Presentaci

1
(No Transcript)
2
Exp. vs. Scale-Free
3
Scale Free Networks

• Scale-free networks are characterized by a
  power-law distribution of a nodes 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
World Wide Web
Nodes WWW documents Links URL links
800 million documents (S. Lawrence, 1999)
ROBOT collects all URLs 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(k500) 10-99
NWWW 109 ? N(k500)10-90
?out 2.45
? in 2.1
P(k500) 10-6
NWWW 109 ? N(k500) 103
Pout(k) k-?out
Pin(k) k- ?in
J. Kleinberg, et. al, Proceedings of the ICCC
(1999)
7
19 degrees
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 ltkgt 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
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
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
24
(No Transcript)
25
Prot Interaction map
Yeast protein network
Nodes proteins Links
physical interactions (binding)
P. Uetz, et al. Nature 403, 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 gt 2 www (in) g
2.1 www (out) g 2.45 actors
g 2.3 citations g
3 power grid g 4 BIOLOGICAL g
lt 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)
C a l Z m n
0.78 2.3 3.48 113.43 25,516,482 449,913 U film actors
0.34 - 4.6 14.44 55,392 7,673 U company directors
0.56 - 7.57 3.92 496,489 253,339 U math coauthorship
0.16 1.5/2.0 4.95 1.44 86,300 59,912 D Email messages
3.2 2,810 U sexual contacts
0.08 1.3/1.4 4.6 30,635 8,830 D Discussion Groups
0.44 2.7 70.13 17,000,000 460,902 U Word co-occurrence info
0.39 2.5 3.31 5.98 31,992 10,697 U Internet tech
0.08 - 18.99 2.67 6,594 4,941 U power grid tech
0.28 - 3.97 7.68 2,359 307 D Neural networks bio
social