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

## Presentation on theme: "1 Complex systems Made of many non-identical elements connected by diverse interactions. NETWORK New York Times Slides: thanks to A-L Barabasi."— Presentation transcript:

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

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

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

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:

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)].

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

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

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

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

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

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

12 Internet-Map

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

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

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

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 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

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

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

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

22 Confirm Power Law Structure

23 But Things Are More Complex Than One Might Think …

24

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

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

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

Similar presentations