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New York Times

New York Times. Complex systems. Made of many non-identical elements connected by diverse interactions. NETWORK. Slides: thanks to A-L Barabasi. (Internet?) Big Ideas (3). Structure in complex networks. Connect with probability p.

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New York Times

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  1. New York Times Complex systems Made of many non-identical elements connected by diverse interactions. NETWORK Slides: thanks to A-L Barabasi

  2. (Internet?) Big Ideas (3) • Structure in complex networks

  3. Connect with probability p p=1/6 N=10 k ~ 1.5 Poisson distribution Erdös-Rényi model(1960) Pál Erdös(1913-1996) - Democratic - Random

  4. Small Worlds • Stanley Milgram ’s experiment • Small Worlds by Watts/Strogatz • (v) = Clustering coefficient of node v = Percentage of neighbours of v connected to each other • Clustering coefficient:

  5. Cluster Coefficient Clustering: My friends will likely know each other! Probability to be connected C»p # of links between 1,2,…n neighbors C = n(n-1)/2 Networks are clustered [large C(p)] but have a small characteristic path length [small L(p)].

  6. Watts-Strogatz Model C(p) : clustering coeff. L(p) : average path length (Watts and Strogatz, Nature 393, 440 (1998))

  7. Web What did we expect? k ~ 6 P(k=500) ~ 10-99 NWWW ~ 109  N(k=500)~10-90 We find: out= 2.45 in = 2.1 P(k=500) ~ 10-6 NWWW ~ 109  N(k=500) ~ 103 Pout(k) ~ k-out Pin(k) ~ k- in J. Kleinberg, et. al, Proceedings of the ICCC (1999)

  8. 19 degrees  Finite size scaling: create a network with N nodes with Pin(k) and Pout(k) < l > = 0.35 + 2.06 log(N) 19 degrees of separation R. Albert et al Nature (99) based on 800 million webpages [S. Lawrence et al Nature (99)] nd.edu < l > IBM A. Broder et al WWW9 (00) 19 degrees of separation 3 l15=2 [125] l17=4 [1346  7] … < l > = ?? 6 1 4 7 5 2

  9. November 2000 March 2001 Power-law Distributions • Gnutella: Node connectivity follows a powerlaw*, i.e. P(k neighbours)  k - * Mapping the Gnutella network: Properties of largescale peer-to-peer systems and implications for system design. M. Ripeanu, A. Iamnitchi, and I. Foster. IEEE Internet Computing Journal 6, 1 (2002), 50-57.

  10. What does it mean? Airlines Poisson distribution Power-law distribution Exponential Network Scale-free Network

  11. Internet INTERNET BACKBONE Nodes: computers, routers Links: physical lines (Faloutsos, Faloutsos and Faloutsos, 1999)

  12. Internet-Map

  13. Actors ACTOR CONNECTIVITIES Nodes: actors Links: cast jointly Days of Thunder (1990) Far and Away (1992) Eyes Wide Shut (1999) N = 212,250 actors k = 28.78 P(k) ~k- =2.3

  14. Citation 25 2212 SCIENCE CITATION INDEX Nodes: papers Links: citations Witten-Sander PRL 1981 1736 PRL papers (1988) P(k) ~k- ( = 3) (S. Redner, 1998)

  15. Coauthorship SCIENCE COAUTHORSHIP Nodes: scientist (authors) Links: write paper together (Newman, 2000, H. Jeong et al 2001)

  16. Food Web Nodes: trophic species Links: trophic interactions R.J. Williams, N.D. Martinez Nature (2000) R. Sole (cond-mat/0011195)

  17. Most real world networks have the same internal structure: Scale-free networks Why? What does it mean?

  18. (2) The attachment is NOT uniform. A node is linked with higher probability to a node that already has a large number of links. Examples : WWW : new documents link to well known sites (CNN, YAHOO, NewYork Times, etc) Citation : well cited papers are more likely to be cited again SCALE-FREE NETWORKS (1) The number of nodes (N) is NOT fixed. Networks continuously expand by the addition of new nodes Examples: WWW : addition of new documents Citation : publication of new papers

  19. BA model Scale-free model P(k) ~k-3 (1)GROWTH: At every timestep we add a new node with m edges (connected to the nodes already present in the system). (2)PREFERENTIAL ATTACHMENT :The probability Π that a new node will be connected to node i depends on the connectivity ki of that node A.-L.Barabási, R. Albert, Science 286, 509 (1999)

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

  21. What Does the Web Really Look Like? • Graph Structure in the Web, Broder et al. • Analysis of 2 Altavista crawls, each with over 200M pages and 1.5 billion links

  22. Confirm Power Law Structure

  23. But Things Are More Complex Than One Might Think …

  24. Reading • Emergence of scaling in random networks, Albert-László Barabási, Réka Albert, Science 286 509-512 (1999) • Search in power-law networks, Lada A. Adamic, Rajan M. Lukose, Amit R. Puniyani and Bernardo A. Huberman, Phys. Rev. E, 64 46135 (2001) • Graph structure in the web, Andrei Broder, Ravi Kumar, Farzin Maghoul, Prabhakar Raghavan, Sridhar Rajagopalan, Raymie Stata, Andrew Tomkins, Janet Wiener, Comput. Netw. 33 309

  25. CMSC 23340-1 (Winter 2005):Course Goals • Primary • Gain deep understanding of fundamental issues that effect design of large-scale networked systems • Map primary contemporary research themes • Gain experience in network research • Secondary • By studying a set of outstanding papers, build knowledge of how to present research • Learn how to read papers & evaluate ideas

  26. How the Class Works • Research papers • Prior to each class, we all read and evaluate two research papers • During each class, we discuss those papers • Project

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