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