Networks, Complexity and Economic Development

Class 2: Scale-Free NetworksCesar A. Hidalgo PhD

Erdös-Rényi model (1960)Lattice

Poisson distribution

WATTS & STROGATZ

High school friendshipJames Moody, American Journal of Sociology 107, 679-716 (2001)

High school dating networkData: Peter S. Bearman, James Moody, and Katherine Stovel. American Journal of Sociology 110, 44-91 (2004)Image: M. Newman

6

Previous Lecture Take Home MessagesNETWORKS-Networks can be used to represent a wide set of systems-The properties of random networks emerge suddenly as a function of connectivity.-The distance between nodes in random networks is small compared to network sizeL log(N)-Networks can exhibit simultaneously: short average path length and high clustering(SMALL WORLD PROPERTY)-The coexistence of these last two properties cannot be explained by random networks-The small world property of networks is not exclusive of “social” networks.

BONUS-Deterministic Systems are not necessarily predictable.-But you shouldn’t always blame the butterfly.

Degree (k)

P(k)

k

Degree Distribution

The Crazy 1990’s

Internet

www

Autonomous Systemi.e. Harvard.edu

"On Power-Law Relationships of the Internet Topology",Michalis Faloutsos, Petros Faloutsos, Christos Faloutsos, ACM SIGCOMM'99, Cambridge, Massachussets,pp 251-262, 1999

Over 3 billion documents

ROBOT: collects all URL’s found in a document and follows them recursively

Nodes: WWW documents Links: URL links

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

Expected

P(k) ~ k-

FoundSca

le-f

ree N

etw

ork

Exponenti

al N

etw

ork

Nodes: scientist (authors) Links: write paper together

(Newman, 2000, A.-L. B. et al 2001)

SCIENCE COAUTHORSHIP

SCIENCE CITATION INDEX

( = 3)

Nodes: papersLinks: citations

(S. Redner, 1998)

P(k) ~k-

1078...

25

H.E. Stanley,...1736 PRL papers (1988)

Swedish sex-web

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

Liljeros et al. Nature 2001

4781 Swedes; 18-74; 59% response rate.

Metabolic NetworkNodes: chemicals (substrates)Links: bio-chemical reactions

Metabolic network

Organisms from all three domains of life have scale-free metabolic networks!

H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000)

Archaea Bacteria Eukaryotes

Protein interaction network

)exp()(~)( 00

k

kkkkkP

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

Nodes: proteins Links: physical interactions (binding)

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

Human Interaction Network

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

Scale-free model

Barabási & Albert, Science 286, 509 (1999)

jj

ii k

kk

)(

(1) Networks continuously expand by the addition of new nodes

WWW : addition of new documents Citation : publication of new papers

GROWTH: add a new node with m linksPREFERENTIAL ATTACHMENT: the probability that a node connects to a node with k links is proportional to k.(2) New nodes prefer to link to highly

connected nodes.

WWW : linking to well known sitesCitation : citing again highly cited papers

Web application:http://www-personal.umich.edu/~ladamic/NetLogo/PrefAndRandAttach.html

Mean Field Theory

γ = 3

t

k

k

kAk

t

k i

j j

ii

i

2)(

ii t

tmtk )(

, with initial condition0)( mtk ii

)(1)(1)())((

02

2

2

2

2

2

tmk

tm

k

tmtP

k

tmtPktkP ititi

33

2

~12))((

)(

kktm

tm

k

ktkPkP

o

i

A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)

k

ii

ii

em

k

m

ekP

tm

tmmtk

tm

mkA

t

k

~)exp()(

1)1

1ln()(

1)(

0

0

Model A growth preferential attachment

Π(ki) : uniform

tN

CttNN

Ntk

Nt

k

N

N

NkA

t

k

N

N

i

ii

i

2~

)2(

)1(2)(

1

21

1)(

)1(2

Model B

growth preferential attachment

P(k) : power law (initially) Gaussian

Yule process

Price Model

Lada A Adamic, Bernardo A HubermanTechnical CommentsPower-Law Distribution of the World Wide WebScience 24 March 2000:Vol. 287. no. 5461, p. 2115DOI: 10.1126/science.287.5461.2115a

WWW

A-L Barabasi, R Albert, H Jeong, G BianconiTechnical Comments

Power-Law Distribution of the World Wide WebScience 24 March 2000:

Vol. 287. no. 5461, p. 2115DOI: 10.1126/science.287.5461.2115a

Movie Actors

Can Latecomers Make It? Fitness Model

SF model: k(t)~t ½ (first mover advantage)Real systems: nodes compete for links -- fitness

Fitness Model: fitness (

k(,t)~t

where =C

G. Bianconi and A.-L. Barabási, Europhyics Letters. 54, 436 (2001).

11/

1)(

Cd

j jj

iii k

kk

)(

Local RulesRandom Walk Model

qe

A VazquezGrowing network with local rules: Preferential attachment, clustering hierarchy, and degree correlations

Physical Review E 67, 056104 (2003)

1-qe

qv

The easiest way to find a hub?

Ask for a friend!!!

Pick a random person and ask that person to name a friend.

Pick a link!Distribution of degrees on the edge of a link is = kP(k)

P(k)=1/kPicking a link and looking for a node at the edge of

it gives you a uniform distribution of degrees!

R. Albert, A.-L. Barabasi, Rev. Mod. Phys 2002

Why scale-free?

F(ax)=bF(x)

What functions satisfy this functional relationship?

F(x)=xP

(ax)P=aPxP=bxp

Tokyo~30 million in metro area

New York~18 million in metro area

Santiago ~ 6 million metro area

Curico~100k people

Size of Cities

Num

ber o

f Citi

es

Tokyo30 million

New York,Mexico City15 million

4 x 8 millioncities

16 x 4 millioncities

P1/x

There is an equivalent number of people living in cities of all sizes!

$50 billion

After Bill enters the arena the average income of the public ~ 1,000,000

Power laws everywhere

Power-law distributions in empirical data, Aaron Clauset, Cosma Rohilla Shalizi, and M. E. J. Newman, submitted to SIAM Review.

Power laws everywhere

Power-law distributions in empirical data, Aaron Clauset, Cosma Rohilla Shalizi, and M. E. J. Newman, submitted to SIAM Review.

Power-Laws are dominated by largest valueAVERAGES

minminmax

minmax

2max

2min

maxmin3

maxminminmax

min

maxminmax

maxmin

minmax2

maxminmax

minmax

minmax

2~2

)11

(21

/1/1/)(

)log(~)log(

/1/1

)/log(/)(

~)/log(

)(/)(

;)(max

min

kkk

kk

kk

kkkAkP

kkkk

kk

kk

kk

kkkAkP

kkk

kkkAkP

kkdkkkPkk

k

Power-Laws are dominated by largest valueMEDIANS

minmin

2max

2

min2

max2

3

minminmax

minmax2

minmax

minmax

2~2

/)(

2~2

/)(

/)(

;2

1)(

min

kkk

kkkkAkP

kkk

kkkkAkP

kkkkAkP

kkdkkPk

med

med

med

k

k

med

med

Power-Laws are dominated by largest valueCOMPARING MEDIANS AND AVERAGES

2/)(

2

)log(

2

)log(/)(

/)(

/

3

max

min

maxmin2

min

max

minmax

max

kAkP

k

k

kkkAkP

k

k

kk

kkAkP

kk med

Power-Laws have diverging VARIANCE

)log(~)log(

/1/1

)/log(/)(

~)/log(

)(/)(

)/log(/)(

;)(

maxminminmax

min

maxminmax

maxmin

minmax

maxminmax

minmax2

minmax

min2

max2

minmax22

3

max

min

max

min

kkkk

kk

kk

kk

kkkAkP

kkk

kkkAkP

kk

kkdkkkAkP

kkdkkPkk

k

k

k

k

F=-GMm/r2

Self-Organized Criticality

Bak, P., Tang, C. and Wiesenfeld, K. (1987). "Self-organized criticality: an explanation of 1 / f noise". Physical Review Letters 59: 381–384.

RobustnessComplex systems maintain their basic functions even under errors and failures (cell mutations; Internet router breakdowns)

node failure

fc

0 1Fraction of removed nodes, f

1

S

Robustness of scale-free networks

1

S

0 1f

fc

Attacks

3 : fc=1(R. Cohen et al PRL, 2000)

Failures

Albert, Jeong, Barabasi, Nature 406 378 (2000)

C

Achilles’ Heel of complex networks

Internet

failure

attack

R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)

SIS Model(compartmental model)

ds/dt = -asi+bidi/dt =asi-bi

ds/dt = -rsi+idi/dt =rsi-i

r=a/b

S+I=1

di/dt =r(1-i)i-idi/dt =ri-ri2-idi/dt=i(r-ri-1)

di/dt=0 i=1-1/r

r= 1

dS/dt > 0dI/dt <0

dS/dt < 0dI/dt > 0

Epidemic Threshold

I

Stable solutionUnstable solution

I=1-1/r

R. Pastor-Satorras and A. Vespignani. Epidemic spreading in scale-free networks.

Physical Review Letters 86, 3200-3203 (2001).

dik/dt =-ik+rk(1-ik)ik’P(k,k’)dik/dt =-ik+rk(1-ik)

ik=rk/(1+rk) (1)

k-1ikkP(k) (2)

(1)(2)

k-1kP(k) rk/(1+rk)

We now have many compartments

Sk , Ik

k-1kP(k) rk/(1+rk)=f()

ffff

df/d|=0≥1 rk2k ≥ 1r ≥ kk2

R. Pastor-Satorras and A. Vespignani. Epidemic spreading in scale-free networks.

Physical Review Letters 86, 3200-3203 (2001).

R. Pastor-Satorras and A. Vespignani. Epidemic spreading in scale-free networks.

Physical Review Letters 86, 3200-3203 (2001).

rc

Infe

cted

There is no epidemic threshold!!!

Take home messages

-Networks might look messy, but are not random.-Many networks in nature are Scale-Free (SF), meaning that just a few nodes have a disproportionately large number of connections.-Power-law distributions are ubiquitous in nature.-While power-laws are associated with critical points in nature, systems can self-organize to this critical state.- There are important dynamical implications of the Scale-Free topology.-SF Networks are more robust to failures, yet are more vulnerable to targeted attacks.-SF Networks have a vanishing epidemic threshold.

GeneratingKoch Curve

Measuring the Dimension of Koch Curve

White Noise

Pink Noise

Brown Noise

Extra Bonus Mandelbrot and

Julia Set

Xn+1=Xn2+C (Mandelbrot set X0 =0)

Main Bulb

Decoration

Antenna