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Networks, Complexity and Economic Development Class 2: Scale-Free Networks Cesar A. Hidalgo PhD

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

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

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

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6 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 shouldnt always blame the butterfly.

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Degree (k) P(k) k Degree Distribution

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The Crazy 1990s

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Internet www 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 251-262, 1999

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Internet-Map

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Over 3 billion documents ROBOT: collects all URLs 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). WWW Expected P(k) ~ k - Found Scale-free Network Exponential Network

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

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SCIENCE CITATION INDEX ( = 3) Nodes: papers Links: citations (S. Redner, 1998) P(k) ~k - 1078... 25 H.E. Stanley,... 1736 PRL papers (1988) Citation

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Swedish sex-web Nodes: people (Females; Males) Links: sexual relationships Liljeros et al. Nature 2001 4781 Swedes; 18-74; 59% response rate.

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Metabolic Network Nodes : chemicals (substrates) Links : bio-chemical reactions Metab-movie

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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 Meta-P(k)

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Protein interaction network H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001) Prot P(k) Nodes: proteins Links: physical interactions (binding)

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

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Scale-free model Barabási & Albert, Science 286, 509 (1999) BA 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: http://www-personal.umich.edu/~ladamic/NetLogo/PrefAndRandAttach.html

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

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Model A growth preferential attachment Π(k i ) : uniform

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

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Yule process Price Model

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Lada A Adamic, Bernardo A Huberman Technical Comments Power-Law Distribution of the World Wide Web Science 24 March 2000: Vol. 287. no. 5461, p. 2115 DOI: 10.1126/science.287.5461.2115a WWW A-L Barabasi, R Albert, H Jeong, G Bianconi Technical Comments Power-Law Distribution of the World Wide Web Science 24 March 2000: Vol. 287. no. 5461, p. 2115 DOI: 10.1126/science.287.5461.2115a Movie Actors

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Can Latecomers Make It? Fitness Model SF model: k(t)~t ½ (f irst 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).

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Local Rules Random Walk Model qeqe A Vazquez Growing network with local rules: Preferential attachment, clustering hierarchy, and degree correlations Physical Review E 67, 056104 (2003) 1-q e qvqv

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

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Why scale-free? F(ax)=bF(x) What functions satisfy this functional relationship? F(x)=x P (ax) P =a P x P =bx p

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Tokyo~30 million in metro area New York~18 million in metro area Santiago ~ 6 million metro area Curico~100k people

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

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$50 billion After Bill enters the arena the average income of the public ~ 1,000,000

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Power laws everywhere Power-law distributions in empirical dataPower-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 dataPower-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 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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F=-GMm/r 2

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

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Robustness Complex systems maintain their basic functions even under errors and failures (cell mutations; Internet router breakdowns) node failure fcfc 01 Fraction of removed nodes, f 1 S Robustness

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Robustness of scale-free networks 1 S 0 1 f fcfc Attacks 3 : f c =1 (R. Cohen et al PRL, 2000) Failures Robust-SF Albert, Jeong, Barabasi, Nature 406 378 (2000) C

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Achilles Heel of complex networks Internet failure attack Achilles Heel R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)

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SIS Model (compartmental model) ds/dt = -asi+bi di/dt =asi-bi ds/dt = -rsi+i di/dt =rsi-i r=a/b S+I=1 di/dt =r(1-i)i-i di/dt =ri-ri 2 -i di/dt=i(r-ri-1) di/dt=0 i=1-1/r

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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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R. Pastor-Satorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters 86, 3200-3203 (2001). di k /dt =-i k +rk(1-i k ) i k P(k,k) di k /dt =-i k +rk(1-i k ) i k =rk /(1+rk ) (1) k -1 i k kP(k) (2) (1) (2) k -1 kP(k) rk /(1+rk ) We now have many compartments S k, I k

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k -1 kP(k) rk /(1+rk )=f( ) f f f f df/d | =0 1 r k 2 k 1 r k k 2 R. Pastor-Satorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters 86, 3200-3203 (2001).

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rcrc Infected There is no epidemic threshold!!!

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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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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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X n+1 =X n 2 +C (Mandelbrot set X 0 =0) Main Bulb Decoration Antenna

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