Exp. vs. Scale-Free

2. Exp. vs. Scale-Free Poisson distribution Power-law distribution Exponential Network 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. WWW L L C C N rand rand 3.1 3.35 0.11 0.00023 153127 WWW 3.65 2.99 0.79 0.00027 225226 Actors 18.7 12.4 0.080 0.005 4914 Power Grid 2.65 2.25 0.28 0.05 282 C. Elegans World Wide Web Nodes: WWW documents Links: URL links 800 million documents (S. Lawrence, 1999) ROBOT:collects all URL’s found in a document and follows them recursively R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999)

6. What did we expect? k ~ 6 P(k=500) ~ 10-99 NWWW ~ 109  N(k=500)~10-90 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)

7. 19 degrees  Finite size scaling: create a network with N nodes with Pin(k) and Pout(k) < l > = 0.35 + 2.06 log(N) < l > 19 degrees of separation 19 degrees of separation R. Albert et al Nature (99) D = 18.59 N = 8x108 domain nd.edu (325.729 pag. 1.469.680 links) A tenfold increase of the WWW D 21 !!!

8. N = 4941 <k> = 2,47 P(k) ~k-

9. 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

10. Scale-Free Networks 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

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

14. Circuitos

15. 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)

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

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

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

19. METABOLISM Bio-chemical reactions Citrate Cycle

20. Boehring-Mennheim

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

22. Meta-P(k) Metabolic network Archaea Bacteria Eukaryotes 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)

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

24. PROTEOME protein-protein interactions

25. Prot Interaction map Yeast protein network Nodes: proteins Links: physical interactions (binding) P. Uetz, et al.Nature403, 623-7 (2000).

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

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- g NON BIOLOGICAL g > 2 www (in) g = 2.1 www (out) g =2.45 actors g = 2.3 citations g = 3 power grid g = 4 BIOLOGICAL g < 2 yeast protein-protein net g =1.5, 1.6, 1.7, 2.5 E. Coli metabolic net g = 1.7, 2.2 yeast gene expression net g = 1.4-1.7 gene functional interaction g = 1.6

29. Some Published Networks (adapted from Newman 2003) social