Self-Similarity of Complex Networks

Maksim Kitsak

Advisor:H. Eugene Stanley

Collaborators:Shlomo Havlin

Gerald PaulZhenhua WuYiping ChenGuanliang Li

Kitsak, Havlin, Paul, Pammolli, Stanley, (submitted), Phys. Rev. E. (2006)

Motivation and Objectives:

• Many real networks are fractals.

• Fractal real networks are shown to have a topology distinct from non-fractal networks.

• Do fractal and non-fractal networks have

different properties? (Transport properties)

• What are the possible applications of these properties?

Networks: Definitions

1) Network is a set of nodes (objects) connected with edges (relations).

2) Degree (k) of a node is a number of edges connected to it.

3) Degree Distribution P(k) is the probability that a randomly chosen node has degree k

4/1)3(

4/2)2(

4/1)1(

P

P

P

Nodes Edges

k=3

k=2

k=2

k=1

Networks: Properties1) Small World property.

Despite the large size, the shortest path between any two nodes is small. (WWW, Internet, Biological)

Number of nodes accessible from a random node (seed) grows exponentially with the distance measured from the seed.

)(lExpn

Networks: Properties2) Degree Distribution

Many real networks have Poisson or power-law degree distribution.

)(kP

kkP )(!

)(kk

ekPk

k

P(k

)

k

Poisson distribution

Scale Free

Lo

g(P

(k))

Log(k)

Power-lawdistribution

Edges connect randomly Preferential Attachment

Networks: Properties3) Self-Similarity

Self-similar network is approximately similar to a part of itself and is fractal. Fractal typically has fractional dimension and doesn’tpossess translational symmetry.

fdLN

fd

Networks: Properties

2) Degree Distribution

Many real networks have Poisson or power-law degree distribution.

)(kP

kkP )(!

)(kk

ekPk

k

3) Self-SimilaritySelf-similar network is approximately similar to a part of itself. Self-similar networks are fractals and have fractal dimension .fd fdLN

It turns out that many real networks possess all three properties

(Small World, Scale-Free, Fractal)!!!

1) Small World property.Despite the large size, the shortest path between any two nodes is small. (WWW, Internet, Biological) )(lExpn

Dimension Calculation: Box Covering Algorithm

A box of size is an imaginary ‘container’ that can hold a part of the network, so

that the shortest path between any 2 nodes Bl

Bji ll ,

What is the minimal number of boxes with size needed to cover the entire network?

Bl

)()( 0llExplN BBB

(Non-Fractal) (Fractal)

BdBBB llN )(

Strategy to calculate dimension of the network:1. Calculate minimal number of boxes needed to cover

the network as a function of their size.2. Analyze obtained function

)( BB lN

Bd1Bl

3BN1Bl

2BN

2Bl

1BN

Fractal analysis with box-covering algorithm

Song, Havlin, Makse, 2005

10-5

10-4

10-3

10-2

10-1

100

100

101

100 10110-4

10-3

10-2

10-1

100 101

(a) WWW (fractal)

dB=4.1

N

B/N

(c) Pharmaceutical (non-fractal)

LB

dB=2.2

NB/N

LB

(b) H. Sapiens (fractal) (d) H. Pylori (non-fractal)

LB

Number of boxes as a function of the box size .BN Bl

Origin of fractals in scale-free networks: Repulsion between hubs

In fractal networks large degree nodes (hubs) tend to connect to small degree nodes and not to each other!

Song, Havlin, Makse, 2005

Fractal network Non-fractal Network

Probability of having a node of degree connected to node of degree . 1k 2k

Transport on networks: Betweenness Centrality

Most of the transport on the network flows along the shortest paths.

Central nodes are critical: if they are blocked – transport becomes inefficient

Betweenness centrality of node :i

ikj kj

kj iiC

, ,

, )()(

Sociology - L.C. Freeman, 1979

)(, ikj Number of shortest paths between nodes and that pass node .ij k

kj , Total number of shortest paths between nodes and . kj

C=0

C=0 C=0

C=2

10-1 100

10-3

10-2

10-1

100

C/C

max

k/kmax

Fractal Model Non-Fractal Model

Transport on networks: Betweenness Centrality

How do we identify nodes with high Centrality?

Is it true that high centrality nodes also have large

degree?

Centrality is weakly correlated with degree in fractal scale-free networks!

Transport on networks: Betweenness CentralityWhy is centrality weakly correlated with degree

in fractal scale-free networks?

Non-Fractal Topology Fractal topology

Due to ‘repulsion between hubs’ small degree nodes appear at

all parts of the fractal network. Thus, their centralities can

have both small and large values.

Centrality-degree correlation in real networks

One can’t compare centralities of networks directly due to uniqueness of real networks.

The network can be compared to its random counterpart !

Rewire 10000 times

Preserve degrees of nodes

Rewired network has degree distribution identical to the original network.Repulsion between hubs is broken by random rewiring. The random network is always non-fractal.

10-1 10010-6

10-5

10-4

10-3

10-2

10-1

100

C/C

max

k/kmax

H. Sapiens (fr) H. Sapiens rewired (nf)

10-1 10010-4

10-3

10-2

10-1

100

C/C

max

k/kmax

H.Pylori (nf) H.Pylori rewired(nf)

Centrality-degree correlation.

Centrality – degree correlation in non-fractal scale-free networks is much stronger than that in SF fractal networks.

Average centrality of small degree nodes in scale-free fractal networks is significantly larger due to repulsion between hubs.

Fractal networks should be more stable to conventional degree attacks.Immunization/Attack strategies should be optimized for fractal networks.

Kitsak, Havlin, Paul, Pammolli, Stanley, 2006

10-3 10-2 10-1 10010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

=2.0

=2.0

P(C

)

C/Cmax

non-fractal model 3nf H.Pylori (non-fractal)

What is the overall Centrality distribution in scale-free networks?

Centrality distribution obeys power-law for both types of networks CCP )(

We show both analytically and numerically that Bd/12

Nodes of fractal networks generally have larger centrality than nodes of non-fractal networks

10-3 10-2 10-1 10010-6

10-5

10-4

10-3

10-2

10-1

100

C/Cmax

=1.54

=1.36

P(C

)

Fractal Model 3f H.Sapiens (fractal)

Kitsak, Havlin, Paul, Pammolli, Stanley, 2006

100 101 10210-5

10-4

10-3

10-2

10-1

100

N=16384E=16383

dB=1.9

NB/N

LB

E=0 E=50 E=300 E=1000 E=5000

Box covering algorithm applied to a fractal network with added random edges

Transition from Fractal to Non-Fractal Behavior.

Real networks are neither pure fractals nor non-fractals due to statistical effects.

What happens if we add random edges to a scale-free fractal network?

l*lFractal behavior Non-fractal behavior

100 101 10210-5

10-4

10-3

10-2

10-1

100

N=16384E=16383

dB=1.9

NB/N

LB

E=0 E=50 E=300 E=1000 E=5000

Scaling Ansatz:

))(/()(),( ** pllFplplN BdB

BduuF )( )1( u

)()( uExpuF )1( uNEp /

How does the crossover length depend on the density of random edges ?

*lp

Under rescaling

all plots should collapse onto a single curve

Crossover length

))(/()()( pblNpalN BB

))(/(1* pbl

How does the crossover length depend on the density of random edges ?

*lp

100 101 10210-5

10-4

10-3

10-2

10-1

100

N=16384E=16383

dB=1.9

NB/N

LB

E=0 E=50 E=300 E=1000 E=5000

Rescale

100 101 10210-5

10-4

10-3

10-2

10-1

100

a(p

)Nb/N

LB/b(p)

E=300 E=1000 E=5000

All plots collapse onto a single curve

2-8 2-6 2-4 2-22-2

2-1

20

21

22

= 0.46

b(p

)

p=E/N

Rescaling parameter as a function of . )( pb p

pl*

We show analytically that

Bd/1

))(/(1* pbl

Summary and Conclusions

1. Centrality-degree correlation is much weaker in fractal networks than in non-fractal.

Fractal networks should be more stable to conventional degree attacks.

Immunization/Attack strategies should be optimized for fractal networks.

2. Power-law centrality distribution

Centralities of nodes are larger in fractal scale-free networks.

fractal networks have different transport properties.

3. Transition from fractal to non-fractal networks.

A crossover is observed from fractal to non-fractal networks.

Relatively small percent of edges is needed to turn fractal network into non-fractal.

Findings of present work have been submitted to Phys. Rev. E.

Centrality-degree correlation.L

og

(C)

Log(K)

(a) Fractal Model

00.62501.2501.8752.5003.1253.7504.3755.000

Lo

g(C

)

Log(k)(b) Non-Fractal Model

00.62501.2501.8752.5003.1253.7504.3755.000

Lo

g(C

)

Log(k)

(c) H.Sapiens (fractal)

00.62501.2501.8752.5003.1253.7504.3755.000

Lo

g(C

)

Log(k)

(d) Pharmaceutical (N-F)

00.62501.2501.8752.5003.1253.7504.3755.000

),(

),(),(

kCP

kCPkCR

R

Transition from Fractal to Non-Fractal Behavior: Analytical Consideration

Consider a fractal network of dimension with nodes and edges.N ED

Suppose random edges are added to the network. E

The probability that a random edge is connected to a given node: Np 2

A cluster of radius centered on a randomly

chosen seed has nodes. Dll )(

l

The probability that a cluster of size contains

a random edge . DlpllP )(l

The crossover length corresponds to *l 1)( * lP

1* Dl Dl /1*

Seed

Centrality distribution: Analytical Consideration

Consider a fractal tree network of dimension with nodes and edges.N ED

A small region with nodes will have a typical diameter . n Dnnl /1

The region will be connected to the rest of the

network via approximately nodes with

centrality .

nlnC

The total number of such nodes in the network: 1/1)(/)( DnnlnNnThe number of nodes with centrality

2/1)()1( Dnnn nC n

Contents:

1. Introduction.

(Definitions, Properties…)

2. Self-Similarity and Fractality.

(Fractal Networks: General Results)

3. Betweenness Centrality in Fractal and Non-Fractal Networks.

(Centrality distribution, Centrality-Degree correlation…)

4. Transition from Fractal to Non-Fractal networks.

(Crossover phenomenon, Scaling…)

5. Summary and Conclusions

Betweenness Centrality

ikj kj

kj iiC

, ,

, )()(

Betweenness Centrality is a measure of

Importance of a node in the network.

C=0

C=0 C=0

C=2

Is it true that larger degree nodes generally have larger centrality?

L.C. Freeman, 1979

10-1 100

10-3

10-2

10-1

100

C/C

max

k/kmax

Fractal Model Non-Fractal Model

Fractal topology

Non-Fractal Topology

Transition from Fractal to Non-Fractal Behavior.

100 101 10210-5

10-4

10-3

10-2

10-1

100

N=16384E=16383

dB=1.9

NB/N

LB

E=0 E=50 E=300 E=1000 E=5000

What happens when random edges are added to a fractal network?

*ll Fractal behavior*ll Non-Fractal behavior

))(/()()( pblNpalN BB

100 101 10210-5

10-4

10-3

10-2

10-1

100

a(p

)Nb/N

LB/b(p)

E=300 E=1000 E=5000

All plots collapse onto a single curve

2-8 2-6 2-4 2-22-2

2-1

20

21

22

= 0.46

b(p

)p=E/N

Scaling Ansatz:

))(/()(),( ** pllFplplN BdB

BduuF )( )1( u

)()( uExpuF )1( u Bdpl /1*

P(k

)

k

Poisson distribution

Scale Free

Log(

P(k

))

Log(k)

Power-lawdistribution

Edges connect randomly Preferential Attachment

Networks: Properties

2) Degree Distribution

Many real networks have Poisson or power-law degree distribution.

)(kP

kkP )(!

)(kk

ekPk

k

3) Self-SimilaritySelf-similar network is approximately similar to a part of itself. Self-similar networks are fractals and have fractal dimension .fd fdLN

It turns out that many real networks possess all three properties

(Small World, Scale-Free, Fractal)!!!

1) Small World property.Despite the large size, the shortest path between any two nodes is small. (WWW, Internet, Biological) )(NLogL