Origin of self-similarity in the growth of complex networks

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Origin of self-similarity in the growth of
complex networks
Shlomo Havlin Bar-Ilan, Israel
2
Are scale-free networks really free-of-scale?
If you had asked me yesterday, I would have said
surely not - said Barabasi. (Science News,
February 2, 2005).
Small World effect shows that distance between
nodes grows logarithmically with N (the network
size) OR Self-similar fractal topology is
defined by a power-law relation
Small world contradicts self-similarity!!!
How can we test if complex networks are fractal?
3
How long is the coastline of Norway?
It depends on the length of your ruler.
Fractals look the same on all scales
scale-invariant.
Box length
Fractal Dimension dB- Box Covering Method
Total no. of boxes
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Boxing in Biology
Boxing in Biology
How to zoom out of a complex network?

• Generate boxes where all
• nodes are within a distance

• Calculate number of boxes, ,
• of size needed to cover the
• network

We need the minimum number of boxes NP-complete
optimization problem!
5
Most efficient tiling of the network
8 node network Easy to solve
4 boxes
5 boxes
300,000 node network Greedy algorithm to find
minimum boxes
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Box covering in yeast protein interaction network
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Most complex networks are Fractal
Biological networks
Metabolic
Protein interaction
43 organisms - all scale
Three domains of life archaea, bacteria, eukaria
E. coli, H. sapiens, yeast
Song, Havlin, Makse, Nature (2005)
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Technological and Social Networks TOO
WWW
Hollywood film actors
212,000 actors
Other bio networks Khang and Bremen groups
Internet is not fractal!
nd.edu domain
300,000 web-pages
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Box Covering flat average
Cluster Growing biased
exponential
power law
Different methods yield different results due to
heterogeneous topology
Box covering reveals the self similarity. Cluster
growth reveals the small world. NO
CONTRADICTION! SAME HUBS ARE USED MANY TIMES IN
CG.
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Turning back the time
Repeatedly BOXING the network is the same as
going back in time from a single
node to present day.
THE RENORMALIZATION SCHEME
renormalization
present day network
ancestral node
1
time evolution
Can we predict the past. ? if not the future.
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Evolution of complex networks
time evolution
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The degree distribution is invariant under
renormalization
WWW
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How does Modularity arise?
The boxes have a physical meaning self-similar
nested communities
How to identify communities in complex networks?
renormalization
present day network
ancestral node
1
time evolution
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Is evolution of the yeast fractal?
present day
Other Fungi
Animals Plants
Archaea Bacteria
Yeast
300 million years ago
Ancestral yeast
Ancestral Fungus
Ancestral Eukaryote
1 billion years ago
Ancestral Prokaryote Cell
3.5 billion years ago
Following the phylogenetic tree of life
1.5 billion years ago
COG database
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Same fractal dimension and scale-free exponent
over 3.5 billion years
P(kk0)
kk0
Suggests that present-day networks could have
been created following a self-similar, fractal
dynamics.
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Theoretical approach
How the communities are linked?
renormalization
k2
k8
s1/4
k degree of the nodes
k degree of the communities
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Theoretical approach to modular networks Scaling
theory to the rescue
The larger the community the smaller their
connectivity
new exponent describing how families link
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Scaling relations
A theoretical prediction relating the different
exponents
distance
boxes
degree
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Scaling relations
The communities also follow a self-similar pattern
WWW
Metabolic
prediction
Scaling relation works
scale-free
fractals
communities/modules
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What is the origin of self-similarity?
Non-fractal networks
Fractal networks

• very compact networks
• hubs connected with other hubs
• strong hub-hub attraction
• assortativity

• less compact networks
• hubs connected with non-hubs
• strong hub-hub repulsion
• dissasortativity

Internet All available models BA model,
hierarchical random scale free, JKK, etc
WWW, PIN, metabolic, genetic, neural networks,
some sociological networks
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Why fractal topology
Appearance of functional modules in E. coli
metabolic network. Most robust network than
non-fractals.
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Quantifying correlations
P(k1,k2) Probability to find a node with k1
links connected with a node of k2 links
Internet map - non fractal
Metabolic map - fractal
high prob.
low prob.
log(k2)
log(k2)
P(k1,k2)
low prob.
high prob.
log(k1)
log(k1)
Hubs connected with hubs
Hubs connected with non-hubs
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Quantify anticorrelation between hubsat all
length scales
fraction of hub-hub connections
hubs
Renormalize
hubs
Hubs connected directly
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Hub-hub connection organized in a self-similar way
non-fractal
The larger de implies more anticorrelation
fractal
(fractal) (non-fractal)
Anticorrelations are essential for fractal
structure
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How to model it? renormalization reverses
time evolution
Song, Havlin, Makse, Nature Physics, 2006
Both mass and degree increase exponentially with
time
time
offspring nodes attached to their parents
renormalize
(m2) in this case
Scale-free
Mode II
Mode I
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How the length increases with time?
Mode I NONFRACTAL
Mode II FRACTAL
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Combine two modes together
Mode I with probability e Mode II with
probability 1-e
time
renormalize
e0.5
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The model reproduces the main features of real
networks
Case 1 e 0.8 FRACTALS Case 2 e 1.0
NON-FRACTALS
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A new principle of network dynamics
less vulnerable to intentional attacks
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Summay

• In contrast to common belief, many real world
  networks are self-similar.
• FRACTALS WWW, Protein interactions, metabolic
  networks, neural networks, collaboration
  networks.
• NON-FRACTALS Internet, all models.
• Communities/modules are self-similar, as well.
• Scaling theory describes the dynamical
  evolution.
• Boxes are related to the functional modules in
  metabolic and protein networks.
• Origin of self similarity anticorrelation
  between hubs
• Fractal networks are less vulnerable than
  non-fractal networks

Positions available jamlab.org
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WWW
nd.edu
300,000 web-pages
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Internet
Faloutsos et al., SIGCOMM 99
Internet connectivity, with selected backbone
ISPs (Internet Service Provider) colored
separately.
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Yeast Protein-Protein Interaction Map

Individual proteins Physical interactions from
the filtered yeast interactome database 2493
high-confidence interactions observed by at least
two methods (yeast two-hybrid). 1379 proteins,
ltkgt 3.6
J. Han et al., Nature (2004)
Modular structure according to function!
Colored according to protein function in the
cell Transcription, Translation, Transcription
control, Protein-fate, Genome maintenance,
Metabolism, Unknown, etc
from MIPS database, mips.gsf.de
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Metabolic network of biochemical reactions in
E.coli

Chemical substrates Biochemical interactions
enzyme-catalyzed reactions that transform one
metabolite into another.
J. Jeong, et al., Nature, 407 651 (2000)
Modular structure according to the
biochemical class of the metabolic products of
the organism.
Colored according to product class Lipids,
essential elements, protein, peptides and amino
acids, coenzymes and prosthetic groups,
carbohydrates, nucleotides and nucleic acids.
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What is the origin of fractality?
Some real networks are not fractal
INTERNET
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What is the origin of fractality?
Can you see the difference?
FRACTAL
NON FRACTAL
Internet map
Yeast protein map
E.coli metabolic map
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Modular network biology
Boxes are related to the biologically relevant
functional modules in the yeast protein
interactome
renormalization
time evolution
translation
transcription
protein-fate
cellular-fate organization
present day network
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Renormalization in Complex Networks
NOW, REGARD EACH BOX AS A SINGLE NODE AND ASK
WHAT IS THE DEGREE DISRIBUTION OF THE
NETWORK OF BOXES AT DIFFERENT SCALES ?
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Renormalization of WWW network with
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Larger distances need fewer boxes
1
2
-dB
fractal
log(NB)
3
non fractal
log(lB)
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Fractals in Nature
Coast lines
Rivers
Mountains
Neurons
Clouds
Lightening
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Fractal learning dynamics of the brain
Calcium imaging of spontaneous action potentials
in neuronal populations of a slice of the brain
of a mouse.
Rafael Yuste and Ikegaya Columbia University
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Time evolution of the network connectivity
t 15 sec
t 30 sec
t 45 sec
t 60 sec
t 75 sec
t 90 sec
t 105 sec
t 120 sec
The weighted network is generated by training a
Hopfield neural network.
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Same scale-free exponent and fractal dimension
over 2 minutes
P(k)
k
Degree distribution P(k) from 30 sec to 120 sec
Fractal Dimension from 30 sec to 120 sec
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Metabolic networks are fractals
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Scale-transformation of the degree
We verify the formula k(t1) s(t1t2)
k(t2) Here we fix t2 120 sec, and take t1 from
30 sec to 105 sec. The linear dependency is
verified for different times t1.
k(t)
k(120)
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Confirmation of the scaling formula for the
degree exponent as a function of the fractal
exponents
From the theory N(t) s(t)??? The inset shows
that both N(t) and s(t) increase
exponentially N(t) exp(0.014t) s(t)
exp(0.021t) This gives rise to the following
scaling relation
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Tolerance to random errors, fragility against
targeted attack
Networks under random or target attack
Largest cluster size Nlargest in the network as p
fraction of nodes are removed under random
failure or targeted attack to the hubs.
random failure
original network
Nlargest/N
targeted attack, network collapse
p
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An finally, a model to put all this together
A multiplicative growth process of the number of
nodes and links
m 2
Analogous to duplication/divergence mechanism in
proteins??
Probability e hubs always connected strong hub
attraction should lead to non-fractal
Probability 1-e hubs never connected strong hub
repulsion should lead to fractal
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Two ways to calculate fractal dimensions
Cluster growing method
Box covering method
In homogeneous systems (all nodes with the
similar k) both definitions
agree
percolation
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Scaling theory to the rescue
Each step the total mass scales with a constant
n, all the degrees scale as a constant s. Assume
the length scale with a constant a, we obtain
And predict the fractal exponents
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Different growth modes lead to different topologie
s
For the both models, each step the total number
of nodes scale as n 2m 1( N(t1) nN(t) ).
Now we investigate the transformation of the
lengths. They show quite different ways for this
two models as following
Mode I L(t1) L(t)2
Then we lead to two different scaling law of N L
smaller
smaller
Mode II L(t1) 2L(t)1
Mode III L(t1) 3L(t)
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Dynamical model
Suppose we have e probability to have mode I, 1-e
probability to have mode II and mode III. Then we
have
or
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Predictions
Model reproduces small world, scale-free and
fractal properties
h.sapiens

• model with e0.2
• repulsion between hubs leads to fractal topology
• small world locally inside well defined
  communities

• model with e1
• attraction between hubs
• non-fractal
• small world globally

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Modular network biology
Boxes are related to the biologically relevant
biochemical modules in the E. coli metabolic
network
renormalization
time evolution
carbohydrates
lipids
nucleotides, nucleic acids
proteins, peptides and aminoacids
coenz. and prosthetic groups
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Graph theoretical representation of a
metabolic network
(a) A pathway (catalyzed by Mg2-dependant
enzymes). (b) All interacting metabolites are
considered equally. (c) For many biological
applications it is useful to ignore co-factors,
such as the high energy-phosphate donor ATP,
which results in a second type of mapping that
connects only the main source metabolites to the
main products.
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Minimum number of boxes to tile the network
renormalization
other covering
1
0
2
0
1
Greedy algorithm to find the minimum coverage
the best local solution.
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Classes of genes in the yeast proteome
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Renormalization following the phylogenetic tree
P. Uetz, et al. Nature 403 (2000).
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BA-model
NOT ALL REAL NETWORKS ARE FRACTALS! ALMOST NO
MODEL!!
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Hierarchy of Scale Free
After Renormalization
With the same
!
Where
THE SCALING TRANSFORMATION OF THE DEGREE
DISTRIBUTION
HOW FAMILIES OF VARIOUS SIZES ARE LINKED?c
From which follows
C. Song, S. Havlin, H. A. Makse, Nature,
(January 2005)
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Fractal and Degree exponents for Various Networks
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We will show that networks areself-similar or
fractals
Why the community missed the self-similar
properties?
The Mandelbrot problem
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Dynamical evolution of networks
Renormalization and Box Covering Approach
NOW, REGARD EACH BOX AS A SINGLE NODE AND ASK
WHAT IS THE DEGREE DISRIBUTION OF THE
NETWORK OF BOXES AT DIFFERENT SCALES ?
65
Weight distribution of the neural network
The weighted network is generated by training a
Hopfield neural network.
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Degree distribution of neural network
The degree distribution at 120 sec for different
weight cutoffs. The inset shows that, after a
certain cutoff3, the degree follows a power law
distribution with the same exponent. The main
figure shows the average for different cutoff gt
3, giving rise to the scaling
P(k) k -??
? 1.67
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Fractal dimension of the neural network
The fractal dimension at 120 sec for different
weight cutoffs. The inset shows that, after a
cutoff3, NB (the number of boxes) follows a
power law distribution with the same exponent.
The figure shows the average for different
cutoffs gt 3, giving rise to the scaling
NB(lB) lB -dB
dB 2.74
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Degree distribution for random and shuffled data
shuffled real data, correct learning law
random (artificial) data, correct learning law
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Degree distribution with wrong learning law
real data, absurd learning law
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How to appy the box coveringmethods in complex
networks
How to embed a networks of infinite dimensions
in a finite dimensional space?
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Biological networks also dont follow the Erdos
model !?
Metabolic networks
Protein Interactions
P(kk0)
kk0
Archaea
Bacteria
Eukaryotes
Same for all three domain of life
J. Jeong, et al., Nature, 407 651 (2000)