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Complex Networks Junio 2006 L. Lacasa, B. Luque y J.C. Nuño Departamentos de Matemática Aplicada Aeronáuticos y Montes Universidad Politécnica de Madrid

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Society Nodes: individuals Links: social relationship (family/work/friends/ etc.) Social networks: Many individuals with diverse social interactions between them.

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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.

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Social networks: Milgram’s experiment 160 letters: From Wichita (Kansas) and Omaha (Nebraska) to Sharon (Mass) Milgram, Psych Today 2, 60 (1967) who is more likely than you 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.

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¡ El mundo es un pañuelo ! C’est petit le monde !! What a Small-World ! “Six degrees of separation”

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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.

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El número de Erdös Fue autor o coautor de art í culos matem á ticos y colabor ó en ellos con un total de 493 coautores distintos. S ó lo un matem á tico en la historia escribi ó m á s p á ginas de matem á ticas originales que Erd ö s. En siglo XVII, el suizo Leonhard Euler, padre de trece ni ñ os, escribi ó ochenta vol ú menes de resultados matem á ticos. Pál Erdös Pál Erdös ( )

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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 Número colaboradores Número colaboradores

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Max von Laue Albert Einstein 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 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 Números de Erdös de premios Nobel de física

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Erdös number Erdös number person Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people

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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

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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)

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Degree Number of edges connected to a node. In-degree Number of incoming edges. Out-degree Number of outgoing edges.

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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?

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Characteristic path length GLOBAL property is the number of edges in the shortest path between vertices i and j (geodesic path). The characteristic path length L of a graph is the average of the for every possible pair (i,j) i j Networks with small values of L are said to have the “Small World property”

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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

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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 Bacon-list

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Rod Steiger Martin Sheen Donald Pleasence #1 #2 #3 #876 Kevin Bacon Bacon-map

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Tree Network

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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.

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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.

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Erdös-Renyi model (1960) Poisson distribution Many properties in these graphs appear quite suddenly, at a threshold value of p = P ER (N) -If P ER ~ c / N with c < 1, then almost all vertices belong to isolated trees. -Cycles of all orders appear at P ER ~ 1/ N

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Random Graphs Model Given N nodes connect each pair with probability p: –P(k) ~ Poisson distribution – = pN. –Most nodes degree ~. – = log(N) / log( ). –Small World property

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Asymptotic behavior LatticeRandom graph

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For many years typical explanation for Small- World property was random graphs –Low diameter: expected distance between two nodes is log N, where 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.

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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)

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Asymptotic behavior LatticeRandom graph

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Power grid NW USA-Canada N = 4914 k max = 19 k aver = 2.67 L = 18.7 C = 0.08 D = 46

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Caenorhabditis elegans Neural system N = 282 k max = 14 k average = 9 L = 2.65 C = 0.28 D = 3

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Real life networks are clustered, large C, but have small average distance L. Duncan J. Watts & Steven H. Strogatz, Nature 393, (1998) LL rand CC N WWW Actors PowerGrid C.Elegans

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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 =10 D = 100 L = 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

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

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Watts-Strogatz Model C(p) : clustering coeff. L(p) : average path length L C p regular SW random

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