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**Classes will begin shortly**

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**Networks, Complexity and Economic Development**

Class 2: Scale-Free Networks Cesar A. Hidalgo PhD

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WATTS & STROGATZ Lattice Erdös-Rényi model (1960) Poisson distribution

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**High school friendship**

James Moody, American Journal of Sociology 107, (2001)

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**High school dating network**

Data: Peter S. Bearman, James Moody, and Katherine Stovel. American Journal of Sociology 110, (2004) Image: M. Newman

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**Previous Lecture Take Home Messages**

NETWORKS -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 size L~ 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.

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Degree (k) Degree Distribution P(k) k

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The Crazy 1990’s

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www Internet Autonomous System i.e. Harvard.edu

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

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Internet-Map

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**WWW World Wide Web Nodes: WWW documents Links: URL links**

Expected Over 3 billion documents Exponential Network ROBOT: collects all URL’s found in a document and follows them recursively P(k) ~ k- Found Scale-free Network R. Albert, H. Jeong, A-L Barabasi, Nature, (1999).

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**Scale-Free Networks Everywhere**

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**Coauthorship SCIENCE COAUTHORSHIP (Newman, 2000, A.-L. B. et al 2001)**

Nodes: scientist (authors) Links: write paper together (Newman, 2000, A.-L. B. et al 2001)

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**SCIENCE CITATION INDEX**

25 H.E. Stanley,... Nodes: papers Links: citations 1736 PRL papers (1988) P(k) ~k- ( = 3) (S. Redner, 1998)

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**Swedish sex-web Nodes: people (Females; Males)**

Links: sexual relationships 4781 Swedes; 18-74; 59% response rate. Liljeros et al. Nature 2001

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**Metab-movie Metabolic Network Nodes: chemicals (substrates)**

Links: bio-chemical reactions Metabolic Network

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**Meta-P(k) Metabolic network Archaea Bacteria Eukaryotes**

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, (2000)

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**Prot P(k) Protein interaction network Nodes: proteins**

Links: physical interactions (binding) H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, (2001)

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**Human Interaction Network**

2,800 Y2H interactions 4,100 binary LC interactions (HPRD, MINT, BIND, DIP, MIPS) Rual et al. Nature 2005; Stelze et al. Cell 2005

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**Explaining Scale-Free Networks**

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**BA model Scale-free model**

(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 links PREFERENTIAL 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 sites Citation : citing again highly cited papers Web application: Barabási & Albert, Science 286, 509 (1999)

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**MFT Mean Field Theory γ = 3**

, with initial condition γ = 3 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)

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Model A growth preferential attachment Π(ki) : uniform

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**Model B P(k) : power law (initially) Gaussian**

growth preferential attachment P(k) : power law (initially) Gaussian

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A note on the BA model Yule process Price Model

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Beyond the BA Model

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**Movie Actors WWW Lada A Adamic, Bernardo A Huberman Technical Comments**

Power-Law Distribution of the World Wide Web Science 24 March 2000: Vol no. 5461, p DOI: /science a A-L Barabasi, R Albert, H Jeong, G Bianconi Technical Comments Power-Law Distribution of the World Wide Web Science 24 March 2000: Vol no. 5461, p DOI: /science a Movie Actors WWW

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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 (h ) k(h,t)~tb(h) where b(h) =h/C G. Bianconi and A.-L. Barabási, Europhyics Letters. 54, 436 (2001).

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**Local Rules Random Walk Model qe qv 1-qe A Vazquez**

Growing network with local rules: Preferential attachment, clustering hierarchy, and degree correlations Physical Review E 67, (2003) Random Walk Model qe qv 1-qe

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**The easiest way to find a hub?**

Ask for a friend!!! Pick a random person and ask that person to name a friend.

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

P(k)=1/k Picking a link and looking for a node at the edge of it gives you a uniform distribution of degrees!

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**More models Other Models**

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

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**Why scale-free? What functions satisfy this functional relationship?**

F(ax)=bF(x) F(x)=xP (ax)P=aPxP=bxp

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Power-Laws Big deal!

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**Tokyo~30 million in metro area**

Santiago ~ 6 million metro area Curico~100k people New York~18 million in metro area

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**P~1/x Number of Cities Size of Cities**

16 x 4 million cities Tokyo ~30 million 4 x 8 million cities New York, Mexico City ~15 million Number of Cities P~1/x Size of Cities There is an equivalent number of people living in cities of all sizes!

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

~ $50 billion

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

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

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**Statistics of Power-Laws**

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**Power-Laws are dominated by largest value**

AVERAGES

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**Power-Laws are dominated by largest value MEDIANS**

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**Power-Laws are dominated by largest value COMPARING MEDIANS AND AVERAGES**

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**Power-Laws have diverging VARIANCE**

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**Why physicists were interested in Power-Laws**

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F=-GMm/r2

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Phase transitions

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**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.

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**Error and Attack Tolerance**

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**Robustness Robustness node failure**

Complex systems maintain their basic functions even under errors and failures (cell mutations; Internet router breakdowns) fc 1 Fraction of removed nodes, f S node failure

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**Robust-SF Robustness of scale-free networks S f Attacks Failures**

1 S f Attacks Failures 3 : fc=1 (R. Cohen et al PRL, 2000) C fc Albert, Jeong, Barabasi, Nature (2000)

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**Achilles Heel Achilles’ Heel of complex networks**

failure attack Internet R. Albert, H. Jeong, A.L. Barabasi, Nature (2000)

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Epidemic Threshold

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**SIS Model ds/dt = -asi+bi di/dt =r(1-i)i-i di/dt =asi-bi**

(compartmental model) S+I=1 di/dt =r(1-i)i-i di/dt =ri-ri2-i di/dt=i(r-ri-1) di/dt=0 -> i=1-1/r ds/dt = -asi+bi di/dt =asi-bi ds/dt = -rsi+i di/dt =rsi-i r=a/b

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**I I=1-1/r r= 1 dS/dt > 0 dI/dt <0 dS/dt < 0 dI/dt > 0**

Epidemic Threshold I Stable solution Unstable solution I=1-1/r

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

We now have many compartments Sk , Ik dik/dt =-ik+rk(1-ik)S ik’P(k,k’) dik/dt =-ik+rk(1-ik)q ik=rkq/(1+rkq) (1) q=<k>-1S ikkP(k) (2) (1)->(2) q=<k>-1S kP(k) rkq/(1+rkq) R. Pastor-Satorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters 86, (2001).

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**df/dq |q=0≥1 -> r<k2>/<k> ≥ 1 r ≥ <k>/<k2>**

q=<k>-1S kP(k) rkq/(1+rkq)=f(q) q=q f(q) df/dq |q=0≥1 -> r<k2>/<k> ≥ 1 r ≥ <k>/<k2> R. Pastor-Satorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters 86, (2001).

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**There is no epidemic threshold!!!**

Infected rc R. Pastor-Satorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters 86, (2001).

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**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.

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Bonus Section Fractals

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**Measuring the Dimension of Koch Curve**

Generating Koch Curve Measuring the Dimension of Koch Curve

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White Noise Pink Noise Brown Noise

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**Extra Bonus Mandelbrot and Julia Set**

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Xn+1=Xn2+C (Mandelbrot set X0 =0) Main Bulb Decoration Antenna

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