Fractality vs self-similarity in scale-free networks

The 2nd KIAS Conference on Stat. Phys., 07/03-06/06

Jin S. Kim, K.-I. Goh, G. Salvi, E. Oh and D. Kim

B. KahngSeoul Nat’l Univ., Korea & CNLS, LANL

Contents

I. Fractal scaling in SF networks [1] K.-I. Goh, G. Salvi, B. Kahng and D. Kim, Skeleton and fractal

scaling in complex networks, PRL 96, 018701 (2006).

[2] J.S. Kim, et al., Fractality in ocmplex networks: Critical and supercritical skeletons, (cond-mat/0605324).

II. Self-similarity in SF networks [1] J.S. Kim, Block-size heterogeneity and renormalization in

scale-free networks, (cond-mat/0605587).

Networks are everywhere

Introduction

• node, link, & degree

Network

Introduction

Random graph model by Erdős & Rényi[Erdos & Renyi 1959]

Put an edge between each vertex pair with probability

1. Poisson degree distribution

2. D ~ lnN

3. Percolation transition at p=1/N

1-α

2-α

4-α

3-α

5-α

6-α

8-α

7-α

Scale-free network: the static model

~ip i

1 1/

( ) ~P k k

Goh et al., PRL (2001).

The number of vertices is fixed as N .

Two vertices are selected with probabilities pi pj.

Song, Havlin, and Makse, Nature

(2005).

Box-covering method:

( ) BdB s sN

Mean mass (number of nodes) within a box:

( ) / ( ) Bds s s s sM N N

Contradictory to the small-worldness:

0/M e ln M

I. Fractal scaling in SF networksI-1. Fractality

Cluster-growing method

Random sequential packing: 1. At each step, a node is selected randomly and served as

a seed.

2. Search the network by distance from the seed and assign newly burned vertices to the new box.

3. Repeat (1) and (2) until all nodes are assigned their respective boxes.

4. is chosen as the smallest number of boxes among all the trials.

B

11B

Nakamura (1986), Evans (1987)

2

3

4

I-2. Box-counting

( )B BN

2 1S B

I-2. Box-countingFractal scaling

dB = 4.1

WWW2, B η γ

5, /( 1)B η τ γ γ

Box mass inhomogeneity

Log Box Size

Log Box Number

dB

Fractal dimension dB

Box-covering method:

I-2. Box-counting

BdBN

Fractal complex networks

www, metabolic networks, PIN (homo sapiens)

PIN (yeast, *), actor network

Non-fractal complex networks

Internet, artificial models (BA model, etc), actor network, etc

Purposes:

1. The origin of the fractal scaling.

2. Construction of a fractal network model.

I-3. Purposes

I-4. Origin

1. Disassortativity, by Yook et al., PRE (2005)

2. Repulsion between hubs, by Song et al., Nat. Phys. (2006).

Fractal network=Skeleton+Shortcuts

Skeleton=Tree based on betweenness centrality

Skeleton Critical branching tree Fractal

By Goh et al., PRL (2006).

1. For a given network, loads (BCs) on each edge are calculated.

2. Generate a spanning tree by following the descending order of edge loads (BCs). Skeleton

What is the skeleton ? Kim, Noh, Jeong PRE (2004)

I-5. Skeleton

Skeleton is an optimal structure for transport in a given network.

Fractal scalings of the original network, skeleton, and random ST

Fractal structures

I-6. Fractal scalings

original skeleton random

Fractal scalings of the original network, skeleton, and random ST

Non-fractal structures

original skeleton random

Network → Skeleton → Tree → Branching tree

Mean branching number0

mbm

m mb

I-7. Branching tree

If 1,b

m then the tree is subcritical

If 1,b

m then the tree is critical

If 1,b

m then the tree is supercritical

Test of the mean branching number: <m>b

WWW metabolic

yeast

Internet BA Static

skeleton

random

BdM 2 > 3

1 2 < < 3

2Bd

γ

γγ

γ

M is the mass within the circle

I-8. Critical branching tree

For the critical branching tree

Cluster-size distribution

/( 1)

3/ 2 1/ 2

3/ 2

(2 < < 3)

( =3 )

( >3 )

( ) (ln )

s

n s s s

s

γ γγ

γ

γ

Goh PRL (2003), Burda PRE (2001)

lnb

m

bM m e

I-9. Supercritical branching tree

For the supercritical branching tree

/( 1)

3/ 2 1/ 2

3/ 2

(2 < < 3)

( =3 )

( >3 )

( ) (ln )

s

n s s s

s

γ γγ

γ

γ

behaves similarly to

but with exponential cutoff.

Cluster-size distribution

Test of the mean branching number: <m>b

WWW metabolic

yeast

www metabolic Yeast PIN

OriginalNetworks

Cluster-growing Exponential Exponential Power law

Box-covering Power law Power law Power law

skeletons Cluster-growing Exponential Power law Power law

Box-covering Power law Power law Power law

random

skeleton

Supercritical Critical

( ) / ( ) BdB B B B BM N N 0/M e

iii) Connect the stubs for the global shortcuts randomly.

ii) Every vertex increases its degree by a factor p; qpki are reserved for global shortcuts, and the rest attempt to connect to local neighbors (local shortcuts).

i) A tree is grown by a random branching process with branching probability:

Resulting network structure is:

i) SF with the degree exponent .

ii) Fractal for q~0 and non-fractal for q>>0.

Model construction rule

I-10. Model construction

( 1) ( 1)

bm m

mb m γ

ς γ

0

1

1 mm

b b

(1 )i ik p k

Networks generated from a critical branching tree

Critical branching tree

+ local shortcuts + global shortcuts

fractal fractal Non-fractal

Fractal scaling and mean branching ratio for the fractal model

Networks generated from a supercritical branching tree

Supercritical branching tree

+ local shortcuts + global shortcuts

Fractal+small world Fractal+small world Non-fractal

Fractal scaling and <m>b for the skeleton of the network generated from a SC tree

1. The distribution of renormalized-degrees under coarse-graining is studied.

2. Modules or boxes are regarded as super-nodes

3. Module-size distribution

4. How is involved in the RG transformation ?

( )mP M M η

Coarse-graining process

II. Self-similarity in SF networks

( ) ?dP k k γ

Random and clustered SF network: (Non-fractal net)

3

'j

j

k kαα

( < )

' ( )

η η γγ

γ η γ

Analytic solution( ) ~

( ) ~

d

m B B

P k k

P M M

γ

η

2,3, and 4η

( )dP k k γ

1

( ) ( ) kd d

k

z P k z

P

1( ) (1 ) +.... if < d z z η η γ P

1( ) (1 ) +.... if > d z z γ η γ P

Derivation

1 2( ) 1 (1 ) (1 ) + ((1 ) ).d z k z a z zγ P O

( ) ( ( ))d m dz z P P P

1, 2,1 .. 11

( ) ( ) ( ) ( )M

M M

d m d j jk k k k jj

P k P M P k k kδ

jjk kα α

Bk M θ

11&

( ) ( )

d m B B

B B

P k dk P M dM

k M k Mηγ θ

and act as relevant parameters in the RG transformation

2B

5B

2,3, and 5B

1+( -1)/ ( < )'

( )

η θ η γγ

γ η γ

( )m B BP M M η

For 2,

2.2, 1B

η θ

For 5,

1.8, 0.6B

η θ

2.2 2.3γ

1 ( 1) /γ η θ

For fractal networks,

WWW and Model

2,3, and 5B

For a nonfractal network,

the Internet

Self-similar

1 (1.8 1) / 0.7

2.1

γ

γ

1.8η 0.7θ

1 ( 1) /γ η θ

2B

Jung et al., PRE (2002)

Scale invariance of the degree

distribution for SF networks

The deterministic model is self-similar, but not fractal !

Fractality and self-similarity are disparate in SF networks.

Skeleton+

Local shortcuts

Summary I

Fractal networks

Branching tree

Critical

Supercritical

Yeast PIN

WWW

Fractal model

[1] Goh et al., PRL 96, 018701 (2006).

[2] J.S. Kim et al., cond-mat/0605324.

Summary II

and act as relevant parameters in the RG transformation.

2. Fractality and self-similarity are disparate in SF networks.

Bk M θ1+( -1)/ ( < )

' ( )

η θ η γγ

γ η γ

( ) ~

( ) ~

d

m B B

P k k

P M M

γ

η

( ) ~dP k k γ

[1] J.S. Kim et al., cond-mat/0605587.