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Classes will begin shortly. Networks, Complexity and Economic Development. Class 2: Scale-Free Networks Cesar A. Hidalgo PhD. WATTS & STROGATZ. Poisson distribution. Lattice. Erdös-Rényi model (1960 ). High school friendship - PowerPoint PPT Presentation
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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