04 CLASS 2012 Scale-Free Property

8/13/2019 04 CLASS 2012 Scale-Free Property

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Class 4: Scale-Free Property

Prof. Albert-Lszl BarabsiDr. Baruch Barzel, Dr. Mauro Martino

Network Science: Scale-F


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lrand

=

logN

log k

Empirical data for real networks

C ~ const

P(k

Regular

network

Erdos-

Renyi

Watts-

Strogatz

Pathlength

Clustering

Degree

l " N1/ D

klog

Nloglrand

klog

Nloglrand

C ~ const

C ~ const

N

kpCrand

P(k)=!(

P(k) = e"

Expone



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Nodes: WWW documentsLinks: URL links

Over 3 billion documents

ROBOT: collects all URLsfound in a document andfollows them recursively

Exp

ected

P(k) ~ k-!Found

R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999).

WORLD WIDE WEB



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Expected

P(k) ~ k-!

Found

R. Albert, H. Jeong, A-L Barabasi, Nature, 40

Degree distribution of the WWW



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The difference between a power law and an exponential distributio

20 40 60 80 1000.20.61

1cx)x(f !=

xc)x(f !=

50.cx)x(f !=

Above a certain x value, the power law is always higher than the exponential.



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semilog

10

100 101 102 103

-410-310-210-1100

loglog

This difference is particularly obvious if we plot them on a log vertical scale: for large xthere are orders of magnitude differences between the two functions.

1cx)x(f !=

xc)x(f !=

cx)x(f !=

xc)x(f !=

50.cx)x(f !=

c)x(f =

The difference between a power law and an exponential distributio



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Over 3 billiondocumentsROBOT:collects all URLsfound in a document and

follows them recursively

Nodes: WWW documentsLinks: URL links

P(k) ~ k-!

Scale-free

Network

Exponential

Network

What does the difference mean? Visual representation.

R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999).Network Science: Scale-F


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WORLD WIDE WEB



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Definition:

Networks with a power law tail in their degree distribution are calledscale-free networks

Where does the name come from?

Critical Phenomena and scale-invariance(a detour)

Slides after Dante R. Chialvo

Scale-free networks: Definition



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|m|

TcT

m:order

param

eter

order disorder

interacting elementary (spins) sitting in a lattic

Neighboring spins like t

in the same direction

If the temperature T is

attraction is not sufficie

there is no net magneti

If the temperature is low

ferromagnetic order seis a phase transition at

T = 0.99 Tc T = 0.999Tc

"

T = Tc T = 1.5 Tc T = 2 Tc

FERROMAGNETIC MATERIALS



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At T = Tc: correlation length diverges

Fluctuations emerge at all scales: scale-free behavior

Scale-free behavior in space



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At T = Tc

Scale-free behavior in time



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1-D

2-D

3-D

Low T High T Solved

Ising 1925

Onsager 194

Provencomputationallintractable - 20

INSING MODEL and universality

Universality:while T

cand many other paramters depend on the details of the system, the critical

exponents, like !or ", do not. The exponents are universal, which means that they depend only on the

dimension of the space and the underlying symmetries of the problem. Hence many different systems

magnets to liquieds and superconductors, share the same exponents.Network Science: Scale-F


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Correlation length diverges at the critical point: thewhole system is correlated!

Scale invariance: there is no characteristic scale forthe fluctuation (scale-free behavior).

! Universality:exponents are independent of thesystems details.

CRITICAL PHENOMENA



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Now we see where the scale-free name comes from.

But whats in the name?

We need to learn a bit more to understand that

soon

Before we get there, let us talk about universality.

SCALE-FREE NETWORKS



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UniversalityHow generic is our finding of a power law degree distribution?



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(Faloutsos, Faloutsos and Faloutsos, 1999)

Nodes: computers, routersLinks: physical lines

INTERNET BACKBONE



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(= 3)

(S. Redner, 1998)

P(k) ~k-

1736 PRL papers (1988)

SCIENCE CITATION INDEX

Nodes: papersLinks: citations

578...

25

H.E. Stanley,...



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(Bacon game for Brainiacs )

Number of links required to connect scto Erd#s, via co-authorship of pap

Erd#s wrote 1500+ papers with 507 coauthors.

Jerry Grossmans (Oakland Univ.) we

allows mathematicians to computeErdos numbers:

http://www.oakland.edu/enp/

Connecting path lengths, amongmathematicians only:

$ average is 4.65$ maximum is 13

Paul Erd"s (1913-1996)

Erdos has better centrality inhis network than Bacon has inhis.

ERDOS NUMBER



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SCIENCE COAUTHORSHIP


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SCIENCE COAUTHORSHIP

M: mathNS: neuro

Nodes: scientist (authors)Links: joint publication

(Newman, 2000, Barabasi et al 2001)


ONLINE COMMUNITIES


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Nodes: online userLinks: email contact

Ebel, Mielsch, Bornholdtz, PRE 2002.

Kiel University log files112 days, N=59,912 nodes

Pussokram.com online community512 days, 25,000 users.

Holme, Edling, Liljeros, 2002.

ONLINE COMMUNITIES

ONLINE COMMUNITIES


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Nodes: online userLinks: email contact

ONLINE COMMUNITIES

Twitter:

Jake HofAlan Mislove, Measurement and Analysis of Online Social Networks

All distribtions show a fat-tail behavior:there are orders of magnitude spread in the degrees


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protein-geninteraction

protein-protinteractions

PROTEOM

GENOM

Citrate Cycle

METABOLI

Bio-chemreactions


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Citrate Cycle

METABOLI

Bio-chemreactions

BOEHRING MENNHEIN


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BOEHRING-MENNHEIN


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protein-geninteraction


PROTEOM

GENOM

Citrate Cycle

METABOLI

Bio-chemreactions

METABOLIC NETWORK


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PROTEOM

METABOLIC NETWORK


TOPOLOGY OF THE PROTEIN NETWORK


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)exp()(~)( 00!

"

k

kkkkkP +

#+ #

H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-4

Nodes: proteinsLinks: physical interactions-binding


C. Elegans Drosophila M.


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Li et al.Science 2004 Giot et al.Science 200

HUMAN INGTERACTION NETWORK


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2,800 Y2H interactions4,100 binary LC interactions(HPRD, MINT, BIND, DIP, MIPS)

Rual et al.Nature 2005; Stelze et al.Cell 2005

HUMAN INGTERACTION NETWORK


ACTOR NETWORK


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Nodes: actorsLinks: cast jointly

N = 212,250 actors"k#= 28.78

P(k) ~k-!

Days of Thunder (1990)Far and Away (1992)Eyes Wide Shut (1999)

!=2.3

SWEDISH SE-WEB


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Nodes:people (Females; Males)Links: sexual relationships

Liljeros et al. Nature 2001

4781 Swedes; 18-74;

59% response rate.


SCALE-FREE NETWORKS


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Many real world networks have a similar architecture:

Scale-free networks

WWW, Internet (routers and domains), electronic circuits, computer software, movieactors, coauthorship networks, sexual web, instant messaging, email web, citations,phone calls, metabolic, protein interaction, protein domains, brain function web,linguistic networks, comic book characters, international trade, bank system,

encryption trust net, energy landscapes, earthquakes, astrophysical network


UNIVERSALITY AGAIN


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Critical phenomena:Universality means that the expare the same for different system

they are independent of details.

Networks:The exponents vary from systesystem.Most are between 2 and 3

Universality:the emergence of common featacross different networks. Like scale-free property.



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Divergences inscale-free

networks


SCALE-FREE DISTRIBUTION: DISCRETE FORMALISM


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pk =Ck"#

pkk=1

"

# =1

C k"#

k=1

$

% =1

C= 1

k"#

k=1

$

%=

1&(#)

Riemann Zeta function

pk =k"

#

$(#)

for k>0 (i.e. we assume that there are no disconnected nodesthe network)


SCALE-FREE DISTRIBUTION: DISCRETE FORMALISM


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pk =Ck"# k=[Kmin,$)

pkk=Kmin

"

# =1

C=1

k"#

k=Kmin

$

%=

1

&(#,Kmin )

Generalized or incomplete Zeta function

pk =k"#

$(#,Kmin )

For some applications we only care about the tail of the degree distribution


SCALE-FREE DISTRIBUTION: CONTINUUM FORMALISM


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C=1

k"#dk

Kmin

$

%=(#"1)Kmin

#"1

P(k) =Ck"#

k=[Kmin

,$)

P(k)Kmin

"

# dk=1

P(k) =("#1)Kmin"#1

k#"


DIVERGENCE OF THE HIGHER MOMENTS


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k=[Kmin ,")P(k) =("#1)Kmin"#1

k#"

m-th moment of the degree distribution: < km>= k

mP(k)dk

Kmin

"

#

< km>= ("#1)Kmin

"#1km#"

dk

Kmin

$

% =("#1)

(m # "+1)Kmin

"#1km#"+1[ ]

Kmin

$

If m-!+1= "

(#"1)

(m " #+1)Kmin

m

If m-!+1>0, the integral diverges.For a fixed !this means that all moments with m>!-1 diverge.


DIVERGENCE OF THE HIGHER MOMENTS


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< km>= ("#1)Kmin

"#1km#$

dk

Kmin

%

& =("#1)

(m # "+1)Kmin

"#1km#"+1[ ]

Kmin

%

For a fixed $this means all moments m>!-1 diverge.

Most degree exponents are smallethan 3

" diverges in the N"#limit


What is the meaning of the observed divergence?


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Most degree exponents are smaller than 3" diverges!!!

What does it mean?

WWW: = 7

Internet: = 3.5

Metabolic: = 7.4

Phone call: = 3.16

"k= (< k

2> # < k>

2)

1 / 2$%

k=< k> "k

The average values are not meaningful, as fluctuations are too large!


FINITE SCALE-FREE NETWORKS


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All real networks are finite " let us explore its consequences."We have an expected maximum degree, Kmax

Estimating Kmax

P(k)dk

Kmax

"

# $ 1

N

Kmax

=KminN

1

"#1

Why: the probability to have a node larger than Kmax should nexceed the prob. to have one node, i.e. 1/N fraction of all nod

P(k)dk

Kmax

"

# = ($%1)Kmin$%1 k%$dkKmax

"

# = ($%1)

(%$+1)Kmin

$%1k%$+1[ ]

Kmax

"=Kmin

$%1

Kmax$%1

& 1N


DISTANCES IN RANDOM GRAPHS


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Random graphs tend to have a tree-like topology with almost constant node degrees.

nr. of first neighbors:

nr. of second neighbors:

nr. of neighbours at distance d:

estimate maximum distance:

klog

Nloglmax

maxl

1l

i

Nk1

kN1

2 kN

Nd " k


Distances in scale-free networks

DISTANCES IN SCALE-FREE NETWORKS


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Size of the biggest hub is of order O(N). Most nodes can be connected withof it, thus the average path length will be independent of the system size.

The average path length increases slower than logarithmically. In a randomnodes have comparable degree, thus most paths will have comparable leng

scale-free network the vast majority of the path go through the few high degreducing the distances between nodes.

Some key models produce !=3, so the result is of particular importance for was first derived by Bollobas and collaborators for the network diameter in t

a dynamical model, but it holds for the average path length as well.

The second moment of the distribution is finite, thus in many ways the netwas a random network. Hence the average path length follows the result that

for the random network model earlier.

Cohen, Havlin Phys. Rev. Lett. 90, 58701(2003); Cohen, Havlin and ben-Avraham, in Handbook of Graphs and Networks, Eds. BShuster (Willy-VCH, NY, 2002) Chap. 4; Confirmed also by: Dorogovtsev et al (2002), Chung and Lu (2002); (Bollobas, Riordan, 21985; Newman, 2001

Ultra

Small

World

Small

World

Kmax

=KminN

1

"#1

SUMMARY OF THE BEHAVIOR OF SCALE-FREE NETWORKS


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!=1 !=2 !=3

diverges


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diverges


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Kmax

=KminN

1

"#1

In order to document a scale-free networks, we need 2-3 orders of magnitude scalingThat is, Kmax~ 10

3

However, that constrains on the system size we require to document it.

For example, to measure an exponent #=5,we need to maximum degree a system sizthe order of

N =K

max

Kmin

"

#$

%

&'

()1

*1012

Onella et al. PNAS 2007

N%

Mobile CallNetwork



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Divergences in sca


CLEANING UP DEGREE DISTRIBUTIONS


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Often it is difficult to determine the best fit to the degree distribution.

Methods of data cleanup:

1. logarithmic binning: bin the k range; use bins of exponentially increasing size(applied to PDF, or probability distribution function)

2. Display the cumulative degree distribution (CDF)

Ex. Determine the degree distribution and

cumulative degree distribution of the graph

on the right.

)Kk(P)Kk(P

)k(P)Kk(PK

kk min

!"=>

=! #=

1

or


CLEANING UP DEGREE DISTRIBUTIONS


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log( )x

cx!"

#

Probability that node

has degree x.

cx"

#

Probability that a node has adegree bigger than x.

)x(P

If the (noncumulative)degree distributiondecays with a slope&>1,

the cumulative degreedistributionwill decay with slope

&-1.Does not apply for

&=1!

)xX(P >


HUMAN INTERACTION NETWORK


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2,800 Y2H interactions

4,100 binary LC interactions

(HPRD, MINT, BIND, DIP, MIP

Rual et al. Nature 2005; Stelze et al. Cell 2005Network Science: Scale-F

HUMAN INTERACTION DATA BY RUAL ET AL.


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(linear scale)

P(k) ~ k!!!'2


HUMAN INTERACTION DATA BY RUAL ET AL.


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(linear scale)

P(k) ~ (k+k0)!!

k0'1.4, !'2.6.


COMMON MISCONCEPTIONS


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Common Misconceptions

-if there is a low-k saturation, it is not scale-fr

-if there is a high-k cutoff, it is not scale-free

Most real networks:

P(k) ~ (k+k0)-"exp(-k/k1)

low-k saturation

High-k cutoff




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)exp()(~)( 00!

"

k

kkkkkP +

#+ #

H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001)

Nodes: proteinsLinks: physical interactions-binding


SUMMARY OF THE BEHAVIOR OF SCALE-FREE NETWORKS

collab


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diverges


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04 CLASS 2012 Scale-Free Property

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