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Classes will begin shortly

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

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Classes will begin shortly

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  1. Classes will begin shortly

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

  3. WATTS & STROGATZ Poisson distribution Lattice Erdös-Rényimodel(1960)

  4. High school friendship James Moody, American Journal of Sociology107, 679-716 (2001)

  5. High school dating network Data: Peter S. Bearman, James Moody, and Katherine Stovel. American Journal of Sociology110, 44-91 (2004) Image: M. Newman

  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 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.

  7. Degree (k) Degree Distribution P(k) k

  8. The Crazy 1990’s

  9. www Internet Autonomous System i.e. Harvard.edu

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

  11. Internet-Map

  12. WWW Expected P(k) ~ k- Found World Wide Web Nodes:WWW documentsLinks:URL links Over 3 billion documents Exponential Network ROBOT:collects all URL’s found in a document and follows them recursively Scale-free Network R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999).

  13. Scale-Free Networks Everywhere

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

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

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

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

  18. 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, 407 651 (2000)

  19. 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, 41-42 (2001)

  20. 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

  21. Explaining Scale-Free Networks

  22. BA model (1) Networks continuously expand by the addition of new nodes WWW : addition of new documents Citation : publication of new papers (2) New nodes prefer to link to highly connected nodes. WWW : linking to well known sitesCitation : citing again highly cited papers Scale-free model 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. Web application: http://www-personal.umich.edu/~ladamic/NetLogo/PrefAndRandAttach.html Barabási & Albert, Science 286, 509 (1999)

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

  24. growth preferential attachment Model A Π(ki) : uniform

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

  26. A note on the BA model Yule process Price Model

  27. Beyond the BA Model

  28. 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. 2115DOI: 10.1126/science.287.5461.2115a 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. 2115DOI: 10.1126/science.287.5461.2115a Movie Actors WWW

  29. Can Latecomers Make It?Fitness ModelSF model: k(t)~t ½(first mover advantage)Real systems: nodes compete for links -- fitnessFitness Model: fitness (h )k(h,t)~tb(h)whereb(h) =h/CG. Bianconi and A.-L. Barabási, Europhyics Letters.54, 436 (2001).

  30. Local Rules A Vazquez Growing network with local rules: Preferential attachment, clustering hierarchy, and degree correlations Physical Review E 67, 056104 (2003) Random Walk Model qe qv 1-qe

  31. The easiest way to find a hub? Ask for a friend!!! Pick a random person and ask that person to name a friend.

  32. 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 ofit gives you a uniform distribution of degrees!

  33. Other Models More models R. Albert, A.-L. Barabasi, Rev. Mod. Phys 2002

  34. Why scale-free? What functions satisfy this functional relationship? F(x)=xP F(ax)=bF(x) (ax)P=aPxP=bxp

  35. Power-LawsBig deal!

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

  37. 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!

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

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

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

  41. Statistics of Power-Laws

  42. Power-Laws are dominated by largest value AVERAGES

  43. Power-Laws are dominated by largest valueMEDIANS

  44. Power-Laws are dominated by largest valueCOMPARING MEDIANS AND AVERAGES

  45. Power-Laws have diverging VARIANCE

  46. Why physicists were interested in Power-Laws

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