Percolation in Complex Networks: Optimal Paths and Minimum

Spanning Trees

. 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

Optimal Distance - Disorder

Random Graph (Erdos-Renyi)Scale Free (Barabasi-Albert)

Small World (Watts-Strogatz)

Z = 4

Optimal Path

~ doptopt r min~ dr

WDSD

11.221.132

11.411.373

11.594

15

1226

mindoptd optdd

0.9 0.90.3

0.2

0.6

0.3

0.1

0.2

0.4

0.4

0.1

0.6

0.4

0.3

0.1

0.8

0.7

0.9

0.4

0.1

0.1

0.3

0.6

0.8

Strong Disorder and Percolation

A

B

Optimal path – weak disorderRandom Graphs and Watts Strogatz Networks

Nl log~min

Nlopt log~

pzn 1

0 ∝Typical short

range neighborhood

Crossover from large

to small world

1/ 0 <nNFor

0/log~~ nNopt eNl

Scale Free – Optimal Path – Weak disorder

55.2 …=λ

)exp(~loglog~

log~32

min

min

llThusNl

NlFor

opt

opt

<< λ

min

min

3~ ( ) log

~ log~

opt

opt

Forl A N

l NThus l l

λλ

>

Optimal path – strong disorderRandom Graphs and Watts Strogatz Networks

CONSTANT SLOPE

0n - typical range of neighborhood

without long range links

0nN - typical number of nodes with

long range links

31

~ Nlopt Analytically and Numerically

LARGE WORLD!!

Compared to the diameter or average shortest path or weak disorder

Nl log~min (small world)

N – total number of nodes

Scale Free – Optimal Path

⎪⎪⎩

⎪⎪⎨

⎧

>

=

<<−−

4

4log

43

~3

1

31

)1/()3(

λ

λ

λλλ

N

NN

N

lopt

Theoretically

+

Numerically

Numerically 32log~ 1 <<− λλ Nlopt

Strong Disorder

Weak Disorder

λallforNlopt log~

Diameter – shortest path

⎪⎩

⎪⎨

⎧

<<=

>

32loglog3loglog/log

3log~min

λλ

λ

NNN

Nl

LARGE WORLD!!

SMALL WORLD!!

Braunstein, Buldyrev, Cohen, Havlin, Stanley, Phys. Rev. Lett. 91, 247901 (2003);Cond-mat/0305051

4=λ

Theoretical Approach – Strong Disorder(i) Distribute random numbers 0<u<1 on the links of the network.

(ii) Strong disorder represented by )exp( ii au=ε with .∞→a

(iii) The largest iu in each path between two nodes dominates the sum.

(iv) The optimal path is the path with the min-max

(vi) The optimal path must therefore be on the percolation cluster at criticality.

A

B

What do we know about percolation clusters

at criticality in networks?

(v) Percolation exists if we remove all links with 1i cu p> −

Theoretical Approach – Strong Disorder

(i) Percolation on random networks is like percolation

in or∞→d .cdd =(ii) Since loops can be neglected the optimal path can

be identified with the shortest path on percolation-only

a single path exist between any nodes.

Mass of infinite cluster fdRS ~ dRN ~

For ER 3/26/4/~ NNNS dd f == (see also Erdos-Renyi, 1960))2(,~ 2 =ldlS

3/12/1 ~~~ NSlloptfor ER, WS and SF with :4>

where

From percolation

Thus, λFor SF with 43 << λ ,, fc dd and ld change due to novel topology: )1/()3(~ −− λλNlopt

A

B

Conclusions

Calculate the length of shortest path:

,6=cd

Random Graphs – Erdos Renyi(1960)

Largest cluster at criticality

21

23

4

4

f

f c

dd dS R N N

S N

λλ λ

λ

−− ≤

≥

∼ ∼ ∼

∼

23S N∼

Scale Free networks

Fractal Dimensions

From the behavior of the critical exponents the fractal dimension of scale-free graphs can be deduced.

Far from the critical point - the dimension is infinite - the mass grows exponentially with the distance.

At criticality - the dimension is finite for λ>3 .2 43

2 4ld

λ λλ

λ

−⎧ <⎪⎪ −= ⎨⎪ ≥⎪⎩

Short path dimension:

dS ∼22 43

4 4fd

λ λλ

λ

−⎧ <⎪⎪ −= ⎨⎪ ≥⎪⎩

Fractal dimension:

fdS R∼ 12 43

6 4cd

λ λλ

λ

−⎧ <⎪⎪ −= ⎨⎪ ≥⎪⎩

Embedding dimension:

(upper critical dimension)

The dimensionality of the graphs depends on the distribution!

Random Graph (Erdos-Renyi)Scale Free (Barabasi-Albert)

Small World (Watts-Strogatz)

Z = 4

!)(

kkekP

kk ⟩⟨

= ⟩⟨−

λ−= AkkP )(

Shortest Paths in Scale Free Networks

loglog

. 2

log 3logloglog

2

3

3

d const

NdN

d N

d N

λ

λ

λ

λ

= =

= =

=

= <

>

<

(Bollobas, Riordan, 2002)

(Bollobas, 1985)(Newman, 2001)

Cohen, Havlin and ben-Avraham, in Handbook of Graphs and Networks

eds. Bornholdt and Shuster (Willy-VCH, NY, 2002) Chap.4

Cohen, Havlin Phys. Rev. Lett. 90, 58701(2003)

Also by: Dorogovtsev, Mendes et al (2002), Chung and Lu (2002)

Ultra Small World

Small World

( ) ~P k k λ−

Same as for ER and WS

Optimal path – weak disorderRandom Graphs and Watts Strogatz Networks

Nl log~min

Nlopt log~

pzn 1

0 ∝Typical short

range neighborhood

Crossover from large

to small world

1/ 0 <nNFor

0/log~~ nNopt eNl

Theoretical Approach – Strong Disorder

(i) Distribute random numbers 0<u<1 on the links of the network.

(ii) Strong disorder represented by )exp( ii au=ε with .∞→a(iii) The largest iu in each path between two nodes dominates the sum.

(iv) The min-max are on the percolation cluster where iu .ci pu(v) The optimal path must therefore be on the percolation cluster at criticality.

(vi) Percolation on random networks is like percolation in or

<

∞→d .cdd =(vii) Since loops can be neglected the optimal path can be identified with the shortest path.

Mass of infinite cluster fdRS ~ dRN ~3/26/4/~ NNNS dd f == (see also Erdos-Renyi, 1960)

Since 2 2~ it follows that ~ , ( 2)r S d =3/12/1 ~~~ NSlloptfor ER, WS and SF with :4>

where

Thus,

Thus, λFor SF with 43 << λ ,, fc dd and ld change due to novel topology: )1/()3(~ −− λλNlopt

Scale Free – Optimal Path – Weak disorder

55.2 …=λ

)exp(~loglog~

log~32

min

min

llThusNl

NlFor

opt

opt

<< λ

min

min

~log~

log)(~3

llThusNl

NAlFor

opt

opt λλ >

Optimal path – strong disorderRandom Graphs and Watts Strogatz Networks

CONSTANT SLOPE

0n - typical range of neighborhood

without long range links

0nN - typical number of nodes with

long range links

31

~ Nlopt Analytically and Numerically

LARGE WORLD!!

Compared to the diameter or average shortest path or weak disorder

Nl log~min (small world)

N – total number of nodes

Scale Free – Optimal Path

⎪⎪⎩

⎪⎪⎨

⎧

>

=

<<−−

4

4log

43

~3

1

31

)1/()3(

λ

λ

λλλ

N

NN

N

lopt

Theoretically

+

Numerically

Numerically 32log~ 1 <<− λλ Nlopt

Strong Disorder

Weak Disorder

λallforNlopt log~

Diameter – shortest path

⎪⎩

⎪⎨

⎧

<<=

>

32loglog3loglog/log

3log~min

λλ

λ

NNN

Nl

LARGE WORLD!!

SMALL WORLD!!

Braunstein et al, Phys. Rev. Lett. 91, 247901 (2003);Cond-mat/0305051

4=λ

Transition from weak to strong disorderFor a given disorder strength a ))exp(( ii au=ε

)(** aNN =

Sreenivasan et al Phys. Rev. E Submitted (2004)

For details see POSTER

alopt /

)/log( 3/1 aN

Nlopt log~)exp(log~ 3/1Nlopt aN

aN<

>3/1

3/1

forfor

Conclusions and Applications• Distance in scale free networks λ<3 : d~loglogN - ultra small world, λ>3 : d~logN.

• Optimal distance – strong disorder – Random Graphs and WS

scale free

•Transition between weak and strong disorder

•Scale Free networks (2<λ<3) are robust to random breakdown.

• Scale Free networks are vulnerable to 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

~ Nlopt31

1

~ 3

~ log 2 3opt

opt

l N for

l N for

λλ

λ

λ

λ

−−

−

> ⇒

< < ⇒{ Large World

Small World

Large World