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

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Modelling, Mining, and Searching Networks

Anthony BonatoRyerson University

Graduate SeminarOctober 2015

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21st Century Graph Theory:Complex Networks

• web graph, social networks, biological networks, internet networks, …

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• a graph G = (V(G),E(G)) consists of a nonempty set of vertices or nodes V, and a set of edges E

nodesedges

• in directed graphs (digraphs) E need not be symmetric

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Degrees• the degree of a node x, written

deg(x)

is the number of edges incident with x

First Theorem of Graph Theory:

V(G)x

|E(G)|2deg(x)

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The web graph

• nodes: web pages

• edges: links

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

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Ryerson

GreenlandTourism

Frommer’s

Four SeasonsHotel

City of Toronto

Nuit Blanche

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Small World Property

• small world networks introduced by Watts & Strogatz in 1998– low distances

between nodes

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Power laws in the web graph• power law degree distribution

(Broder et al, 01)

2 some ,, bniN bni

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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

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SPA model (Aiello,Bonato,Cooper,Janssen,Prałat, 09)

• 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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Biological networks: proteomics

nodes: proteins

edges:

biochemical interactions

Yeast: 2401 nodes11000 edges

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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

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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

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Social Networks

nodes: people

edges: social interaction

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

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Bieber to Pope Francis on

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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

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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?

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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

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6 Dimensions of Separation in Facebook and LinkedIn

Cops and Robbers

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C

C

C

R

Cops and Robbers

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C

C

C

R

Cops and Robbers

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C

C

C

R

cop number c(G) ≤ 3

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)|

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Applications of Cops and Robbers

• robotics– mobile computing– missile-defense– gaming

• counter-terrorism– intercepting messages

or agents

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(n1/2).

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Good guys vs bad guys games in graphs

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slow medium fast helicopter

slow traps, tandem-win

medium robot vacuum Cops and Robbers edge searching eternal security

fast cleaning distance k Cops and Robbers

Cops and Robbers on disjoint edge sets

The Angel and Devil

helicopter seepage Helicopter Cops and Robbers, Marshals, The Angel and Devil,Firefighter

Hex

badgood

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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

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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

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Drop in office hours

• Wednesday, November 4, 10 am – 12 pm• Thursday, November 5, 10 am – 12 pm

• Yeates School of Graduate Studies • 11th floor of 1 Dundas St West, YDI – 1117

• Come to say hello, chat, discuss thesis topics

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AM8204 – Topics in Discrete Mathematics

• Winter 2014• 6 weeks each: complex networks, graph

searching• project based• Prequisite: AM8002 (or permission from

me)

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Graphs at Ryerson (G@R)