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Masters 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 - PowerPoint PPT Presentation

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Models of the web graph

Networks - Bonato1Modelling, Mining, and Searching NetworksAnthony BonatoRyerson UniversityMasters SeminarNovember 2012Networks - Bonato221st Century Graph Theory:Complex Networksweb 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

nodesedges directed graphs (digraphs)Networks - Bonato4Degreesthe degree of a node x, writtendeg(x)is the number of edges incident with x

First Theorem of Graph Theory:

Networks - Bonato5The web graphnodes: web pages

edges: links

over 1 trillion nodes, with billions of nodes added each day

5Networks - Bonato6RyersonGreenlandTourismFrommersFour SeasonsHotelCity of TorontoNuit BlancheNetworks - Bonato7Small World Propertysmall world networks introduced by social scientists Watts & Strogatz in 1998low distances between nodes

Networks - Bonato8Power laws in the web graphpower law degree distribution

(Broder et al, 01)

Geometric modelswe introduced a stochastic network model which simulates power law degree distributions and other propertiesSpatially Preferred Attachment (SPA) Modelnodes have a region of influence whose volume is a function of their degreeNetworks - Bonato9

SPA model (Aiello,Bonato,Cooper,Janssen,Praat, 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 - Bonato12Biological networks: proteomicsnodes: proteins

edges: biochemical interactions

Yeast: 2401 nodes11000 edgesProtein networksproteins are essential macromolecules of lifeunderstanding their function and role in disease is of importanceprotein-protein interaction networks (PPI)nodes: proteinsedges: biochemical interactionNetworks - Bonato13

Domination sets in PPI (Milenkovic, Memisevic, Bonato, Przulj, 2011)dominating sets in graphs

we found that dominating sets inPPI networks are vital for normalcellular functioning and signallingdominating sets capture biologically vital proteins and drug targetsmight eventually lead to new drug therapies

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Networks - Bonato15Social Networksnodes: people

edges: social interaction(eg friendship)

Networks - Bonato16On-line Social Networks (OSNs)Facebook, Twitter, LinkedIn, Google+

Lady Gaga is the centre of TwitterverseNetworks - Bonato17

Dalai LamaLady Gaga

Anderson CooperQueen Rania of JordanArnold Schwarzenegger

6 Degrees in Facebook?1 billion users, > 70 billion friendship links(Backstrom et al., 2012)4 degrees of separation in Facebookwhen considering another person in the world, a friend of your friend knows a friend of their friend, on averagesimilar results for Twitter and other OSNsNetworks - Bonato19

Dimension of an OSNdimension 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 - Bonato20GEO-P model (Bonato, Janssen, Praat, 2012)reverse engineering approachgiven network data GEO-P model predicts dimension of an OSN; i.e. the smallest number of attributes needed to identify usersthat is, given the graph structure, we can (theoretically) recover the social spaceNetworks - Bonato21

6 Dimensions of SeparationOSNDimensionYouTube6Twitter4Flickr4Cyworld7Networks - Bonato22Cops and Robbers

Networks - Bonato23CCCRCops and Robbers

Networks - Bonato24CCCRCops and Robbers

Networks - Bonato25CCCRcop number c(G) 3Cops and Robbersplayed on reflexive undirected graphs Gtwo players Cops C and robber R play at alternate time-steps (cops first) with perfect informationplayers 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 captureminimum number of cops needed to capture the robber is the cop number c(G)well-defined as c(G) |V(G)|Networks - Bonato26Applications of Cops and Robbersmoving target searchmissile-defensegaming

counter-terrorismintercepting messages or agentsNetworks - 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!

Meyniels Conjecture: c(G) = O(n1/2).

Networks - Bonato28Networks - Bonato29

Example of a variantThe 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

CCCRThesis topicswhat precisely is a community in a complex network? biological network modelsmore exploration of dominating sets in PPIfit GEO-P model to OSN datamachine learning techniquesnew models for complex networksCops and Robbers games Meyniels conjecture, random graphs, variations: good vs bad guy games in graphs

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Good guys vs bad guys games in graphs32slowmediumfasthelicopterslowtraps, tandem-winmediumrobot vacuumCops and Robbersedge searchingeternal securityfastcleaningdistance k Cops and RobbersCops and Robbers on disjoint edge setsThe Angel and DevilhelicopterseepageHelicopter Cops and Robbers, Marshals, The Angel and Devil,FirefighterHexbadgoodNetworks - BonatoBrief biographyover 80 papers, two books, two edited proceedings, with 40 collaborators (many of which are my students)over 250K in research funding in past 6 yearsgrants from NSERC, Mprime, and Ryersonsupervised 8 masters students, 2 doctoral, and 7 post-docsover 30 invited addresses world-wide (India, China, Europe, North America)won 2011 and 2009 Ryerson Research awardseditor-in-Chief of journal Internet Mathematics; editor of Contributions to Discrete Mathematics

Networks - Bonato33AM8204 Topics in Discrete MathematicsWinter 20126 weeks each: complex networks, graph searchingproject basedPrequisite: AM8002 (or permission from me)Networks - Bonato34Graphs at Ryerson (G@R)Networks - Bonato35