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).
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
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)
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 !!!
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
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
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
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)