Complex Networks
Junio 2006
L. Lacasa, B. Luque y J.C. Nuño
Departamentos de Matemática Aplicada Aeronáuticos y Montes
Universidad Politécnica de Madrid
Society
Nodes: individuals
Links: social
relationship
(family/work/friends/
etc.)
Social networks:
Many individuals with diverse
social interactions between them.
Social networks
• Contacts and Influences
Poll & Kochen (1958)
– How great is the chance that two
people chosen at random from the
population will have a friend in common?
– How far are people aware of the available lines of contact?
• The Small-World Problem – Milgram (1967)
– How many intermediaries are needed to move a letter from
person A to person B through a chain of acquaintances?
– Letter-sending experiment: starting in Nebraska/Kansas,
with a target person in Boston.
Social networks: Milgram’s experiment
160 letters: From Wichita (Kansas) and Omaha (Nebraska)
If you do not know the target person on a personal basis,
do not try to contact him directly. Instead, mail this
folder to a personal acquaintance who is more likely than
you to know the target person.
to Sharon (Mass)
Milgram, Psych Today 2, 60 (1967)
“Six degrees
of separation”
¡El mundo es un pañuelo!
C’est petit le monde !!
What a Small-World !
The Small World concept in
simple terms describes the fact
despite their often large size,
in most networks there is a
relatively short path between
any two nodes.
El número de Erdös
Pál Erdös
(1913-1996)
Fue autor o coautor de
1.475 artículos
matemáticos y colaboró en
ellos con un total de 493
coautores distintos. Sólo un
matemático en la historia
escribió más páginas de
matemáticas originales que
Erdös. En siglo XVII, el
suizo Leonhard Euler,
padre de trece niños,
escribió ochenta volúmenes
de resultados matemáticos.
Números de Erdös de
científicos famosos
http://www.oakland.edu/enp/
Walter Alvarez
Rudolf Carnap
Jule G. Charney
Noam Chomsky
Freeman J. Dyson
George Gamow
Stephen Hawking
Pascual Jordan
Theodore von Kármán
John Maynard Smith
Oskar Morgenstern
J. Robert Oppenheimer
Roger Penrose
Jean Piaget
Karl Popper
Claude E. Shannon
Arnold Sommerfeld
Edward Teller
George Uhlenbeck
John A. Wheeler
Número 1- 504 colaboradores
Número 2- 6593 colaboradores
geology
philosophy
meteorology
linguistics
quantum physics
nuclear physics and cosmology
relativity and cosmology
quantum physics
aeronautical engineering
biology
economics
nuclear physics
relativity and cosmology
psychology
philosophy
electrical engineering
atomic physics
nuclear physics
atomic physics
nuclear physics
7
4
4
4
2
5
4
4
4
4
4
4
3
3
4
3
5
4
2
3
Números de Erdös de
premios Nobel de física
Max von Laue
1914
Albert Einstein
1921
Niels Bohr
Louis de Broglie
Werner Heisenberg
Paul A. Dirac
Erwin Schrödinger
Enrico Fermi
Ernest O. Lawrence
Otto Stern
Isidor I. Rabi
Wolfgang Pauli
Frits Zernike
Max Born
Willis E. Lamb
John Bardeen
Walter H. Brattain
William B. Shockley
Chen Ning Yang
Tsung-dao Lee
Emilio Segrè
1922
1929
1932
1933
1933
1938
1939
1943
1944
1945
1953
1954
1955
1956
1956
1956
1957
1957
1959
4
2
5
5
4
4
8
3
6
3
4
3
6
3
3
5
6
6
4
5
4
Owen Chamberlain
Robert Hofstadter
Eugene Wigner
Richard P. Feynman
Julian S. Schwinger
Hans A. Bethe
Luis W. Alvarez
Murray Gell-Mann
John Bardeen
Leon N. Cooper
John R. Schrieffer
Aage Bohr
Ben Mottelson
Leo J. Rainwater
Steven Weinberg
Sheldon Lee Glashow
Abdus Salam
S. Chandrasekhar
Norman F. Ramsey
1959
1961
1963
1965
1965
1967
1968
1969
1972
1972
1972
1975
1975
1975
1979
1979
1979
1983
1989
5
5
4
4
4
4
6
3
5
6
5
5
5
7
4
2
3
4
3
Erdös number
Erdös number 0 --- 1 person
Erdös number 1 --- 504 people
Erdös number 2 --- 6593 people
Erdös number 3 --- 33605 people
Erdös number 4 --- 83642 people
Erdös number 5 --- 87760 people
Erdös number 6 --- 40014 people
Erdös number 7 --- 11591 people
Erdös number 8 --- 3146 people
Erdös number 9 --- 819 people
Erdös number 10 --- 244 people
Erdös number 11 --- 68 people
Erdös number 12 --- 23 people
Erdös number 13 --- 5 people
• Graph: a pair of sets G = {P,E}
where P is a set of nodes, and
E is a set of edges that connect
2 elements of P.
• Degree of a node: the number of edges
incident on the node
i
Degree of node i = 5
Type of Edges
• Directed
• edges have a direction, only go one way
(citations, one way streets)
• Undirected
• no direction (committee membership, twoway streets)
• Weighted
• Not all edges are equal. (Friendships)
• Degree
• Number of edges connected to a node.
• In-degree
• Number of incoming edges.
• Out-degree
• Number of outgoing edges.
Network parameters
Diameter
Maximum distance between any pair of nodes.
Characteristic path length
Connectivity
Number of neighbours of a given node:
k := degree.
P(k) := Probability of having k neighbours.
Clustering
Are neighbours of a node also neighbours among
them?
Characteristic path length  GLOBAL property
• L( i , j ) is the number of edges in the shortest path
between vertices i and j (geodesic path).
i
L(i , j )  2
j
• The characteristic path length L of a graph is the
average of the L( i , j ) for every possible pair (i,j)
Networks with small values of L are said to have the
“Small World property”
Austin Powers:
The spy who
shagged me
Bacon’s
Game
Let’s make
it legal
Robert Wagner
Wild Things
What Price Glory
Internet Movie Database
Barry Norton
A Few
Good Man
http://www.cs.virginia.edu/oracle/
Monsieur
Verdoux
Why Kevin Bacon?
Measure the average distance between Kevin Bacon and all other actors.
Kevin Bacon
Is Kevin Bacon
the most
connected actor?
NO!
No. of movies : 46
No. of actors : 1811
Average separation: 2.79
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
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
# 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
Kevin Bacon
Kevin
Bacon
2.786981
2.786981
46
46
1811
1811
Rank
Name
1
2
3
4
5
6
7
8
9
10
11
12
…
876
876
…
#1 Rod Steiger
#876
Kevin Bacon
Donald
#2
Pleasence
#3 Martin Sheen
Tree Network
Random Network:
The typical distance between any two nodes in a random graph scales as the
logarithm of the number of nodes. Then the Small World concept is not an
indication of a particular organizing principle.
Random graphs – Erdos & Renyi (1960)
• Start with N nodes and for each pair of nodes, with
probability p, add a link between them.
• For large N, there is a giant connected component if the
average connectivity (number of links per node) is larger
than 1.
• The average path length L in the giant component scales
as L  lnN.
Minimal number of links one needs to follow to go
from one node to another, on average.
Erdös-Renyi model (1960)
Many properties in these graphs appear quite
suddenly, at a threshold value of p = PER(N)
-If PER ~ c / N with c < 1,
then almost all vertices
belong to isolated trees.
-Cycles of all orders
appear at PER ~ 1/ N
P(k )  e  pN
k
( pN ) k

k

 e  k 
k!
k!
Poisson distribution
Random Graphs Model
Given N nodes connect each pair with
probability p:
– P(k) ~ Poisson distribution
– <k> = pN.
– Most nodes degree ~ <k>.
– <L> = log(N) / log(<k>).
– Small World property
Asymptotic behavior
Lattice
L( N )  N
Random graph
1/ d
L( N )  log N
• For many years typical explanation for SmallWorld property was random graphs
– Low diameter: expected distance between two
nodes is log<k>N, where <k> is the average
outdegree and N the number of nodes.
– When pairs or vertices are selected uniformly at
random they are connected by a short path with
high probability.
• But there are some inaccuracies
– If A and B have a common friend C it is more likely
that they themselves will be friends! (clustering).
– Many real world networks exhibit this clustering
property. Random networks are NOT clustered.
Clustering coefficient
Local propierty:
C(v) =
C(v) = 4/6
# of links between neighbors
n(n-1)/2
C is the average over all C(v)
Clustering: My friends will know each other with high probability!
(typical example: social networks)
Asymptotic behavior
Lattice
L( N )  N 1/ d
C ( N )  const.
Random graph
L( N )  log N
C(N )  N
1
Power grid NW USA-Canada
N = 4914
kmax = 19
kaver = 2.67
L = 18.7
C = 0.08
D = 46
Caenorhabditis elegans
Neural system
N = 282
kmax = 14
kaverage = 9
L = 2.65
C = 0.28
D = 3
Duncan J. Watts & Steven H. Strogatz,
Nature 393, 440-442 (1998)
Real life networks are clustered, large C, but have small
average distance L.
WWW
Actors
Power Grid
C. Elegans
L
3.1
3.65
18.7
2.65
Lrand
3.35
2.99
12.4
2.25
C
0.11
0.79
0.080
0.28
Crand
0.00023
0.00027
0.005
0.05
N
153127
225226
4914
282
Structured network
Random network
• high clustering Small-world network • small clustering
• high clustering
• large diameter
• small diameter
• regular
• small diameter
• almost regular
N = 1000 k =10
D = 100 L = 49.51
C = 0.67
N =1000 k = 8-13
D = 14 d = 11.1
C = 0.63
N =1000 k = 5-18
D = 5 L = 4.46
C = 0.01
Duncan J. Watts & Steven H. Strogatz, Nature 393, 440-442 (1998)
Watts-Strogatz Model
regular
SW
random
C
L
p
C(p) : clustering coeff.
L(p) : average path length