Scaling, renormalization and self-similarity in complex networks

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Scaling, renormalization and self-similarity in
complex networks
Chaoming Song (CCNY) Lazaros Gallos (CCNY) Shlomo
Havlin (Bar-Ilan, Israel)
Protein interaction network
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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!
AIM How the network behaves under a scale
transformation. Implications for 1.
Dynamics 2.
Modularity 3. Universality
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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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Biological networks
Protein Homology
Tree of life
Similarities between sequence of Amino-acids
(BLAST) Network of 5 million proteins 1.2 TB of
data growing at 50GB Per month. Adai et al. J Mol
Biol (2004)
Complex network of species Representing their
evolucionary history 90,000 species
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Introduction to fractals
In Nature there exist many examples of random
fractals
Coast lines
Rivers
Mountains
Neurons
Clouds
Lightening
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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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Most efficient tiling of the network
8 node network Easy to solve
4 boxes
5 boxes
300,000 node network Mapping to graph colouring
problem. NP-complete Greedy algorithm to find
minimum boxes
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Burning algorithms
Song et al. JSTAT (2007)
1. Compact box burning CBB
2. Maximum mass burning MEMB Burning from
the hubs with the radius r
Minimazing the number of boxes is analogous to
maximizing the mass of each box implications
for modularity
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Box covering in yeast protein interaction network
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Metabolic networks are fractals
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More topological fractals
WWW
1. Protein homology network 2. Tree of life
(taxonomy) 3. Genetic networks (Meyer-Ortmanns,
Khang) 4. Neural networks (Yuste)
nd.edu domain
300,000 web-pages
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Two universality classes
Fractal networks WWW Biological networks
protein interactions, metabolic, genetic
(Meyer-Ortmanns, Khang), taxonomy, tree of life,
protein homology network, neural activity
network. Non-Fractal networks Internet
(routers and AS level) Social networks (citations
(Khang), IMDB) Models based on uncorrelated
preferential attachment
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Two ways to calculate fractal dimensions
Cluster growing method
Box covering method
In homogeneous systems (all nodes with similar k)
both definitions agree
percolation
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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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Statistical properties are invariant under
renormalization
WWW
PIN
Internet
E.coli
Internet is not fractal, dB--gt infinity but it
is renormalizable
FRACTALS
NON-FRACTALS
Self-similarity Invariant under renormalization
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DYNAMICS 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
time evolution
Can we predict the past. ? if not the future.
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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
time evolution
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Classes of genes in the yeast proteome
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Same fractal dimension and scale-free exponent
over 3.5 billion years
Suggests that present-day networks could have
been created following a self-similar, fractal
dynamics.
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Renormalization following the phylogenetic tree
P. Uetz, et al. Nature 403 (2000).
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Emergence of modularity in metabolic networks
Appearance of functional modules in E. coli
metabolic network. Most robust network than
non-fractals.
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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 topological fractality?
Can you see the difference?
FRACTAL
NON FRACTAL
Internet map
Yeast protein map
E.coli metabolic map
Compact cluster
HINT the key to understand fractals is in the
degree correlations P(k1,k2) not in P(k)
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Quantifying correlations
P(k1,k2) Probability to find a node with k1
links connected with a node of k2 links
Gallos et al. (2007)
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 non-hubs
Hubs connected with hubs
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Quantify anticorrelation between hubsat all
length scales
Hub-Hub Correlation function fraction of hub-hub
connections
hubs
Renormalize
hubs
Hubs connected directly
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Hub-hub correlations organized in a self-similar
way
non-fractal
The larger de implies more anticorrelations
fractal
(fractal) (non-fractal)
Anticorrelations are essential for fractal
structures
Exponent de determines the joint probability
distribution
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What is the origin of fractality?
Non-fractal networks
Fractal networks

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

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

Internet, social All available models BA model,
hierarchical random scale free, JKK, etc
WWW, PIN, metabolic, genetic, neural networks,
protein homology, taxonomy
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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
offspring nodes attached to their parents
(m2) in this case
time
renormalize
Scale-free
Mode II
Mode I
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How does the length increase with time?
Mode I NONFRACTAL SMALL WORLD
Mode II FRACTAL
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Combine two modes together
Mode I with probability e Mode II with
probability 1-e
renormalize
time
e0.5
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Model
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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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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Model predicts all exponents in terms of growth
rates
Each step the total mass scales with a constant
n, all the degrees scale with a constant s. The
length scales with a constant a, we obtain
We predict the fractal exponents
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Predictions
Model reproduces local small world, scale-free
and fractality
yeast

• FRACTAL
• repulsion between hubs leads to fractal topology
• small world locally inside well defined
  communities

• NON-FRACTAL
• attraction between hubs
• non-fractal
• small world globally

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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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Summary of scaling exponents and scaling relations
Mass
Links
Hub-hub correlations
Modularity ratio Modularity exponent
Number of links outside modules
Number of hub-hub links
Number of links inside modules
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Modularity is also scale-invariant
Protein Homology
Yeast protein interaction
Similarities between sequence of amino-acids
(BLAST) Network of 5 million proteins 1.2 TB of
data growing at 50GB per month. Adai et al. J Mol
Biol (2004)
Ultramodularity
Large modularity
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Time evolution in yeast network
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Self-similar learning dynamics of the brain
Calcium imaging of spontaneous action potentials
in large neuronal populations of a slice of the
medial prefrontal cortex of a b rain slice of
mouse.
John Cage minimalist avant-garde music
Rafael Yuste and Ikegaya
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Time evolution of the network
t 15 sec
t 30 sec
t 45 sec
t 60 sec
t 75 sec
t 90 sec
t 105 sec
t 120 sec
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The degree exponents and fractal dimension
are invariant under the time evolution
The degree distribution P(k) is invariant under
evolution. The plots go from 30 sec to 120 sec
The fractal dimension is also invariant under
evolution from 30 sec to 120 sec
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Scale-transformation of 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.
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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)?-1 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 of the network under random failure and
intentional attack
We plot the largest cluster size as a function of
the fraction p of nodes removed
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Summary

• In contrast to common belief, many real world
  networks are self-similar.
• FRACTALS WWW, Protein interactions, metabolic
  networks, neural networks, homology networks,
  tree of life.
• NON-FRACTALS Internet, social, 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

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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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Burning algorithms
Song et al. JSTAT (2007)
Compact box burning CBB
Maximum excluded mass burning MEMB Burning from
the hubs with the radius r
Minimazing the number of boxes is analogous to
maximizing the mass of each box Modularity