Networks - Bonato1 Modelling, Mining, and Searching Networks Anthony Bonato Ryerson University Master’s Seminar November 2012

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

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 directed graphs (digraphs)

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:

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

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

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

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

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

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

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

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

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 - Bonato14

Networks - Bonato15 Social Networks nodes: people edges: social interaction (eg friendship)

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

Lady Gaga is the centre of Twitterverse Networks - Bonato17 Dalai Lama Lady Gaga Anderson Cooper Queen Rania of Jordan Arnold Schwarzenegger

6 degrees of separation Networks - Bonato18 Stanley Milgram: famous chain letter experiment in 1967

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 - Bonato19

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 - Bonato20

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 - Bonato21

6 Dimensions of Separation OSNDimension YouTube6 Twitter4 Flickr4 Cyworld7 Networks - Bonato22

Cops and Robbers Networks - Bonato23 C C C R

Cops and Robbers Networks - Bonato24 C C C R

Cops and Robbers Networks - Bonato25 C C C R cop number c(G) ≤ 3

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 - Bonato26

Applications of Cops and Robbers moving target search –missile-defense –gaming counter-terrorism –intercepting messages or agents Networks - Bonato27

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 - Bonato28

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Example of a variant The robber fights back! 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. Networks - Bonato30 C C C R

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 - Bonato31

Good guys vs bad guys games in graphs 32 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

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 - Bonato33

AM8204 – Topics in Discrete Mathematics Winter weeks each: complex networks, graph searching project based Prequisite: AM8002 (or permission from me) Networks - Bonato34

Graphs at Ryerson Networks - Bonato35