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Redes sicomoro

En esta charla se hace una introducción a los conceptos básicos de las redes complejas. Iniciaremos definiendo el concepto de grafo. Tras esta breve descripción veremos algunos ejemplos de diferentes ámbitos donde se han aplicado las redes complejas con éxito. En la última parte de la introducción, profundizaremos en las redes utilizando un caso de estudio. Tras esta introducción, veremos los primeros modelos de redes y algunos de los casos donde se aplicaron. Terminaremos viendo algunas sorprendentes propiedades de las redes complejas, como "Small world" para finalizar con un ejemplo práctico en donde se pueden aplicar las redes, en particular grafos de colores.

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Redes sicomoro

  1. 1. Redes Complejas, introducción Javier Galeano. Universidad Politécnica de Madrid @galeanojav
  2. 2. The large blue butterfly became extinct in England
  3. 3. ¿What do they have in common? A butterfly and a rabbit
  4. 4. Myxomatosis desease nearly killed the whole rabbit population in the area
  5. 5. Maculinea arion Myrmica sabuleti
  6. 6. These three different species share the habitat and a complex network
  7. 7. The fate of Saddam and network science
  8. 8. The fate of Saddam and network science
  9. 9. The fate of Saddam and network science
  10. 10. WHAT IS A GRAPH ?
  11. 11. • 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
  12. 12. Components of a Complex System
  13. 13. ! components: nodes, vertices N Components of a Complex System
  14. 14. ! components: nodes, vertices N Components of a Complex System
  15. 15. ! components: nodes, vertices N ! system: network, graph (N,L) Components of a Complex System ! interactions: links, edges L
  16. 16. Academic expertise
  17. 17. Academic expertise
  18. 18. Academic expertise
  19. 19. Academic expertise
  20. 20. A common language
  21. 21. A common language Peter Mary Albert Albert co-worker friendbrothers friend
  22. 22. 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
  23. 23. 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
  24. 24. CASO DE ESTUDIO
  25. 25. Caso real ..como la vida misma Primer día de colegio con 3 años A unos meses de entrar en el Instituto
  26. 26. “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.
  27. 27. “Chains of affection: The structure of adolescent romantic and sexual networks” Cada año en USA millones de personas descubren que son portadores de SDT
  28. 28. la tasa de infección en los adolescentes es muy superior a los otros grupos
  29. 29. “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. Entrevistaron a 832 adolescentes del instituto “Jefferson”
  30. 30. “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. Entrevistaron a 832 adolescentes del instituto “Jefferson”
  31. 31. “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. Entrevistaron a 832 adolescentes del instituto “Jefferson”
  32. 32. “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. Entrevistaron a 832 adolescentes del instituto “Jefferson”
  33. 33. “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. Entrevistaron a 832 adolescentes del instituto “Jefferson”
  34. 34. “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. Entrevistaron a 832 adolescentes del instituto “Jefferson”
  35. 35. “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. Entrevistaron a 832 adolescentes del instituto “Jefferson”
  36. 36. “Chains of affection: The structure of adolescent romantic and sexual networks” Male Female 12 9 63 2 Figure 2. Structure of Romantic and Sexual Contact at Jefferson
  37. 37. “Chains of affection: The structure of adolescent romantic and sexual networks” Male Female 12 9 63 2 Figure 2. Structure of Romantic and Sexual Contact at Jefferson
  38. 38. “Chains of affection: The structure of adolescent romantic and sexual networks” Male Female 12 9 63 2 Figure 2. Structure of Romantic and Sexual Contact at Jefferson
  39. 39. “Chains of affection: The structure of adolescent romantic and sexual networks” Male Female 12 9 63 2 Figure 2. Structure of Romantic and Sexual Contact at Jefferson
  40. 40. 12 63 Figure 2. Structure of Romantic and Sexual Contact at Je “Chains of affection: The structure of adolescent romantic and sexual networks” 288 adolescentes (52%) unidos en una componente gigante que involucra intercambios de fluidos.
  41. 41. 12 63 Figure 2. Structure of Romantic and Sexual Contact at Je “Chains of affection: The structure of adolescent romantic and sexual networks” Conexión con muchas ramas con falta de ciclos.
  42. 42. 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.
  43. 43. First Models
  44. 44. • Leonhard Euler wrote a paper on Seven Bridges of Könisberg and published in 1736. Very beginning of Graph Theory
  45. 45. 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)
  46. 46. Random network model p=1/6 N=12
  47. 47. Random network model p=0.03 N=100
  48. 48. 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.
  49. 49. Degree distribution k = 3×1+ 2 × 2 +1× 3 6 = 1,66 3 2 1
  50. 50. Degree distribution k = 1 k = 1 k = 1 k = 3×1+ 2 × 2 +1× 3 6 = 1,66 3 2 1
  51. 51. 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
  52. 52. 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
  53. 53. 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
  54. 54. “El mundo es un pañuelo”
  55. 55. Small World (Experimento de Milgram)
  56. 56. 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.
  57. 57. 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. •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.
  58. 58. Small World (Experimento de Milgram) 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.
  59. 59. Small World (Experimento de Milgram) • 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.
  60. 60. Small World (Experimento de Milgram) • 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!!!
  61. 61. 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)
  62. 62. 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/
  63. 63. 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 Erdös number
  64. 64. Which is my Erdös number?
  65. 65. 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.
  66. 66. 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. • I searched my Erdös number.
  67. 67. Which is my Erdös number? My Erdos Number is =5 • MathSciNet will automatically find a path in their database from you to Paul Erdös, or between any two people you wish. • I searched my Erdös number.
  68. 68. The oracle of Bacon Santiago Segura
  69. 69. The oracle of Bacon Santiago Segura Hellboy (2004)
  70. 70. The oracle of Bacon Santiago Segura Hellboy (2004) John Hurt
  71. 71. The oracle of Bacon Santiago Segura Hellboy (2004) John Hurt Infierno en Alabama (2012)
  72. 72. Santiago Segura Hellboy (2004) John Hurt Infierno en Alabama (2012) Kevin Bacon Bacon number = 2
  73. 73. Erdös number
  74. 74. Erdös number Natalie Portman Erdos number = 2 Bacon number = 5 Erdos-Bacon number = 7
  75. 75. 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
  76. 76. Real networks are not random!!
  77. 77. The Web of Human Sexual Contacts Liljeros et al. Nature 2001
  78. 78. The Web of Human Sexual Contacts
  79. 79. The Web of Human Sexual Contacts En el año 1996 4781 suecos entrevistados Entre 18 y 74 años Respondieron un 59 % (2810)
  80. 80. The Web of Human Sexual Contacts
  81. 81. The Web of Human Sexual Contacts
  82. 82. The Web of Human Sexual Contacts El Julio Iglesias sueco
  83. 83. The Web of Human Sexual Contacts El Julio Iglesias sueco
  84. 84. 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
  85. 85. Degree distributions • Miremos las distribuciones de grados de algunas redes
  86. 86. Redes scale-free A.-L.Barabási, R. Albert, Science 286, 509 (1999)
  87. 87. Redes scale-free • Crecimiento de la red: Cada paso se añade un nodo con un determinado número m de enlaces. A.-L.Barabási, R. Albert, Science 286, 509 (1999)
  88. 88. Redes scale-free • Crecimiento de la red: Cada paso se añade un nodo con un determinado número m de enlaces. • Preferential Attachment: la probabilidad Π de un nuevo nodo depende de la conectividad A.-L.Barabási, R. Albert, Science 286, 509 (1999)
  89. 89. Redes scale-free • Crecimiento de la red: Cada paso se añade un nodo con un determinado número m de enlaces. • Preferential Attachment: la probabilidad Π de un nuevo nodo depende de la conectividad A.-L.Barabási, R. Albert, Science 286, 509 (1999)
  90. 90. Redes scale-free • Crecimiento de la red: Cada paso se añade un nodo con un determinado número m de enlaces. • Preferential Attachment: la probabilidad Π de un nuevo nodo depende de la conectividad P(k) ~k-3 A.-L.Barabási, R. Albert, Science 286, 509 (1999)
  91. 91. OTRO CASO DE ESTUDIO
  92. 92. • 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
  93. 93. 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
  94. 94. 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
  95. 95. Incompatibilidades MR EA PI JC PSCL AR
  96. 96. Incompatibilidades MR EA PI JC PSCL AR
  97. 97. Incompatibilidades • Pablo Iglesias • Pedro Sanchez • Cayo Lara • Mariano Rajoy • Albert Rivera • Esperanza Aguirre • Ciudadano Juan Carlos
  98. 98. GENERAL AUDIENCE
  99. 99. Thex DOCUMENTARY
  100. 100. 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).
  101. 101. 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).
  102. 102. • Science: Special Issue for the 10 year anniversary of Barabasi&Albert 1999 paper.   JOURNAL

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