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Generalized Percolation in Complex Networks
Shlomo Havlin
Daniel ben- Avraham Clarkson University
Boston Universtiy
Reuven Cohen Tomer KaliskyAlex Rozenfeld
Bar-Ilan University
Eugene StanleyLidia BraunsteinSameet Sreenivasan
Gerry Paul
Outline Percolation – Graph Theory: Introduction Complex Networks: Theory vs Experiment Degree Distribution, Critical Concentration, Distance Generalized Networks: Broad Degree Distribution –
Anomalous physics Applications: Efficient Immunization Strategy Optimal path - Optimal transport Optimize Network Stability
References
Cohen et al Phys. Rev. Lett. 85, 4626 (2000); 86, 3682 (2001) Rozenfeld et al Phys. Rev. Lett. 89, 218701 (2002)Cohen and Havlin Phys. Rev. Lett. 90, 58705 (2003)
Cohen et al Phys. Rev. Lett. 91,247901 (2003)Phys. Rev. Lett. 91,247901 (2003)Braunstein et al
Paul et al Europhys. J. B (in press) (cond-mat/0404331)
Percolation and Immunization
0 . 3p r e m o v e o r i m m u n e
0 . 5p r e m o v e o r i m m u n e
Network collapse
Network exists
Percolation theory
Prob. that a site belongs to a spanning cluster
P
cp p0
1 c~P pp
Random Graph Theory
Developed in the 1960’s by Erdos and Renyi. (Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 1960).
Discusses the ensemble of graphs with N vertices and M edges (2M links).
D istribution o f connectiv ity per vertex is P o issonian (exponential), where k is the num ber o f links :
P k ec
kc
k
( )!
, c k
M
N
2
D istance d=log N -- SM AL L W O R L D
Known Results
Phase transition at average connectivity, :
k 1 No spanning cluster (giant component) of order logN k 1 A spanning cluster exists (unique) of order N
k 1 The largest cluster is of order N 2 3/
k 1
S i z e o f t h e s p a n n i n g c l u s t e r i s d e t e r m i n e d b y t h e s e l f - c o n s i s t e n t e q u a t i o n :
P e k P
1
B e h a v i o r o f t h e s p a n n i n g c l u s t e r s i z e n e a r t h e t r a n s i t i o n i s l i n e a r :
)( ppP c , 1 , w h e r e p i s t h e p r o b a b i l i t y o f d e l e t i n g a s i t e ,
1 1 /cp k
Percolation on a Cayley Tree
Contains no loops
Connectivity of each node is fixed (zconnections)
Behavior of the spanning cluster size near the transition is linear:
)( ppP c , 1
Critical threshold:
1
1cpz
In Real World - Many Networks are non-Poissonian
P k e
k
kk
k
( )!
( )
0
c k m k KP k
o t h e r w i s e
Internet Network
Networks in Physics
Erdös Theory is Not Valid Distribution
Generalization of Erdös Theory: Cohen, Erez, ben-Avraham, Havlin, PRL 85, 4626 (2000)
Epidemiology Theory: Vespignani, Pastor-Satoral, PRL (2001), PRE (2001)
Modelling: Albert, Jeong, Barabasi (Nature 2000) Cohen, Havlin,
Phys. Rev. Lett. 90, 58701(2003)
1k
11
cp
Infectious disease
Malaria 99%Measles 90-95%Whooping cough 90-95%Fifths disease 90-95%Chicken pox 85-90%Mumps 85-90%Rubella 82-87%Poliomyelitis 82-87%Diphtheria 82-87%Scarlet fever 82-87%Smallpox 70-80%
INTERNET 99%
Critical concentration
Stability and Immunization
Critical concentration 30-50%
Distance
Almost constant (Metabolic Networks,
Jeong et. al. (Nature, 2000))
Nd log~
Distance in Scale Free Networks
log log
. 2
log3
log log
log
2
3
3
d const
Nd
N
d N
d N
Cohen, Havlin Phys. Rev. Lett. 90, 58701(2003)
Cohen, Havlin and ben-Avraham, in Handbook of Graphs and Networks
Eds. Bornholdt and Shuster (Willy-VCH, NY, 2002) Chap. 4
Confirmed also by: Dorogovtsev et al (2002), Chung and Lu (2002)
Ultra Small World
Small World
(Bollobas, Riordan, 2002)
(Bollobas, 1985)(Newman, 2001)
( ) ~P k k
Efficient Immunization Strategies:
Acquaintance Immunization
Critical Threshold Scale Free
Cohen et al Phys. Rev. Lett. 91 , 168701 (2003)
1
11p
:
1
11p
22
0
2
0
0
k
k
kk
k
kK
PoissonFor
k
kK
K
c
c
cp
Random
Acquaintance
Intentional
General result:
robust
vulnerable
Poor immunization
Efficient immunization
Not only critical thresholds but also critical exponents are different!
THE UNIVERSALITY CLASS DEPENDS ON THE WAY CRITICALITY REACHED
Critical ExponentsUsing the properties of power series (generating functions) near a singular point
(Abelian methods), the behavior near the critical point can be studied.
(Diff. Eq. Molloy & Reed (1998) Gen. Func. Newman Callaway PRL(2000), PRE(2001))
For random breakdown the behavior near criticality in scale-free networks is different than for random graphs or from mean field percolation. For intentional attack-same as mean-field.
P p pc ~ ( )
12 3
31
3 43
1 4
(known mean field)
Size of the infinite cluster:
n ss ~
2 34
2
2.5 4
(known mean field)
Distribution of finite clusters at criticality:
Even for networks with , where and are finite, the critical exponents change from the known mean-field result . The order of the phase transition and the exponents are determined by .
13 4 k
2k
3k
.Optimal Distance - Disorder
lmin= 2(ACB)
lopt= 3(ADEB)
D
A
E
BC
2
35
4 1
.
. . . = weight = price, quality, time…..
minimal optimal path
Weak disorder (WD) – all contribute to the sum (narrow distribution)
Strong disorder (SD)– a single term dominates the sum (broad distribution)
SD – example: Broadcasting video over the Internet.
A transmission at constant high rate is needed.
The narrowest band width link in the path
between transmitter and receiver controls the rate.
i
iw
iw
iw
Path fromA to B
Random Graph (Erdos-Renyi)Scale Free (Barabasi-Albert)
Small World (Watts-Strogatz)
Z = 4
Optimal path – strong disorder
Random Graphs and Watts Strogatz Networks
CONSTANT SLOPE
0n - typical range of neighborhood
without long range links
0n
N- typical number of nodes with
long range links
31
~ Nlopt Analytically and Numerically
LARGE WORLD!!
Compared to the diameter or average shortest path
Nl log~min (small world)
N – total number of nodes
Mapping to shortest path at critical percolation
Scale Free +ER – Optimal Path
ER4N
4NlogN
43N
~l
31
31
)1/()3(
opt
Theoretically
+
Numerically
Numerically 32log~ 1 Nlopt
Strong Disorder
Weak Disorder
allforNlopt log~
Diameter – shortest path
32loglog
3loglog/log
3log
~min
N
NN
N
lLARGE WORLD!!
SMALL WORLD!!
For 2 3 min~ exp( )optl l
Braunstein et al Phys. Rev. Lett. 91, 247901 (2003)
Paul et al. Europhys. J. B 38, 187 (2004), (cond-mat/0404331)
Simultaneous waves of targeted and random attacks
SF
Bimodal
- Fraction of targeted
- Fraction of random
Bi-modal
SF
Tanizawa et al. Optimization of Network Robustness to Waves of Targeted and Random Attacks (Cond-mat/0406567)
r = 0.001—0.15 from left to right
Bimodal: fraction of (1-r) having k links and r having links
1
Optimal Bimodal : )/(2 rt ppr
r=0.001
0.01
rrkk /)1(2
Optimal Networks
rt pp /For
22
k
kCondition for connectivity:
,1, rt pp is the only parameter
tp
rp
kkr
kr
K
k
ppk
kkPkP
0
0
)1()()( 00
P(k) changes also due to targeted attack
P(k) changes:
cf - critical threshold
Optimal Bimodal
Given: N=100, ,
and
i.e, 10 “hubs” of degree
using
1.005.0/ rpp rt
rrkk /)1(2
Specific example:
11 k1.2 k
122 k
Conclusions and Applications
•Generalized percolation , >4 – Erdos-Renyi, <4 – novel topology –novel physics.
•Distance in scale free networks <3 : d~loglogN - ultra small world, >3 : d~logN.
•Optimal distance – strong disorder – ER, WS and SF (>4)
scale free
• Scale Free networks (2<λ<3) are robust to random breakdown.
• Scale Free networks are vulnerable to targeted attack on the highly connected nodes.
• Efficient immunization is possible without knowledge of topology, using Acquaintance Immunization.
• The critical exponents for scale-free directed and non-directed networks are different than those in exponential networks – different universality class!
•Large networks can have their connectivity distribution optimized for maximum robustness to random breakdown and/or intentional attack.
31
~ Nlopt3
1
1
~ 3
~ log 2 3
opt
opt
l N for
l N for
Large World
Small World
Large World
-λP(k)~k