Complex NetworksJunio 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 InfluencesPoll & 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)

to Sharon (Mass)

Milgram, Psych Today 2, 60 (1967)

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 who is more likely than youthan you to know the target person.

¡El mundo es un pañuelo!C’est petit le monde !!What a Small-World !

“Six degrees

of separation”

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

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.

Pál ErdösPál Erdös (1913-1996)

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

Números de Erdös de

científicos famososNúmero 1- 504 colaboradoresNúmero 2- 6593 colaboradores

http://www.oakland.edu/enp/

Max von Laue 1914 4

Albert Einstein 1921 2Niels Bohr 1922 5Louis de Broglie 1929 5Werner Heisenberg 1932 4Paul A. Dirac 1933 4Erwin Schrödinger 1933 8Enrico Fermi 1938 3Ernest O. Lawrence 1939 6Otto Stern 1943 3Isidor I. Rabi 1944 4Wolfgang Pauli 1945 3Frits Zernike 1953 6 Max Born 1954 3 Willis E. Lamb 1955 3John Bardeen 1956 5Walter H. Brattain 1956 6William B. Shockley 1956 6Chen Ning Yang 1957 4 Tsung-dao Lee 1957 5Emilio Segrè 1959 4

Owen Chamberlain 1959 5Robert Hofstadter 1961 5Eugene Wigner 1963 4Richard P. Feynman 1965 4Julian S. Schwinger 1965 4Hans A. Bethe 1967 4Luis W. Alvarez 1968 6Murray Gell-Mann 1969 3John Bardeen 1972 5Leon N. Cooper 1972 6John R. Schrieffer 1972 5Aage Bohr 1975 5Ben Mottelson 1975 5Leo J. Rainwater 1975 7Steven Weinberg 1979 4Sheldon Lee Glashow 1979 2Abdus Salam 1979 3S. Chandrasekhar 1983 4 Norman F. Ramsey 1989 3

Números de Erdös de

premios Nobel de física

Erdös numberErdös 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, two-

way 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 DiameterMaximum 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

• is the number of edges in the shortest path between vertices i and j (geodesic path).

( , )i jL

• The characteristic path length L of a graph is the average of the for every possible pair (i,j)( , )i jL

( , ) 2i jL

i

j

Networks with small values of L are said to have the “Small World property”

A Few Good Man

Robert Wagner

Austin Powers: The spy who shagged me

Wild Things

Let’s make it legal

Barry Norton

What Price Glory

Monsieur Verdoux

Bacon’s Game

Internet Movie Database

http://www.cs.virginia.edu/oracle/

Rank NameAveragedistance

# ofmovies

# oflinks

1 Rod Steiger 2.537527 112 25622 Donald Pleasence 2.542376 180 28743 Martin Sheen 2.551210 136 35014 Christopher Lee 2.552497 201 29935 Robert Mitchum 2.557181 136 29056 Charlton Heston 2.566284 104 25527 Eddie Albert 2.567036 112 33338 Robert Vaughn 2.570193 126 27619 Donald Sutherland 2.577880 107 2865

10 John Gielgud 2.578980 122 294211 Anthony Quinn 2.579750 146 297812 James Earl Jones 2.584440 112 3787…

876 Kevin Bacon 2.786981 46 1811…

Why Kevin Bacon?Measure the average distance between Kevin Bacon and all other actors.

No. of movies : 46 No. of actors : 1811 Average separation: 2.79

Kevin Bacon

Is Kevin Bacon the most

connected actor?

NO!

876 Kevin Bacon 2.786981 46 1811

Rod Steiger

Martin Sheen

Donald Pleasence

#1

#2

#3

#876Kevin Bacon

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 ln N.

Minimal number of links one needs to follow to go from one node to another, on average.

Erdös-Renyi model (1960)

Poisson distribution

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

!!

)()(

k

ke

k

pNekP

kk

kpN

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

dNNL /1)( NNL log)(

Lattice

Random graph

• For many years typical explanation for Small-World 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) =# of links between neighbors

n(n-1)/2

Clustering: My friends will know each other with high probability!(typical example: social networks)

C(v) = 4/6

C is the average over all C(v)

Asymptotic behavior

.)(

)( /1

constNC

NNL d

1)(

log)(

NNC

NNL

Lattice

Random graph

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

Real life networks are clustered, large C, but have small average distance L.

Duncan J. Watts & Steven H. Strogatz, Nature 393, 440-442 (1998)

L Lrand C Crand NWWW 3.1 3.35 0.11 0.00023 153127Actors 3.65 2.99 0.79 0.00027 225226Power Grid18.7 12.4 0.080 0.005 4914C. Elegans 2.65 2.25 0.28 0.05 282

Structured network• high clustering• large diameter• regular

Random network• small clustering• small diameter

Small-world network• high clustering• small diameter• almost regular

N = 1000 k =10D = 100 L = 49.51C = 0.67

N =1000 k = 8-13D = 14 d = 11.1C = 0.63

N =1000 k = 5-18D = 5 L = 4.46C = 0.01

Duncan J. Watts & Steven H. Strogatz, Nature 393, 440-442 (1998)

Watts-Strogatz Model

C(p) : clustering coeff. L(p) : average path length

L

C

p

regular SW random