Scaling, renormalization and self-similarity in complex networks

Chaoming Song (CCNY)Chaoming Song (CCNY)Lazaros Gallos (CCNY)Lazaros Gallos (CCNY)Shlomo Havlin (Bar-Ilan, Israel)Shlomo Havlin (Bar-Ilan, Israel)

Hernan A. MakseHernan A. Makse

Levich Institute and Physics Dept.Levich Institute and Physics Dept.City College of New YorkCity College of New York

Protein interaction networkProtein interaction network

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).

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N ~ e<l>/ l0

Small world contradicts self-similarity!!!

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:

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N ~ ld B

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< l > ~ ln(N)

How the network behaves under a scale transformation.

WWW nd.edu

300,000 web-pages

R. Albert, et al., Nature (1999)

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P(k) ~ k−γ

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γ=2.3

Internet connectivity, with selected backbone ISPs (Internet Service Provider) colored separately.

Faloutsos et al., SIGCOMM ’99

Internet

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P(k) ~ k−γ

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γ=2.4

k

P(k

)

J. Han et al., Nature (2004)

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, <k> = 3.6

Colored according to protein function in the cell:Transcription, Translation, Transcription control, Protein-fate, Genome maintenance, Metabolism, Unknown, etc

Modular structure according to function!

from MIPS database, mips.gsf.de

Metabolic network of biochemical reactions in E.coli

Chemical substrates

Biochemical interactions: enzyme-catalyzed reactions that transform one metabolite into another.

Modular structureaccording to the biochemicalclass of the metabolic productsof the organism.

Colored according to product class:Lipids, essential elements, protein, peptides and amino acids, coenzymes and prosthetic groups, carbohydrates, nucleotides and nucleic acids.

J. Jeong, et al., Nature, 407 651 (2000)

How long is the coastline of Norway?It depends on the length of your ruler.

Fractal Dimension dB-Box Covering Method

Fractals look the same on all scales = `scale-invariant’.

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lB

€

NB

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NB (lB ) ~ lB−d B

Box length

Total no. of boxes

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

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NB (lB ) ~ lB−d B€

lB

€

NB

€

lB

We need the minimum number of boxes: NP-complete optimization problem!We need the minimum number of boxes: NP-complete optimization problem!

Boxing in Biology

Most efficient tiling of the network

4 boxes

5 boxes

1

0

0

1

2

8 node network: Easy to solve

300,000 node network: Mapping to graph colouring problem. Greedy algorithm to find minimum boxes

Larger distances need fewer boxes

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NB (lB ) ~ lB−d B

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NB = 4

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NB = 3

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NB = 2 -dB

log(lB)lo

g(N

B)

1

2

3fractal

non fractal

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dB → ∞

Box covering in yeast: protein interaction network

Most complex networks are Fractal

Metabolic Protein interaction

Song, Havlin, Makse, Nature (2005)

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N(lB ) ~ lB−dB

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dB = 2.3

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dB = 3.5

Biological networks

Three domains of life: archaea, bacteria, eukaria

E. coli, H. sapiens, yeast

43 organisms - all scale

Metabolic networks are fractals

Technological and Social Networks TOO

WWW

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N(lB ) ~ lB−dB

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dB = 6.25

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dB = 4.15

nd.edu domain

Hollywood film actors

212,000 actors

300,000 web-pages

Other bio networks: Khang and Bremen groups Internet is not fractal!

Two ways to calculate fractal dimensions

Box covering method Cluster growing method

In homogeneous systems (all nodes with similar k) both definitions agree:

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NB (lB ) ~ lB−d B

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MC (lC ) ~ lCd f

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< MB >= N /NB ~ lBd B

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dB = d f

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lC

€

lB

percolation

Box Covering= flat average Cluster Growing = biased

power law

Different methods yield different results due to heterogeneous topology

exponential

Box covering reveals the self similarity. Cluster growth reveals the small world. NO CONTRADICTION! SAME HUBS ARE USED MANY TIMES IN CG.

Is evolution of the yeast fractal?

Ancestral Prokaryote Cell

YeastOtherFungi

Ancestral yeast

Animals+ Plants

Ancestral Fungus

Archaea + Bacteria

Ancestral Eukaryote

presentday

~ 300 million years ago

1 billion years ago

1.5 billion years ago

Following the phylogenetic tree of life:

3.5 billion years ago

COG databasevon Mering, et al Nature (2002)

Suggests that present-day networks could have been created following a self-similar, fractal dynamics.

Same fractal dimension and scale-freeexponent over 3.5 billion years…

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dB = 2.3

Renormalization in Complex Networks

NOW, REGARD EACH BOX AS A SINGLE NODEAND ASK WHAT IS THE DEGREEDISRIBUTION OF THE NETWORKOF BOXES AT DIFFERENT SCALES ?

Renormalization of WWW network with

l

3B =l

The degree distribution is invariant under renormalization

Internet is not fractal dB--> infinityBut it is renormalizable

Turning back the timeRepeatedly BOXING the network is the same as going back

in time: from a single node to present day.

renormalization

time evolution

Can we “predict” the past…. ? if not the future.

ancestral node

present daynetwork

THE RENORMALIZATION SCHEME

1

time evolution

Evolution of complex networks

opening boxes

How does Modularity arise?The boxes have a physical meaning =

self-similar nested communities

time evolution

ancestral node

present daynetwork

renormalization

1

How to identify communities in complex networks?

Emergence of Modularity in PINBoxes are related to the biologically relevant functional modules

in the yeast protein interactome

time evolution renormalization

present day network

translation transcription protein-fate cellular-fateorganization

ancestralcell

Emergence of modularity in metabolic networks

Appearance of functional modules in E. coli metabolic network.Most robust network than non-fractals.

Theoretical approachHow the communities/modules are linked?

k: degree of the nodes

k’=2renormalization

s=1/4k=8

k’: degree of the communities

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k' = s(lB ) knode degree

community degree factor<1

Theoretical approach to modular networks: Scaling theory to the rescue

WWW

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s(lB ) ~ lB−d k

The larger the communitythe smaller their connectivity

new exponent describing how families link

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k' = s(lB ) k

Scaling relations

A theoretical prediction relating the different exponents

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NB (lB ) = lB−dB

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s(lB ) = lB−d k€

lB

€

dB

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γ=1+dB

dk

new scaling relation

boxes

distance

degree

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dknew exponent

Scaling relationsThe communities also follow a self-similar pattern

Network dB dk 1+ dB/dk

WWW 4.1 2.5 2.6 2.6

Actor 6.3 5.3 2.2 2.2

E. coli (PIN) 2.3 2.1 2.1 2.2

H. sapiens (PIN) 2.3 2.2 2.0 2.1

43 Metabolic 3.5 3.2 2.1 2.2

WWW Metabolic

Scaling relationworks

€

s(lB )

€

lB

€

lB

€

s(lB )

fractalsfractals communities/modulescommunities/modules

scale-freescale-free

predictionprediction

Why fractality?Some real networks are not fractal

Other models fail too: Erdos-Renyi, hierarchical model, fitness model, JKK model, pseudo-fractals models, etc.

The Barabasi-Albert model of preferentialattachment does not generate fractal networks

All the models fail to predict self-similarity

INTERNET

What is the origin of self-similarity?

HINT: the key to understand fractals is in the degreecorrelations P(k1,k2) not in P(k)

Can you see the difference?

Internet map Yeast protein map

E.coli metabolic map

NON FRACTAL FRACTAL

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

log(k1)log(k1)

log(

k 2)

log(

k 2)

P(k1,k2)

low prob.

low prob.

high prob.

high prob.

Hubs connected with hubs Hubs connected with non-hubs

Quantify anticorrelation between hubsat all length scales

hubs

hubs

Renormalize

Hubs connected directly

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ε(l B ) = 2 /3

Hub-Hub Correlation function: fraction of hub-hub connections

Hub-hub connection organized in a self-similar way

The larger de implies more anticorrelation

(fractal) (non-fractal)

Anticorrelations are essential for fractal structure

non-fractal

fractal

What is the origin of self-similarity?

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

Non-fractal networks

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

Fractal networks

InternetAll available models: BA model, hierarchicalrandom scale free, JKK, etc

WWW, PIN, metabolic, genetic, neural networks, some sociological networks

How to model it? renormalization reverses time evolution

Mode IIMode I

tim

e

)()1( tnNtN =+)()1( tsktk ii =+

Both mass and degree increase exponentially with time

Scale-free: γ−kkP ~)(

s

n

ln

ln1+=γ

offspring nodes attached to their parentsimk

(m=2) in this case

reno

rmal

ize

Song, Havlin, Makse, Nature Physics, 2006

How does the length increase with time?

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N(t) ~ eL(t ) / l0

l0 = 2 /ln n

dB = ∞

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N(t) ~ L(t)d B

dB = ln n /ln3

Mode II: FRACTALMode I: NONFRACTAL

SMALL WORLD

2)()1( +=+ tLtL )(3)1( tLtL =+

)()1( tnNtN =+ ntetN ln~)(

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L(t) = 2 t

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L(t) = e ln 3 t

Combine two modes together

))()(23()1( 00 LtLeLtL +−=++

tim

e )1/(0 eeL −=

)23ln(/ln

)23ln(/ln

eed

end

e

B

−−=−=

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NB (t) ~ lB (t)−dB

ε(t) ~ lB (t)−d e

e=0.5

Mode I with probability e Mode II with probability 1-e

reno

rmal

ize

PredictionsModel reproduces local small world, scale-free and fractality

model with e=1• attraction between hubs• non-fractal• small world globally

model with e=0.2• repulsion between hubs leads to fractal topology• small world locally inside well defined communities

h.sapiensyeast

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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lB (t) + l0 = at

N(t) = n t

k(t) = st

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N(lB ) ~ lBd B

s(lB ) ~ lBd k

Model predicts all exponents in terms ofgrowth 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:

a

nd B ln

ln=

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γ=1+lnn

ln s=1+

dB

dka

sd k ln

ln=

We predict the fractal exponents:

Time evolution in yeast network

Multiplicative and exponential growth in yeast PINLength-scales, number of conserved proteins and degree

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k(t) = eα k t

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dB =α N

α l

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γ=1+α N

α k

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dk =α k

α l

€

N(t) = eα N t

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l(t) = eα l t

A new principle of network dynamics 1930solid-state physicsbig world

1960Erdos-Renyi model small world

democracy=socialism

1999BA model “rich-get-richer”=

capitalism

2005fractal model“rich-get-richer”

at the expense of the “poor”=globalization

less vulnerable to intentional attacks

Summary

• 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

m = 2

An finally, a model to put all this together

A multiplicative growth processof the number of nodes and links

Probability ehubs always connected

strong hub attractionshould lead to non-fractal

Probability 1-ehubs never connectedstrong hub repulsionshould lead to fractal

Analogous to duplication/divergence

mechanism in proteins??

For the both models, each step the total number of nodes scale as n = 2m +1( N(t+1) = nN(t) ). Now we investigate the transformation of the lengths. They show quite different ways for this two models as following:

2)()1( +=+ tLtL aa )(3)1( tLtL bb =+Then we lead to two different scaling law of N ~ L

)2/ln(~ 0/ 0 nLeN LL

aa =

)3ln/ln(~ ndLN Bd

bbB =

Mode III: L(t+1) =3L(t)Mode II: L(t+1) = 2L(t)+1

Mode I: L(t+1) = L(t)+2

smaller

smaller

Different growth modes lead to differenttopologies

Suppose we have e probability to have mode I, 1-e probability to have mode II and mode III. Then we have:

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L(t +1) = (3− 2 e)L(t) + 2 e

]2/)1(,[ eefep +⊂+=

taltL ~))(( 0+

pa 23−=

or

)1/(0 ppl −=

Dynamical model

Graph theoretical representation of a metabolicGraph theoretical representation of a metabolicnetworknetwork

(a) A (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.

Classes of genes in the yeast proteome

Renormalization following the phylogenetic treeRenormalization following the phylogenetic tree

P. Uetz, et al. Nature 403 (2000).