2. Exp. vs. Scale-Free Poisson distribution Exponential Network Power-law distribution Scale-free Network

3. Scale Free Networks Scale-free networks are characterized by a power-law distribution of a node’s degree. There is a few hubs hold together numerous small degree nodes. Scale free imply any function f(x) that remains unchanged within a multiplicative factor under a rescaling of x: f(ax) = b f(x).

4. WWW (2000)

5. World Wide Web 800 million documents (S. Lawrence, 1999) ROBOT: collects all URL’s 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

6.  k  ~ 6 P( k=500 ) ~ 10 -99 N WWW ~ 10 9  N(k=500)~10 -90 What did we expect? 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)

7.  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 19 degrees of separation R. Albert et al Nature (99) 19 degrees D = 18.59 N = 8x10 8 domain nd.edu (325.729 pag. 1.469.680 links) A tenfold increase of the WWW D 21 !!!

8. N = 4941 = 2,47 P(k) ~k - 

9. 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

10. P(k) = k -  A: actors N = 212.250 k = 28.78  = 2.3 B: WWW N = 325.729 k = 5.46  = 2.67 C: power grid N= 494 k = 2.67  = 4 Scale-Free Networks

11. Communication networks The Earth is developing an electronic nervous system, a network with diverse nodes and links are -computers -routers -satellites -phone lines -TV cables -EM waves Llamadas telefónicas: N = 50 millones de nodos. Exponentes in/out = 2,1

12. Internet-Map

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

14. Circuitos

15. 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

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

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

18. protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions Bio-Map

19. Citrate Cycle METABOLISM Bio-chemical reactions

20. Boehring-Mennheim

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

22. Metabolic network Organisms from all three domains of life are scale-free networks! H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000) ArchaeaBacteriaEukaryotes Meta-P(k)

23. protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions Bio-Map

24. protein-protein interactions PROTEOME

25. Yeast protein network Nodes : proteins Links : physical interactions (binding) P. Uetz, et al. Nature 403, 623-7 (2000). Prot Interaction map

26. Topology of the protein network H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001) Prot P(k)

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

28. Real networks for which we know the topology: P(k) ~ k -  NON BIOLOGICAL  > 2 www (in)  = 2.1 www (out)  =2.45 actors  = 2.3 citations  = 3 power grid  = 4 BIOLOGICAL  < 2 yeast protein-protein net  =1.5, 1.6, 1.7, 2.5 E. Coli metabolic net  = 1.7, 2.2 yeast gene expression net  = 1.4-1.7 gene functional interaction  = 1.6

29. Some Published Networks ( adapted from Newman 2003 ) CαlZmn 0.782.33.48113.4325,516,482449,913Ufilm actors 0.34-4.614.4455,3927,673Ucompany directors 0.56-7.573.92496,489253,339Umath coauthorship 0.161.5/2. 0 4.951.4486,30059,912DEmail messages 3.22,810Usexual contacts 0.081.3/1. 4 4.630,6358,830DDiscussion Groups 0.442.770.1317,000,000460,902UWord co- occurrence info 0.392.53.315.9831,99210,697UInternettech 0.08-18.992.676,5944,941Upower grid 0.28-3.977.682,359307DNeural networksbio social