fractality vs self-similarity in scale-free networks
DESCRIPTION
Fractality vs self-similarity in scale-free networks. B. Kahng Seoul Nat’l Univ., Korea & CNLS, LANL. Jin S. Kim, K.-I. Goh, G. Salvi, E. Oh and D. Kim. The 2 nd KIAS Conference on Stat. Phys., 07/03-06/06. Contents I. Fractal scaling in SF networks - PowerPoint PPT PresentationTRANSCRIPT
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.