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Networks in Nature

Networks in Nature. e-Tutor ricerca e azione I piccoli mondi dell’e-learnig Milano, 7 novembre 2003. Fabrizio Coccetti e Guido Caldarelli Cecile Caretta, Diego Garlaschelli, Luciano Pietronero, Vito Servedio, Federico Squartini

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Networks in Nature

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  1. Networks in Nature e-Tutor ricerca e azione I piccoli mondi dell’e-learnig Milano, 7 novembre 2003 Fabrizio Coccetti e Guido Caldarelli Cecile Caretta, Diego Garlaschelli, Luciano Pietronero, Vito Servedio, Federico Squartini Centro Studi e Ricerche e Museo Storico della Fisica “Enrico Fermi” Università di Roma “La Sapienza” Fabrizio Coccetti - Guido Caldarelli - et al.

  2. Il messaggio da ricordare Nella maggior parte delle reti reali: • Effetto Small World(il mondo è piccolo) • Struttura Scale-free Fabrizio Coccetti - Guido Caldarelli - et al.

  3. Agenda • Esperimento di Stanley Milgram (1967) • Small World • Il problema dei ponti di Königsberg • Teoria dei Grafi • Strutture scale-free: Internet • Esempi Vari Fabrizio Coccetti - Guido Caldarelli - et al.

  4. L’esperimento di Milgram (1967) E’ possibile consegnare un messaggio ad un agente di cambio a Chicago partendo da persone prese a caso nel Nebraska ? Fabrizio Coccetti - Guido Caldarelli - et al.

  5. Il mondo è piccolo ! In media meno di 6 passaggi !! Sei gradi di separazione Fabrizio Coccetti - Guido Caldarelli - et al.

  6. La struttura delle reti sociali According to Mark Granovetter the shortcuts are the “weak links” Fabrizio Coccetti - Guido Caldarelli - et al.

  7. Il mondo è piccolo La distanza massima tra due punti del sistema è un numero “piccolo”. • Sistemi sociali (relazioni di amicizia, …) • Sistemi di trasporto • Sistemi Informatici (Internet, …) • Sistemi Biologici (proteine, …) • Sistemi ecologici (catene alimentari, …) Fabrizio Coccetti - Guido Caldarelli - et al.

  8. I ponti di Königsberg E` possibile visitare tutte le parti della città di Königsberg passando tutti i ponti sul Pregel una sola volta ? NO! Leonard Euler (1736) mostrò che per essere un punto di passaggio un vertice deve avere un numero pari di archi (collegamenti). Solo i punti di partenza o di fine possono avere un numero dispari di collegamenti. Questo non è il caso di Königsberg. Fabrizio Coccetti - Guido Caldarelli - et al.

  9. 1736 (Königsberg) Tutti i vertici hanno grado dispari. No way Il problema dipende dal tempo ? 2003 (Kaliningrad) Solo B e C hanno grado dispari. OK Fabrizio Coccetti - Guido Caldarelli - et al.

  10. Topologia dei Grafi Un Grafo G(v,e) è un oggetto composto da v vertici and e archi Degree k(In-degree kinandout-degree kout) = number of edges (oriented) per vertex Distance d= minimum number of edges amongst two vertices (in the connected region ) Diameter D= Maximum of the distances ( in the connected region !) Clustering= cliques distribution, or clustering coefficient Usually many quantities are needed In order to “classify” a network Fabrizio Coccetti - Guido Caldarelli - et al.

  11. Topologia dei Grafi (segue) Degree frequency density P(k)= how many times you find a vertex whose degree is k P(k) k Degree Correlation Knn (k) = average degree of a neighbour of a vertex with degree k Clustering Coefficient (k) = the average value of c for a vertex whose degree is k Fabrizio Coccetti - Guido Caldarelli - et al.

  12. Topologia dei Grafi (segue) P(A) A Centrality betweenness b(k)= The probability that a vertex whose degree is k has betweenness b betweenness of Vis the number of distances between any pair of vertices passing through V V TREES ONLY!!! P(A) = Probability Density for subbranches of size A 1 1 1 1 Size distribution: Allometric relations: 1 1 3 5 1 1 C(A) 11 5 2 3 1 1 22 8 A 10 33 Fabrizio Coccetti - Guido Caldarelli - et al.

  13. Aggregation in networks • Low level properties : • Degree : the number of nearest neighbours, not their properties. (B.A. Model) • High level properties : • How the individuality of nodes influence the formation of edges between them. (Our model) • Example: Proprietà , , , … Is there a correlation between properties of adjacent nodes? • Maslov, Sneppen, Zaliznyak, cond-mat/0205379. • Catania et al., Am. J. of Public Health, 82: 284-287 (1992) . Fabrizio Coccetti - Guido Caldarelli - et al.

  14. Assortativity In a first approximation: Property of the node = its degree Assortative networks Disassortative networks • Real networks display one of these two tendencies, • “similar” networks display “similar” behaviours. Techological networks Social networks Fabrizio Coccetti - Guido Caldarelli - et al.

  15. Fabrizio Coccetti - Guido Caldarelli - et al.

  16. Internet Autonomous Systems Exchange Point border routers Peering hosts Routers home networks Fabrizio Coccetti - Guido Caldarelli - et al.

  17. Struttura Scale-free • La maggior parte dei nodi ha poche connessioni • Pochi nodi hanno moltissime connessioni • Buona resistenza a guasti random • Scarsa resistenza ad attacchi pianificati E’ una struttura diversa da reti random o reti regolari Fabrizio Coccetti - Guido Caldarelli - et al.

  18. “Food Web” (ecological network) Set of interconnected food chains resulting in a much more complex topology: Fabrizio Coccetti - Guido Caldarelli - et al.

  19. Ecosystems around the world Lazio Utah Iran Amazonia Peruvian and Atacama Desert Argentina we focus our attention on plants in order to obtain a good universality of the results we have chosen a great variety of climatic environments Ecosystem= Set of all living organisms and environmental properties of a restricted geographic area Fabrizio Coccetti - Guido Caldarelli - et al.

  20. phylum subphylum class subclass order family genus species Connected graph without loops or double-linked nodes • From Linnean trees to graph theory Linnean Tree= hierarchical structure organized on different levels, called taxonomic levels, representing: • classification and identification of different plants • history of the evolution of different species A Linnean tree already has the topological structure of a tree graph • each node in the graph represents a different taxa • (specie, genus, family, and so on). All nodes are • organized on levels representing the taxonomic one • all link are up-down directed and each one • represents the belonging of a taxon to the relative • upper level taxon Fabrizio Coccetti - Guido Caldarelli - et al.

  21. ~ 2.5  0.2 • Scale-free properties Degree distribution: P(k) k The best results for the exponent value are given by ecosystems with greater number of species. For smaller networks its value can increase reaching  = 2.8 - 2.9. Fabrizio Coccetti - Guido Caldarelli - et al.

  22. Protein Interaction Network of Yeast Saccaromyces Cerevisiae Lievito di Birra Fabrizio Coccetti - Guido Caldarelli - et al.

  23. Portfolio Composition Investors or Companies not traded at Borsa di Milano (Italy) Fabrizio Coccetti - Guido Caldarelli - et al. Companies traded at Borsa di Milano (Italy)

  24. Portfolio Composition Fabrizio Coccetti - Guido Caldarelli - et al.

  25. Portfolio Composition Fabrizio Coccetti - Guido Caldarelli - et al.

  26. Portfolio Composition Fabrizio Coccetti - Guido Caldarelli - et al.

  27. Board of Directors Fabrizio Coccetti - Guido Caldarelli - et al.

  28. COSINCOevolution and Self-organisation In dynamical Networks RTD Shared Cost Contract IST-2001-33555 http://www.cosin.org • Nodes 6 in 5 countries • Period of Activity: April 2002-April 2005 • Budget: 1.256 M€ • Persons financed: 8-10 researchers • Human resources: 371.5 Persons/months EU countries Non EU countries EU COSIN participant Non EU COSIN participant Fabrizio Coccetti - Guido Caldarelli - et al.

  29. References • http://www.cosin.org/ • http://www.cosin.org/Publications.html • http://pil.phys.uniroma1.it/~gcalda/Publications.html • http://www.nd.edu/~alb/public.html • http://www1.cs.columbia.edu/~sanders/graphtheory/people/Bollobas.B.html Fabrizio Coccetti - Guido Caldarelli - et al.

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