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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 1.475 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 (1913-1996)

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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 1- 504 colaboradores Número 2- 6593 colaboradores http://www.oakland.edu/enp/

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Max von Laue 1914 4 Albert Einstein 1921 2 Niels Bohr 1922 5 Louis de Broglie 1929 5 Werner Heisenberg 1932 4 Paul A. Dirac 1933 4 Erwin Schrödinger 1933 8 Enrico Fermi 1938 3 Ernest O. Lawrence 1939 6 Otto Stern 1943 3 Isidor I. Rabi 1944 4 Wolfgang Pauli 1945 3 Frits Zernike 1953 6 Max Born 1954 3 Willis E. Lamb 1955 3 John Bardeen 1956 5 Walter H. Brattain 1956 6 William B. Shockley 1956 6 Chen Ning Yang 1957 4 Tsung-dao Lee 1957 5 Emilio Segrè 1959 4 Owen Chamberlain 1959 5 Robert Hofstadter 1961 5 Eugene Wigner 1963 4 Richard P. Feynman 1965 4 Julian S. Schwinger 1965 4 Hans A. Bethe 1967 4 Luis W. Alvarez 1968 6 Murray Gell-Mann 1969 3 John Bardeen 1972 5 Leon N. Cooper 1972 6 John R. Schrieffer 1972 5 Aage Bohr 1975 5 Ben Mottelson 1975 5 Leo J. Rainwater 1975 7 Steven Weinberg 1979 4 Sheldon Lee Glashow 1979 2 Abdus Salam 1979 3 S. Chandrasekhar 1983 4 Norman F. Ramsey 1989 3 Números de Erdös de premios Nobel de física

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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

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

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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 2.786981 46 1811 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, 440-442 (1998) LL rand CC N WWW 3.13.350.110.00023153127 Actors 3.652.990.790.00027225226 PowerGrid 18.712.40.0800.0054914 C.Elegans 2.652.250.280.05282

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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 = 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

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