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

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

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

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

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

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

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7 k ~ 6 P( k=500 ) ~ 10 -99 N WWW ~ 10 9 N(k=500)~10 -90 What did we expect? We find: P out (k) ~ k - out P( k=500 ) ~ 10 -6 out = 2.45 in = 2.1 P in (k) ~ k - in N WWW ~ 10 9 N(k=500) ~ 10 3 J. Kleinberg, et. al, Proceedings of the ICCC (1999) Web

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8 Finite size scaling: create a network with N nodes with P in (k) and P out (k) = 0.35 + 2.06 log(N) 19 degrees of separation l 15 =2 [1 2 5] l 17 =4 [1 3 4 6 7] … = ?? 1 2 3 4 5 6 7 nd.edu 19 degrees of separation R. Albert et al Nature (99) based on 800 million webpages [S. Lawrence et al Nature (99)] A. Broder et al WWW9 (00) IBM 19 degrees

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9 Power-law Distributions l 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. November 2000March 2001

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

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

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12 Internet-Map

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

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

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

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

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

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18 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 (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

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19 Scale-free model (1) GROWTH : A t 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 k i of that node A.-L.Barabási, R. Albert, Science 286, 509 (1999) P(k) ~k -3 BA model

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

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

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22 Confirm Power Law Structure

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23 But Things Are More Complex Than One Might Think …

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25 Reading l Emergence of scaling in random networks, Albert-László Barabási, Réka Albert, Science 286 509-512 (1999) Emergence of scaling in random networks l Search in power-law networks, Lada A. Adamic, Rajan M. Lukose, Amit R. Puniyani and Bernardo A. Huberman, Phys. Rev. E, 64 46135 (2001) Search in power-law networks l 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 Graph structure in the web

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26 CMSC 23340-1 (Winter 2005): Course Goals l 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 l Secondary –By studying a set of outstanding papers, build knowledge of how to present research –Learn how to read papers & evaluate ideas

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27 How the Class Works l Research papers –Prior to each class, we all read and evaluate two research papers –During each class, we discuss those papers l Project

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