Transcript S. Boccaletti Homoclinic chaos: experimental

Stefano Boccaletti
Complex networks in science and society
*Istituto Nazionale di Ottica Applicata - Largo E.
Fermi, 6 - 50125 Florence, ITALY
*CNR-Istituto dei Sistemi Complessi
* MIND- Mediterranean Institute for Nonlinear
Dynamics
Coworkers:
Dong-Uk Hwang, Mario Chavez, Andreas Amann,Vito latora
Hector Mancini, Jean Bragard, Louis Pecora, Juergen Kurths
Dedicated to the memory of Carlos Pérez Garcia
PAMPLONA 2005
Summary
•WHAT IS A NETWORK?
•WHAT IS A COMPLEX NETWORK?
•THE STRUCTURE OF COMPLEX NETWORKS
•THE MODELS OF COMPLEX NETWORKS
Do you want to know more?

S.Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang
COMPLEX NETWORKS: STRUCTURE AND DYNAMICS
212 pages, 856 References
TO APPEAR SOON IN PHYSICS REPORTS
For preprints write to
[email protected]
Society
Nodes: individuals
Links: social relationship
(family/work/friendship/etc.)
S. Milgram (1967)
Six Degrees of Separation
John Guare
Social networks: Many individuals with diverse
social interactions between them.
Communication networks
The Earth is developing an electronic nervous system,
a network with diverse nodes and links are
-computers
-phone lines
-routers
-TV cables
-satellites
-EM waves
INTERNET BACKBONE
Erdös-Rényi model
(1960)
Pál Erdös
(1913-1996)
Poisson distribution
Connect with probability p
ARE COMPLEX NETWORKS REALLY RANDOM?
Road and Airline networks
Poisson distribution
Exponential Network
Power-law distribution
Scale-free Network
SCIENCE CITATION INDEX
Nodes: papers
Links: citations
25
Witten-Sander
PRL 1981
2212
P(k) ~k-
SCIENCE COAUTHORSHIP
Nodes: scientist (authors)
Links: write paper together
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
Centrality: Why Kevin Bacon?
Measure the average distance between Kevin Bacon and all other actors.
No. of movies : 46
No. of actors : 1811
Kevin Bacon
Average separation: 2.79
Is Kevin Bacon
the most
connected actor?
NO!
Rank
Name
1
2
3
4
5
6
7
8
9
10
11
12
…
876
876
…
Rod Steiger
Donald Pleasence
Martin Sheen
Christopher Lee
Robert Mitchum
Charlton Heston
Eddie Albert
Robert Vaughn
Donald Sutherland
John Gielgud
Anthony Quinn
James Earl Jones
Kevin Bacon
Kevin Bacon
Average
distance
2.537527
2.542376
2.551210
2.552497
2.557181
2.566284
2.567036
2.570193
2.577880
2.578980
2.579750
2.584440
2.786981
2.786981
# of
movies
112
180
136
201
136
104
112
126
107
122
146
112
# of
links
2562
2874
3501
2993
2905
2552
3333
2761
2865
2942
2978
3787
46 1811
46
1811
#1 Rod Steiger
#876
Kevin Bacon
#2
Donald
Pleasence
#3 Martin Sheen
FOOD WEBS
Nodes: trophic species
Links: trophic interactions
R.J. Williams, N.D. Martinez Nature (2000)
SEX WEBS
Nodes: people (Females; Males)
Links: sexual relationships
4781 Swedes; 18-74;
59% response rate.
Liljeros et al. Nature 2001
Metabolic Networks I
Nodes: chemicals (substrates)
Links: bio-chemical reactions
Metabolic Networks II
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)
Protein networks I
Nodes: proteins
Links: physical interactions (binding)
P. Uetz, et al. Nature 403, 623-7 (2000).
Protein networks II
P(k ) ~ (k  k0 )  exp(
k  k0
)
k
H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001)
Nature 408 307 (2000)
…
“One way to understand the p53
network is to compare it to the
Internet.
The cell, like the Internet, appears to
be a ‘scale-free network’.”
p53 network (mammals)
Network
C
Crand
L
N
WWW
0.1078
0.00023
3.1
153127
Internet
0.18-0.3
0.001
3.7-3.76
30156209
Actor
0.79
0.00027
3.65
225226
Coauthorship
0.43
0.00018
5.9
52909
Metabolic
0.32
0.026
2.9
282
Foodweb
0.22
0.06
2.43
134
C. elegance
0.28
0.05
2.65
282
WWW
(in)
 = 2.1
Internet
Actor
Citation
index
 = 2. 5
 = 2.3
=3
Sex
Web
Cellular Phone call
linguistics
network network
 = 3.5  = 2.1
 = 2.1
 = 2.8
Watts-Strogatz Model
(Watts and Strogatz, Nature 393,
440 (1998))
C(p) : clustering coeff.
L(p) : average path length
BA - Scale-free model
(1) GROWTH :
At every timestep we add a new node with m edges
(connected to the nodes already present in the system).
(2) PREFERENTIAL ATTACHMENT :
The probability Π that a new node will be connected to
node i depends on the connectivity ki of that node
P(k) ~k-3
A.-L.Barabási, R. Albert, Science 286, 509 (1999)
Robustness
Complex systems maintain their basic
functions even under errors and failures
(cell  mutations; Internet  router
breakdowns)
1
S
fc
0
1
Fraction of removed nodes, f
node failure
Achilles’ Heel of complex networks
failure
attack
Internet
R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)
Yeast protein network
- lethality and topological position -
Highly connected proteins are more essential (lethal)...
H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001)