1. 1 Modelling, Mining, and Searching Networks Anthony Bonato Ryerson University Graduate Seminar October 2015

2. Networks - Bonato2 21 st Century Graph Theory: Complex Networks web graph, social networks, biological networks, internet networks, …

3. Networks - Bonato3 a graph G = (V(G),E(G)) consists of a nonempty set of vertices or nodes V, and a set of edges E nodes edges in directed graphs (digraphs) E need not be symmetric

4. Networks - Bonato4 Degrees the degree of a node x, written deg(x) is the number of edges incident with x First Theorem of Graph Theory:

5. Networks - Bonato5 The web graph nodes: web pages edges: links over 1 trillion nodes, with billions of nodes added each day

6. Networks - Bonato6 Ryerson Greenland Tourism Frommer’s Four Seasons Hotel City of Toronto Nuit Blanche

7. Networks - Bonato7 Small World Property small world networks introduced by Watts & Strogatz in 1998 –low distances between nodes

8. Networks - Bonato8 Power laws in the web graph power law degree distribution (Broder et al, 01)

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 - Bonato9

10. SPA model (Aiello,Bonato,Cooper,Janssen,Prałat, 09) Networks - Bonato10 as nodes are born, they are more likely to enter a region of influence with larger volume (degree) over time, a power law degree distribution results

11. Networks - Bonato11

12. Networks - Bonato12 Biological networks: proteomics nodes: proteins edges: biochemical interactions Yeast: 2401 nodes 11000 edges

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 - Bonato13

14. Domination sets in PPI (Milenkovic, Memisevic, Bonato, Przulj, 2011) PLOS ONE 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 - Bonato14

15. Networks - Bonato15 Social Networks nodes: people edges: social interaction

16. Networks - Bonato16 On-line Social Networks (OSNs) Facebook, Twitter, LinkedIn, Google+…

17. 17 Bieber to Pope Francis on Networks - Bonato

18. 6 Degrees in Facebook? 1.15 billion users (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 18Networks - Bonato

19. 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 - Bonato19

20. GEO-P model (Bonato et al, 2014): PLOS ONE reverse engineering approach –given network data GEO-P model predicts dimension of an OSN to be around log n, where n is the number of users that is, given the graph structure, we can (theoretically) recover the social space Networks - Bonato20

21. 6 Dimensions of Separation in Facebook and LinkedIn Networks - Bonato21

22. Cops and Robbers Networks - Bonato22 C C C R

23. Cops and Robbers Networks - Bonato23 C C C R

24. Cops and Robbers Networks - Bonato24 C C C R cop number c(G) ≤ 3

25. Cops and Robbers minimum number of cops needed to capture the robber is the cop number c(G) –well-defined as c(G) ≤ |V(G)| Networks - Bonato25

26. Applications of Cops and Robbers robotics –mobile computing –missile-defense –gaming counter-terrorism –intercepting messages or agents Networks - Bonato26

27. 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(n 1/2 ). Networks - Bonato27

28. Good guys vs bad guys games in graphs 28 slowmediumfasthelicopter slowtraps, tandem-win mediumrobot vacuumCops and Robbersedge searchingeternal security fastcleaningdistance k Cops and Robbers Cops and Robbers on disjoint edge sets The Angel and Devil helicopterseepageHelicopter Cops and Robbers, Marshals, The Angel and Devil, Firefighter Hex bad good Networks - Bonato

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30. Thesis topics new models of complex networks biological network models Banking/financial networks fitting models to data Cops and Robbers games –Meyniel’s conjecture, random graphs, variations: good vs bad guy games in graphs Networks - Bonato30

31. Brief biography over 90 papers, two original books, 7 edited proceedings books, with 61 collaborators (many of which are my students) over 480K lifetime research –grants from NSERC, MITACS, Mprime, and Ryerson –FOS accelerator (additional support available in Y1) supervised 12 masters students, 2 doctoral, and 13 post-docs over 30 invited addresses world-wide (India, China, Europe, North America) won 2011 and 2009 Ryerson SRC awards for research excellence won 2013 an inaugural YSGS Outstanding Contribution to Graduate Education Award editor-in-Chief of journal Internet Mathematics; editor of Contributions to Discrete Mathematics Networks - Bonato31

32. Drop in office hours Wednesday, November 4, 10 am – 12 pm Thursday, November 5, 10 am – 12 pm Yeates School of Graduate Studies 11 th floor of 1 Dundas St West, YDI – 1117 Come to say hello, chat, discuss thesis topics Networks - Bonato32

33. AM8204 – Topics in Discrete Mathematics Winter 2014 6 weeks each: complex networks, graph searching project based Prequisite: AM8002 (or permission from me) Networks - Bonato33

34. Graphs at Ryerson (G@R) Networks - Bonato34