Claim: once we correct for the changes in Kmax and P(k), we are back to the robustness problem.

That is, attack is nothing but a robustness of the network with a new K’max and f’.
1
f ' 1
 '1
K min

2  
3
 2

K
 max K min
3 
K max

 k '2 
 k2 


'


 k'
(1  f c )  k  1  f c
 3
3 2
2   1
Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
fc
2
1
2 
 3

2
K min f c 1 1


3 
Network Science: Robustness Cascades March 23, 2011
Attack threshold for arbitrary P(k)
Attack problem: we remove a fraction f of the hubs.
At what threshold fc will the network fall apart (no giant component)?
fc
2-g
1-g
2 -g
æ 3-g
ö
=2+
K min ç f c 1-g -1÷
è
ø
3 -g
fc
•fc depends on γ; it reaches its max for γ<3
•fc depends on Kmin (m in the figure)
•Most important: fc is tiny. Its maximum reaches
only 6%, i.e. the removal of 6% of nodes can
destroy the network in an attack mode.
•Internet: γ=2.1, so 4.7% is the threshold.
Figure: Pastor-Satorras & Vespignani, Evolution and
Structure of the Internet: Fig 6.12
Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).
Network Science: Robustness Cascades March 23, 2011
Application: ER random graphs
Consider a random graph with connection probability p such that
at least a giant connected component is present in the graph.
Find the critical fraction of removed
nodes such that the giant connected
component is destroyed.
fc  1 
1
k02
k0
 1
1
1
1
 1
pN
k0
S
Minimum
damage
l
The higher the original average degree, the larger damage the
Network Science: Robustness Cascades
network can survive.
March 23, 2011
Achilles’ Heel of scale-free networks
1
Attacks
Failures
  3 : fc=1
S
(R. Cohen et al PRL, 2000)
0
fc
Albert, Jeong, Barabási, Nature 406 378 (2000)
f
1
Network Science: Robustness Cascades March 23, 2011
Achilles’ Heel of complex networks
failure
attack
Internet
R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)
Network Science: Robustness Cascades March 23, 2011
Historical Detour: Paul Baran and Internet
1958
Network Science: Robustness Cascades March 23, 2011
Numerical simulations of network resilience
Two networks with equal number of nodes and edges
• ER random graph
• scale-free network (BA model)
Study the properties of the network as an increasing fraction f of the
nodes are removed.
Node selection: random (errors)
the node with the largest number of edges (attack)
Measures: the fraction of nodes in the largest connected cluster, S
the average distance between nodes in the largest
cluster, l
R. Albert, H. Jeong, A.-L. Barabási, Nature 406, 378 (2000)
Network Science: Robustness Cascades March 23, 2011
Scale-free networks are more error tolerant, but
also more vulnerable to attacks
S
• squares: random failure
• circles: targeted attack
l
Failures: little effect on
the integrity of the
network.
Attacks: fast breakdown
Network Science: Robustness Cascades March 23, 2011
Real scale-free networks show the same dual
behavior
S
S
l
l
• blue squares: random failure
• red circles: targeted attack
• open symbols: S
• filled symbols: l
• break down if 5% of the nodes are eliminated selectively (always
the highest degree node)
• resilient to the random failure of 50% of the nodes.
Similar results have been obtained for metabolic networks and
food webs.
Network Science: Robustness Cascades
March 23, 2011
Cascades
Potentially large events triggered by small initial shocks
• Information cascades
social and economic
systems
diffusion of innovations
• Cascading failures
infrastructural networks
complex organizations
Network Science: Robustness Cascades March 23, 2011
Cascading Failures in Nature and Technology
Blackout
Earthquake
Flows of physical quantities
Cascades depend on
•
•
•
•
•
•
•
congestions
instabilities
Overloads
Avalanche
Structure of the network
Properties of the flow
Properties of the net elements
Science: Robustness Cascades
BreakdownNetwork
mechanism
March 23, 2011
Northeast Blackout of 2003
Origin
A 3,500 MW power surge (towards Ontario)
affected the transmission grid at 4:10:39 p.m.
EDT. (Aug-14-2003)
Before the blackout
After the blackout
Consequences
More than 508 generating 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
March 23, 2011
Network Science: Robustness Cascades March 23, 2011
Cascades Size Distribution of Blackouts
Unserved energy/power magnitude (S) distribution
P(S) ~ S −α, 1< α < 2
Probability of energy
unserved during North
American blackouts
1984 to 1998.
Source
Exponent
Quantity
North America
2.0
Power
Sweden
1.6
Energy
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 March 23, 2011
Cascades Size Distribution of Earthquakes
Earthquake size S distribution
Earthquakes during 1977–2000.
P(S) ~ S −α,α ≈ 1.67
Network Science: Robustness Cascades
Y. Y. Kagan, Phys. Earth Planet. Inter. 135 (2–3), 173–209 (2003)
March 23, 2011
Failure Propagation Model
Undercritical
<k>
Critical
Network
falls apart
(<k>=1)
Overcritical
Initial Setup
•Random graph with N nodes
•Initially each node is functional.
Cascade
•Initiated by the failure of one node.
•fi : fraction of failed neighbors of node i. Node i
fails if fi is greater than a global threshold φ.
φ =0.4
● Overcritical
□ Critical
f = 1/2
f = 1/2
f=0
f = 1/2
f = 1/3
f = 2/3
Erdos-Renyi network
P(S) ~ S −3/2
D. Watts, PNAS 99, 5766-5771 (2002)
Network Science: Robustness Cascades March 23, 2011
Overload Model
Critical
Undercritical
Overcritical
Lmin
Overcritical
Critical
Undercritical
Initial Conditions
•N Components (complete graph)
•Each components has random initial load Li
drawn at random uniformly from [Lmin, 1].
Cascade
•Initiated by the failure of one component.
•Component fail when its load exceeds 1.
•When a component fails, a fixed amount P is
transferred to all the rests.
Li =0.95
Li =0.8
P=0.15
Li =0.7
Li =0.9
Li =0.85
Li =1.05
P(S) ~ S −3/2
Network
Science:
Robustness
Cascades
I. Dobson, B. A. Carreras, D. E. Newman, Probab. Eng. Inform. Sci.
19,
15-32
(2005)
March 23, 2011
Self-organized Criticality (BTW Sandpile Model)
Homogenous case
Scale-free network
Initial Setup
•Random graph with N nodes
•Each node i has height hi = 0.
Cascade
•At each time step, a grain is added at a randomly
chosen node i: hi ← hi +1
•If the height at the node i reaches a prescribed
threshold zi = ki, then it becomes unstable and all
the grains at the node topple to its adjacent
nodes: hi = 0 and hj ← hj +1
•if i and j are connected.
Homogenous network: <k2> converged
P(S) ~ S −3/2
Scale-free network : pk ~ k-γ (2<γ<3)
P(S) ~ S −γ/(γ −1)
Network Science: Robustness Cascades March 23, 2011
K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003)
Branching Process Model
Branching Process
Starting from a initial node, each node in
generation t produces k number of
offspring nodes in the next t + 1
generation, where k is selected randomly
from a fixed probability distribution qk=pk-1.
Hypothesis
Fix <k>=1 to be
critical  power
law P(S)
• No loops (tree structure)
• No correlation between branches
Narrow distribution: <k2> converged
P(S) ~ S −3/2
Fat tailed distribution: qk ~ k-γ (2<γ<3)
P(S) ~ S −γ/(γ −1)
K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003)Network Science: Robustness Cascades
March 23, 2011
Short Summary of Models: Universality
Models
Networks
Exponents
Failure Prorogation Model
ER
1.5
Overload Model
Complete Graph
1.5
BTW Sandpile Model
ER/SF
1.5 (ER)
γ/(γ - 1)(SF)
Branching Process Model
ER/SF
1.5 (ER)
γ/(γ - 1)(SF)
Universal for homogenous networks
P(S) ~ S −3/2
Same exponent for percolation too
(random failure, attacking, etc.)
Network Science: Robustness Cascades March 23, 2011
Explanation of the 3/2 Universality
Simplest Case: q0 = q2 = 1/2, <k> = 1
S: number of nodes
X: number of open branches
S = 2, X = 0
k= 0
½ chance
S = S+1
X = X -1
S = 2, X = 2
S=1
X=1
k=2
½ chance
S = S+1
X = X+1
X
X >0, Branching process stops
when X = 0
S
Dead
X
Explanation of 3/2 Universality
Dead
S
Equivalent to 1D random walk model, where X and S are the position and time , respectively.
Question: what is the probability that X = 0 after S steps?
First return probability ~ S-3/2
M. Ding, W. Yang, Phys. Rev. E. 52, 207-213 (1995)
Size Distribution of Branching Process (Cavity Method)
S1
k=0
S2
k=1
S=1
P (S ) 
S1
k=2
S = 1+S1+S2
S = 1+S1
q0 (1)  q1  P( S1 ) (1  S1  S ) q2
S1
 P( S
1
) P( S 2 ) (1  S1  S 2  S )  
S1 , S 2
æ
P(S) = å qk ç
ç å P(S1 )P(S2 )
k
è S1 , Sk
ö
P(Sk )d (1 + å S j - S)÷
÷
j =1
ø
k
Network
Science: Robustness Cascades
K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Physica A 346, 93-103
(2005)
March 23, 2011
Solving the Equation by Generating Function
Definition:
GS(x) = ΣS=0 P(S)xS
Gk(x) = Σk=0 qkxk
Property:
GS(1) = Gk(1) = 1
GS’(1) = <S>, Gk’(1) = <k>
k



P ( S )   qk   P ( S1 ) P ( S 2 )  P ( S k ) (1   S j  S ) 

k
j 1
 S1 ,S k


1  S j
j
GS ( x )   q k 
P
(
S
)

P
(
S
)
x

1
k
 S ,S
k
k
 1
 xGk (GS ( x))
Phase Transition
<S> = GS’(1) = 1+ Gk’(1) GS’(1) = 1 + <k> <S>, then
<S> = 1/(1- <k>)
The average size <S> diverges at <k>c = 1




q
k
xGS ( x ) k
k
Network Science: Robustness Cascades March 23, 2011
Finding the Critical Exponent from Expansion
Definition:
GS(x) = ΣS=0 P(S)xS
Gk(x) = Σk=0 qkxk
Theorem:
If P(k) ~ k-γ (2<γ<3), then for δx < 0, |δx| << 1
G(1+ δx) = 1 + <k>δx + <k(k-1)/2> (δx)2 + … + O(|δx|γ - 1)
P(S) ~ S −α,1< α < 2
GS(1+ δx) ≈ 1 + A|δx|α -1
Homogenous case: <k2> converged
<k> = 1, <k2> < ∞
Gk(1+ δx) ≈ 1 + δx + Bδx2
Inhomogeneous case: <k2> diverged
<k> = 1, qk ~ k-γ (2<γ<3)
Gk(1+ δx) ≈ 1 + δx + B|δx|γ - 1
Network Science: Robustness Cascades March 23, 2011
Critical Exponent for Homogenous Case
Homogenous case
Gk(1+ δx) ≈ 1 + δx + Bδx2
GS(1+ δx) ≈ 1 + A|δx|α -1
GS(x) = xGk(GS(x))
GS(x) ≈ 1 + A|δx|α -1
xGk(GS(x)) ≈ (1+δx)[1+ (GS(1+δx)-1) + B (GS(1+δx)-1)2]
≈ (1+δx)[1+ A|δx|α -1 + AB|δx|2α -2]
= 1 + A|δx|α -1 + AB|δx|2α -2 + δx + O(|δx|α)
The lowest order reads AB|δx|2α -2 + δx = 0, which requires
2α -2 = 1and A = 1/B. Or,
α = 3/2
Network Science: Robustness Cascades March 23, 2011
Critical Exponent for Inhomogeneous Case
Inhomogeneous case
Gk(1+ δx) ≈ 1 + δx + B|δx|γ - 1
GS(1+ δx) ≈ 1 + A|δx|α -1
GS(x) = xGk(GS(x))
GS(x) ≈ 1 + A|δx|α -1
xGk(GS(x)) ≈ (1+δx)[1+ (GS(1+δx)-1) + B |GS(1+δx)-1|γ -1]
≈ (1+δx)[1+ A|δx|α -1 + AB|δx|(α -1)(γ -1)]
= 1 + A|δx|α -1 + AB|δx|(α -1)(γ -1) + δx + O(|δx|α)
The lowest order reads AB|δx|(α -1)(γ -1) + δx = 0, which requires
(α -1)(γ -1) = 1and A = 1/B. Or,
α = γ/(γ −1)
Network Science: Robustness Cascades March 23, 2011
Compare the Prediction with the Real Data
P( S ) ~ S

 3
2 3
 3 / 2,
,  
 /(  1),
Blackout
Source
Exponent
Quantity
North America
2.0
Power
Sweden
1.6
Energy
Norway
1.7
Power
New Zealand
1.6
Energy
China
1.8
Energy
Earthquake α ≈ 1.67
I. Dobson, B. A. Carreras, V. E. Lynch, D. E. Newman, CHAOS 17, 026103 (2007)
Y. Y. Kagan, Phys. Earth Planet. Inter. 135 (2–3), 173–209 (2003)Network Science: Robustness Cascades
March 23, 2011
The end
Network Science: Evolving Network Models February 14, 2011