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

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).

WWW (2000)

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, (1999) WWW

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

 Finite size scaling: create a network with N nodes with P in (k) and P out (k) = log(N) 19 degrees of separation 19 degrees of separation R. Albert et al Nature (99) 19 degrees D = N = 8x10 8 domain nd.edu ( pag links) A tenfold increase of the WWW D 21 !!!

N = 4941 = 2,47 P(k) ~k - 

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

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

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

Internet-Map

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

Circuitos

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

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

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

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

Citrate Cycle METABOLISM Bio-chemical reactions

Boehring-Mennheim

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

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, (2000) ArchaeaBacteriaEukaryotes Meta-P(k)

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

protein-protein interactions PROTEOME

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

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

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

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  = gene functional interaction  = 1.6

Some Published Networks ( adapted from Newman 2003 ) CαlZmn ,516,482449,913Ufilm actors ,3927,673Ucompany directors ,489253,339Umath coauthorship / ,30059,912D messages 3.22,810Usexual contacts / ,6358,830DDiscussion Groups ,000,000460,902UWord co- occurrence info ,99210,697UInternettech ,5944,941Upower grid ,359307DNeural networksbio social