Fractality vs selfsimilarity in scalefree networks

1
Fractality vs self-similarity in scale-free
networks
B. KahngSeoul Natl Univ., Korea CNLS, LANL
Jin S. Kim, K.-I. Goh, G. Salvi, E. Oh and D. Kim
The 2nd KIAS Conference on Stat. Phys.,
07/03-06/06
2
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).
3
Introduction
Introduction
Network

• node, link, degree

Networks are everywhere
4
Random graph model by Erdos Rényi
Erdos Renyi 1959
Put an edge between each vertex pair with
probability p

• Poisson degree distribution
• D lnN
• Percolation transition at p1/N

5
Scale-free network the static model
5-a
6-a
1-a
4-a
The number of vertices is fixed as N .
2-a
8-a
3-a
7-a
Two vertices are selected with probabilities pi
pj.
Goh et al., PRL (2001).
6
I-1. Fractality
I. Fractal scaling in SF networks
Song, Havlin, and Makse, Nature (2005).
Box-covering method
Mean mass (number of nodes) within a box
Contradictory to the small-worldness
? Cluster-growing method
7
I-2. Box-counting
Random sequential packing
Nakamura (1986), Evans (1987)

• At each step, a node is selected randomly and
  served as a seed.
• Search the network by distance from the
  seed and assign newly burned vertices to the new
  box.
• Repeat (1) and (2) until all nodes are assigned
  their respective boxes.
• is chosen as the smallest number of
  boxes among all the trials.

3
1
2
4
8
I-2. Box-counting
Fractal scaling
dB 4.1
9
I-2. Box-counting
Box-covering method
Fractal dimension dB
10
I-3. Purposes
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.
11
I-4. Origin

• Disassortativity, by Yook et al., PRE (2005)
• Repulsion between hubs, by Song et al., Nat.
  Phys. (2006).

Fractal networkSkeletonShortcuts SkeletonTree
based on betweenness centrality Skeleton ?
Critical branching tree ? Fractal By Goh et al.,
PRL (2006).
12
I-5. Skeleton
What is the skeleton ? Kim, Noh, Jeong
PRE (2004)

• For a given network, loads (BCs) on each edge are
  calculated.
• Generate a spanning tree by following the
  descending order of edge loads (BCs). ?Skeleton

Skeleton is an optimal structure for transport in
a given network.
13
(No Transcript)
14
I-6. Fractal scalings
Fractal scalings of the original network,
skeleton, and random ST
Fractal structures
15
Fractal scalings of the original network,
skeleton, and random ST
Non-fractal structures
16
I-7. Branching tree
Network ? Skeleton ? Tree ? Branching tree
Mean branching number
If
then the tree is critical
If
then the tree is supercritical
17
(No Transcript)
18
Test of the mean branching number ltmgtb
skeleton
yeast
WWW
metabolic
random
Static
BA
Internet
19
I-8. Critical branching tree
For the critical branching tree
M is the mass within the circle
Goh PRL (2003), Burda PRE (2001)
Cluster-size distribution
20
I-9. Supercritical branching tree
For the supercritical branching tree
Cluster-size distribution
behaves similarly to
but with exponential cutoff.
21
Test of the mean branching number ltmgtb
skeleton
WWW
metabolic
yeast
random
Critical
Supercritical
22
I-10. Model construction
Model construction rule
i) A tree is grown by a random branching process
with branching probability
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).
iii) Connect the stubs for the global shortcuts
randomly.

• Resulting network structure is
• SF with the degree exponent g.
• Fractal for q0 and non-fractal for qgtgt0.

23
(No Transcript)
24
Networks generated from a critical branching tree
25
Fractal scaling and mean branching ratio for the
fractal model
26
Networks generated from a supercritical branching
tree
27
Fractal scaling and ltmgtb for the skeleton of the
network generated from a SC tree
28
II. Self-similarity in SF networks

• The distribution of renormalized-degrees under
  coarse-graining is studied.
• Modules or boxes are regarded as super-nodes
• Module-size distribution
• How is h involved in the RG transformation ?

Coarse-graining process
29
Random and clustered SF network (Non-fractal net)
Analytic solution
30
Derivation

31
h and q act as relevant parameters in the RG
transformation
32
For fractal networks, WWW and Model
33
For a nonfractal network, the Internet
? Self-similar
34
Scale invariance of the degree distribution
for SF networks
Jung et al., PRE (2002)
35
The deterministic model is self-similar, but not
fractal !
Fractality and self-similarity are disparate in
SF networks.
36
Summary I
Skeleton Local shortcuts
Branching tree
Fractal networks
Fractal model
Yeast PIN WWW
1 Goh et al., PRL 96, 018701 (2006). 2 J.S.
Kim et al., cond-mat/0605324.
37
Summary II
1. h and q act as relevant parameters in the RG
transformation. 2. Fractality and self-similarity
are disparate in SF networks.
1 J.S. Kim et al., cond-mat/0605587.
38
(No Transcript)
39
(No Transcript)