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Modelling, Mining, and Searching Networks

Master’s Seminar November 2012. Modelling, Mining, and Searching Networks. Anthony Bonato Ryerson University. 21 st Century Graph Theory: Complex Networks. web graph, social networks, biological networks, internet networks , …. a graph G = (V(G),E(G )) consists of a nonempty set

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Modelling, Mining, and Searching Networks

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  1. Master’sSeminar November 2012 Modelling, Mining, and Searching Networks Anthony Bonato Ryerson University Networks - Bonato

  2. 21st Century Graph Theory:Complex Networks • web graph, social networks, biological networks, internet networks, … Networks - Bonato

  3. a graphG = (V(G),E(G)) • consists of a nonempty set • of vertices or nodesV, and • a set of edgesE nodes edges • directed graphs (digraphs) Networks - Bonato

  4. Degrees • the degree of a node x, written deg(x) is the number of edges incident with x First Theorem of Graph Theory: Networks - Bonato

  5. nodes: web pages edges: links over 1 trillion nodes, with billions of nodes added each day The web graph Networks - Bonato

  6. Nuit Blanche Ryerson City of Toronto Four Seasons Hotel Frommer’s Greenland Tourism Networks - Bonato

  7. Small World Property • small world networks introduced by social scientists Watts & Strogatz in 1998 • low distances between nodes Networks - Bonato

  8. Power laws in the web graph • power law degree distribution (Broder et al, 01) Networks - Bonato

  9. Geometric models • we introduced a stochastic network model which simulates power law degree distributions and other properties • Spatially Preferred Attachment (SPA) Model • nodes have a region of influence whose volume is a function of their degree Networks - Bonato

  10. SPA model (Aiello,Bonato,Cooper,Janssen,Prałat, 09) • as nodes are born, • they are more • likely to enter a region of influencewith larger volume (degree) • over time, a • power law • degree • distribution • results Networks - Bonato

  11. Networks - Bonato

  12. nodes: proteins edges: biochemical interactions Biological networks: proteomics Yeast: 2401 nodes 11000 edges Networks - Bonato

  13. Protein networks • proteins are essential macromolecules of life • understanding their function and role in disease is of importance • protein-protein interaction networks (PPI) • nodes: proteins • edges: biochemical interaction Networks - Bonato

  14. Domination sets in PPI(Milenkovic, Memisevic, Bonato, Przulj, 2011) • dominating sets in graphs • we found that dominating sets in PPI networks are vital for normal cellular functioning and signalling • dominating sets capture biologically vital proteins and drug targets • might eventually lead to new drug therapies Networks - Bonato

  15. Social Networks nodes: people edges: social interaction (eg friendship) Networks - Bonato

  16. On-line Social Networks (OSNs)Facebook, Twitter, LinkedIn, Google+… Networks - Bonato

  17. Lady Gaga is the centre of Twitterverse Dalai Lama Arnold Schwarzenegger Queen Rania of Jordan Anderson Cooper Lady Gaga Networks - Bonato

  18. 6 degrees of separation • Stanley Milgram: famous chain letter experiment in 1967 Networks - Bonato

  19. 6 Degrees in Facebook? • 1 billion users, > 70 billion friendship links • (Backstrom et al., 2012) • 4 degrees of separation in Facebook • when considering another person in the world, a friend of your friend knows a friend of their friend, on average • similar results for Twitter and other OSNs Networks - Bonato

  20. Dimension of an OSN • dimension of OSN: minimum number of attributes needed to classify nodes • like game of “20 Questions”: each question narrows range of possibilities • what is a credible mathematical formula for the dimension of an OSN? Networks - Bonato

  21. GEO-P model (Bonato, Janssen, Prałat, 2012) • reverse engineering approach • given network data GEO-P model predicts dimension of an OSN; i.e. the smallest number of attributes needed to identify users • that is, given the graph structure, we can (theoretically) recover the social space Networks - Bonato

  22. 6 Dimensions of Separation Networks - Bonato

  23. Cops and Robbers C C R C Networks - Bonato

  24. Cops and Robbers C C R C Networks - Bonato

  25. Cops and Robbers C R C C cop number c(G) ≤ 3 Networks - Bonato

  26. Cops and Robbers • played on reflexive undirected graphs G • two players Cops C and robber R play at alternate time-steps (cops first) with perfect information • players move to vertices along edges; allowed to moved to neighbors or pass • cops try to capture (i.e. land on) the robber, while robber tries to evade capture • minimum number of cops needed to capture the robber is the cop number c(G) • well-defined as c(G) ≤ |V(G)| Networks - Bonato

  27. Applications of Cops and Robbers • moving target search • missile-defense • gaming • counter-terrorism • intercepting messages or agents Networks - Bonato

  28. How big can the cop number be? • if the graph G with order n is disconnected, then the cop number can be as n • if G is connected, then no one knows how big the cop number can be! • Meyniel’s Conjecture: c(G) = O(n1/2). Networks - Bonato

  29. Networks - Bonato

  30. Example of a variantThe robber fights back! • robber can attackneighbouring cop • one more cop needed in this graph (check) • Conjecture: For any graph with this modified game, one more cop needed than for usual cop number. C C R C Networks - Bonato

  31. Thesis topics • what precisely is a community in a complex network? • biological network models • more exploration of dominating sets in PPI • fit GEO-P model to OSN data • machine learning techniques • new models for complex networks • Cops and Robbers games • Meyniel’s conjecture, random graphs, variations: good vs bad guy games in graphs Networks - Bonato

  32. Good guys vs bad guys games in graphs bad good Networks - Bonato

  33. Brief biography • over 80 papers, two books, two edited proceedings, with 40 collaborators (many of which are my students) • over 250K in research funding in past 6 years • grants from NSERC, Mprime, and Ryerson • supervised 8 masters students, 2 doctoral, and 7 post-docs • over 30 invited addresses world-wide (India, China, Europe, North America) • won 2011 and 2009 Ryerson Research awards • editor-in-Chief of journal Internet Mathematics; editor of Contributions to Discrete Mathematics Networks - Bonato

  34. AM8204 – Topics in Discrete Mathematics • Winter 2012 • 6 weeks each: complex networks, graph searching • project based • Prequisite: AM8002 (or permission from me) Networks - Bonato

  35. Graphs at Ryerson (G@R) Networks - Bonato

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