Redes sicomoro

Redes Complejas,
introducción
Javier Galeano.
Universidad Politécnica de Madrid
@galeanojav

The large blue butterﬂy became extinct in England

¿What do they have in common?
A butterﬂy
and a rabbit

Myxomatosis desease nearly killed the whole
rabbit population in the area

Maculinea arion
Myrmica sabuleti

These three different species share the habitat
and a complex network

The fate of Saddam
and network science

The fate of
Saddam and
network
science

• Graph G:

• 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.
Deﬁnition

Components of a Complex System

! components: nodes, vertices N

! components: nodes, vertices N
! system: network, graph (N,L)
! interactions: links, edges L

A common language
Peter Mary
Albert
Albert
co-worker
friendbrothers
friend

A common language
Peter Mary
Albert
Albert
co-worker
friendbrothers
friend Movie 1
Movie 3
Movie 2
Actor 3
Actor 1 Actor 2
Actor 4

A common language
Peter Mary
Albert
Albert
co-worker
friendbrothers
friend
Protein
1
Protein
2 Protein
5
Protein
9
Movie 1
Movie 3
Movie 2
Actor 3
Actor 1 Actor 2
Actor 4
N=4!
L=4

Caso real ..como la vida misma
Primer día de colegio con 3 años A unos meses de entrar en el Instituto

“Chains of affection: The structure of adolescent
romantic and sexual networks”
Bearman PS, Moody J, Stovel K.
American Journal of Sociology, Vol. 100, No. 1.

Cada año en USA millones de personas
descubren que son portadores de SDT

la tasa de infección en los adolescentes
es muy superior a los otros grupos

Bearman PS, Moody J, Stovel K.
American Journal of Sociology, Vol. 100, No. 1.
Entrevistaron a 832 adolescentes
del instituto “Jefferson”

Male
Female
12 9
63
2
Figure 2. Structure of Romantic and Sexual Contact at Jefferson

12
63
Figure 2. Structure of Romantic and Sexual Contact at Je
288 adolescentes (52%) unidos en una
componente gigante que involucra
intercambios de fluidos.

12
63
Figure 2. Structure of Romantic and Sexual Contact at Je
Conexión con muchas
ramas con falta de ciclos.

The adolescent rules
homofilia
Una
prohibición: “
No salir con el
antiguo novio
de la actual
novia de tu
anterior novio.
crea una
componente
gigante
Prohibe los
ciclos de
cuatro pasos.

• Leonhard Euler wrote a paper on Seven Bridges of
Könisberg and published in 1736.
Very beginning of Graph Theory

A random graph is a graph of N nodes where each node pair of nodes is
connected by a preset probability P
Random network model
Image 3.2a
Pál Erdős (1913-1996)
Hungarian mathematician known for both his eccentricity and exceptional
scientific output, having published more papers than any other mathema-
tician in the history of mathematics. His productivity had its roots in his
fondness for collaboration: he co-authored papers with over five hundred
mathematicians, inspiring the concept of Erdős number. His legendarily
personality and profound professional impact has inspired two biographies
Image 3.2b
Alfréd Rényi (1921-1970)
Hungarian mathematician with fundamental contributions to combina-
torics, graph theory, and number theory. His impact goes beyond mathe-
matics: the Rényi entropy is widely used in chaos theory and the random
network model he co-developed is at the heart of network science. He is
remembered through the hotbed of Hungarian mathematics, the Alfréd
Rényi Institute of Mathematics in Budapest. He once said, “A mathemati-
Pál Erdös (1913-1996) Alfréd Rényi (1921-1970)

p=1/6
N=12

p=0.03
N=100

Degree
• Grado de un nodo: es el número
de enlaces en un nodo. El nodo i
(en azul) tiene grado 5
i
• Grado medio <K>: es el promedio de
los grados de una red.

Degree distribution
k =
3×1+ 2 × 2 +1× 3
6
= 1,66
3
2
1

Degree distribution
k = 1
k = 1
k = 1
k =
3×1+ 2 × 2 +1× 3
6
= 1,66
3
2
1

Degree distribution
k = 2
k = 2
k = 1
k = 1
k = 1
k =
3×1+ 2 × 2 +1× 3
6
= 1,66
3
2
1

Degree distribution
k = 3
k = 2
k = 2
k = 1
k = 1
k = 1
k =
3×1+ 2 × 2 +1× 3
6
= 1,66
3
2
1

Degree distribution
k = 3
k = 2
k = 2
k = 1
k = 1
k = 1
k
f (k)
1 2 3k =
3×1+ 2 × 2 +1× 3
6
= 1,66
3
2
1

Small World (Experimento de Milgram)

• En 1967, Milgram eligió
individuos en ciudades
norteamericanas de Omaha y
Wichita para ser el inicio y
Sharon para el ﬁn de una
cadena de correspondencia.

• En 1967, Milgram eligió
individuos en ciudades
norteamericanas de Omaha y
Wichita para ser el inicio y
Sharon para el ﬁn de una
cadena de correspondencia.
•A los individuos de Omaha y Wichita se les enviaba la información básica
sobre el estudio y acerca del destinatario ﬁnal en Sharon, al que tenían que
enviar una carta.

HOW TO TAKE PART IN THIS STUDY
1. ADD YOUR NAME TO THE ROSTER AT THE BOTTOM OF THIS SHEET, so that the next
person who receives this letter will know who it came from.
2. DETACH ONE POSTCARD. FILL IT AND RETURN IT TO HARVARD UNIVERSITY. No stamp
is needed. The postcard is very important. It allows us to keep track of the progress of the folder as
it moves toward the target person.
3. IF YOU KNOW THE TARGET PERSON ON A PERSONAL BASIS, MAIL THIS FOLDER
DIRECTLY TO HIM (HER). Do this only if you have previously met the target person and know each
other on a first name basis.
4. IF YOU DO NOT KNOW THE TARGET PERSON ON A PERSONAL BASIS, DO NOT TRY TO
CONTACT HIM DIRECTLY. INSTEAD, MAIL THIS FOLDER (POST CARDS AND ALL) TO A
PERSONAL ACQUAINTANCE WHO IS MORE LIKELY THAN YOU TO KNOW THE TARGET
PERSON. You may send the folder to a friend, relative or acquaintance, but it must be someone
you know on a first name basis.

• En algunos casos, los paquetes
alcanzaban su destinatario en apenas
uno o dos pasos, mientras que algunas
cadenas estaban compuestas por hasta
9 ó 10 eslabones. Pero la longitud
promedio de la cadena de conexiones
ﬂuctuaba entre las 5,5 y las 6 personas.

• En algunos casos, los paquetes
alcanzaban su destinatario en apenas
uno o dos pasos, mientras que algunas
cadenas estaban compuestas por hasta
9 ó 10 eslabones. Pero la longitud
promedio de la cadena de conexiones
ﬂuctuaba entre las 5,5 y las 6 personas.
• ! El mundo es un pañuelo!

• C’est petit le monde!

• What a Small-World!!!

Fue autor o coautor de 1.475 artículos
matemáticos y colaboró en ellos con un
total de 493 coautores distintos.
El número de Erdös (1913-1996)

Walter Alvarez geology 7
Rudolf Carnap philosophy 4
Jule G. Charney meteorology 4
Noam Chomsky linguistics 4
Freeman J. Dyson quantum physics 2
George Gamow nuclear physics and cosmology 5
Stephen Hawking relativity and cosmology 4
Pascual Jordan quantum physics 4
Theodore von Kármán aeronautical engineering 4
John Maynard Smith biology 4
Oskar Morgenstern economics 4
J. Robert Oppenheimer nuclear physics 4
Roger Penrose relativity and cosmology 3
Jean Piaget psychology 3
Karl Popper philosophy 4
Claude E. Shannon electrical engineering 3
Arnold Sommerfeld atomic physics 5
Edward Teller nuclear physics 4
George Uhlenbeck atomic physics 2
John A. Wheeler nuclear physics 3
Números de Erdös de
cientíﬁcos famosos
http://www.oakland.edu/enp/

Erdös number 0 --- 1 person

Erdös number 1 --- 504 people












Erdös number

Which is my Erdös number?
• MathSciNet will automatically ﬁnd a path in their
database from you to Paul Erdös, or between any two
people you wish.

people you wish.
• I searched my Erdös number.

My Erdos Number is =5
people you wish.
• I searched my Erdös number.

The oracle of Bacon
Santiago Segura

The oracle of Bacon
Santiago Segura
Hellboy (2004)

The oracle of Bacon
Santiago Segura
Hellboy (2004)
John Hurt

The oracle of Bacon
Santiago Segura
Hellboy (2004)
John Hurt
Inﬁerno en Alabama (2012)

Santiago Segura
Hellboy (2004)
John Hurt
Inﬁerno en Alabama (2012)
Kevin Bacon
Bacon number = 2

Erdös number
Natalie Portman
Erdos number = 2
Bacon number = 5
Erdos-Bacon number = 7

Longitud de paso característica
• L(ij) : es el número de enlaces en el
paso más corto entre los nodos i j
(paso geodésico)
• La longitud de paso característica de un grafo es el
promedio de todas las longitudes características para
cada posible par de nodos.

• Las redes con un valor pequeño de L se dice que
tienen la propiedad de Small World.
i
j

Real networks are not random!!

The Web of
Human Sexual Contacts
Liljeros et al. Nature 2001

The Web of
Human Sexual
Contacts

The Web of
En el año 1996
4781 suecos entrevistados
Entre 18 y 74 años
Respondieron un 59 % (2810)

The Web of

The Web of
El Julio Iglesias sueco

Degree distributions
• Miremos las distribuciones de grados de algunas redes
ER Model
ER Model WS Model actors power grid www
• Las redes de tipo
random graph o WS
model son redes
homogéneas.
• Pero hay muchas redes que tienen una
distribución en ley de potencia. Redes
Scale-free

Degree distributions
• Miremos las distribuciones de grados de algunas redes

Redes scale-free
A.-L.Barabási, R. Albert,
Science 286, 509 (1999)

Redes scale-free
• Crecimiento de la red: Cada paso se añade un nodo
con un determinado número m de enlaces.
Science 286, 509 (1999)

Redes scale-free
• Preferential Attachment: la probabilidad Π de un
nuevo nodo depende de la conectividad
Science 286, 509 (1999)

Redes scale-free
• Preferential Attachment: la probabilidad Π de un
nuevo nodo depende de la conectividad
P(k) ~k-3
Science 286, 509 (1999)

• Los políticos escapan a un desastre nuclear y como en Galáctica tienen
que huir en busca de un nuevo planeta
El grafo de colores
• Mariano Rajoy

• Esperanza Aguirre

• Pablo Iglesias

• Pedro Sanchez

• Ciudadano Juan Carlos

• Cayo Lara

• Albert Rivera

Incompatibilidades
Mariano
Rajoy
Esperanza
Aguirre
Pablo
Iglesias
Juan
Carlos
Pedro
Sanchéz
Cayo
Lara
Albert
Rivera
Pedro
Sánchez
Pablo
Iglesias
Esperanza
Aguirre
Cayo Lara
Mariano
Rajoy
Juan
Carlos
Pablo
Iglesias
Cayo Lara Cayo Lara
Albert
Rivera
Pablo
Iglesias
Esperanza
Aguirre
Mariano
Rajoy
Cayo
Lara
Esperanza
Aguirre
Pedro
Sánchez
Albert
Rivera
Esperanza
Aguirre
Juan
Carlos
Mariano
Rajoy
Albert
Rivera

Incompatibilidades
Mariano
Rajoy
Espe
Aguirre
Pablo
Iglesias
Juan
Carlos
Pedro
Sanchéz
Cayo
Lara
Albert
Rivera
Pedro
Sánchez
Pablo
Iglesias
Espe
Aguirre
Cayo
Lara
Mariano
Rajoy
Juan
Carlos
Pablo
Iglesias
Cayo Lara Cayo Lara
Albert
Rivera
Pablo
Iglesias
Espe
Aguirre
Mariano
Rajoy
Cayo
Lara
Espe
Aguirre
Pedro
Sánchez
Albert
Rivera
Espe
Aguirre
Juan
Carlos
Mariano
Rajoy
Albert
Rivera MR
EA
PI
JC
PSCL
AR

Incompatibilidades
MR
EA
PI
JC
PSCL
AR

Incompatibilidades
• Pablo Iglesias

• Pedro Sanchez

• Cayo Lara
• Mariano Rajoy

• Albert Rivera
• Esperanza Aguirre

• Ciudadano Juan Carlos

BOOKS
Handbook of Graphs and Networks: From
the Genome to the Internet (Wiley-VCH,
2003).
S. N. Dorogovtsev and J. F. F. Mendes,
Evolution of Networks: From Biological
Nets to the Internet and WWW (Oxford
University Press, 2003).
S. Goldsmith, W. D. Eggers, Governing by
Network: The New Shape of the Public
Sector (Brookings Institution Press,
2004).
P. Csermely, Weak Links: The Universal Key
to the Stability of Networks and Complex
Systems (The Frontiers Collection) (Springer,
2006), rst edn.
M. Newman, A.-L. Barabasi, D. J. Watts, The
Structure and Dynamics of Networks:
(Princeton Studies in Complexity) (Princeton
University Press, 2006), rst edn.
L. L. F. Chung, Complex Graphs and
Networks (CBMS Regional Conference Series
in Mathematics) (American Mathematical
Society, 2006).

BOOKS
R. Pastor-Satorras, A. Vespignani,
Evolution and Structure of the Internet: A
Statistical Physics Approach (Cambridge
University Press, 2007), rst edn.
F. Kopos, Biological Networks (Complex
Systems and Interdisciplinary Science)
(World Scientic Publishing Company,
2007), rst edn.
B. H. Junker, F. Schreiber, Analysis of
Biological Networks (Wiley Series in
Bioinformatics) (Wiley-Interscience,
2008).
T. G. Lewis, Network Science: Theory and
Applications (Wiley, 2009).
E. Ben Naim, H. Frauenfelder, Z.Torotzai,
Complex Networks (Lecture Notes in Physics)
(Springer, 2010), rst edn.
M. O. Jackson, Social and Economic
Networks (Princeton University Press, 2010).

• Science:
Special Issue for the 10 year
anniversary of
Barabasi&Albert 1999 paper.

JOURNAL

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