Complex networks and their models

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

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

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

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

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

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

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

11. Complex networks - Bonato

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

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

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

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

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

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 - 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? Complex networks - Bonato

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 - Bonato

22. 6 Dimensions of Separation Complex networks - Bonato

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

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

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

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

27. Complex networks - Bonato

28. The robber fights back! (Haidar’s thesis) • 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 Complex networks - Bonato

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 - Bonato

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

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