1. Complex networks - Bonato1 Complex networks and their models Anthony Bonato Ryerson University Graduate Seminar October 2011

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

3. Complex networks - Bonato3 a graph G = (V(G),E(G)) = (V,E) consists of a nonempty set of vertices or nodes V, and a set of edges E, which is a symmetric binary relation on V (usually irreflexive) nodes edges in directed graphs (digraphs) E need not be symmetric

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

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

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

8. Complex 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 Complex networks - Bonato9

10. SPA model (Aiello,Bonato,Cooper,Janssen,Prałat, 09) Complex 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. Complex networks - Bonato11

12. Complex 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 Complex networks - Bonato13

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 Complex networks - Bonato14

15. Complex networks - Bonato15 Social Networks nodes: people edges: social interaction (eg friendship)

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

17. Lady Gaga is the centre of Twitterverse Complex networks - Bonato17 Dalai Lama Lady Gaga Anderson Cooper Queen Rania of Jordan Arnold Schwarzenegger

18. 6 degrees of separation Complex networks - Bonato18 Stanley Milgram: famous chain letter experiment in 1967

19. 6 Degrees in Twitter Java et al. (2009) –6 degrees of separation in Twitter other researchers found similar results in Facebook, LinkedIn, … Complex networks - Bonato19

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? Complex networks - Bonato20

21. GEO-P model (Bonato, Janssen, Prałat, 2010) 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 Complex networks - Bonato21

22. 6 Dimensions of Separation OSNDimension YouTube6 Twitter4 Flickr4 Cyworld7 Complex networks - Bonato22

23. Cops and Robbers Complex networks - Bonato23 C C C R

24. Cops and Robbers Complex networks - Bonato24 C C C R

25. Cops and Robbers Complex networks - Bonato25 C C C R cop number c(G) ≤ 3

26. Applications of Cops and Robbers moving target search –missile-defense –gaming counter-terrorism –intercepting messages or agents Complex networks - Bonato26

27. Complex networks - Bonato27

28. The robber fights back! (Haidar’s thesis) robber can attack neighbouring 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. Complex networks - Bonato28 C C C R

29. 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 –is theoretical estimate of the dimension of an OSN accurate? Cops and Robbers games –Meyniel’s conjecture, random graphs, variations (good vs bad guy games in graphs) Complex networks - Bonato29

30. Complex networks - Bonato30 preprints, reprints, contact: Google: “Anthony Bonato”

31. Brief biography over 70 papers and two books, with 40 collaborators (many of which are my students) over 200K in research funding in past 6 years –grants from NSERC, Mprime, and Ryerson supervised 6 masters students, 2 doctoral, and 8 post- docs over 25 invited addresses world-wide over last 6 years won a 2009 Ryerson Research award editor-in-Chief of journal Internet Mathematics Complex networks - Bonato31

32. Graphs at Ryerson (G@R) Complex networks - Bonato32