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

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Modelling, Mining, and Searching Networks

Anthony BonatoRyerson University

Master’s SeminarNovember 2012

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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

• directed graphs (digraphs)

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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 social scientists 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)

• 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(eg friendship)

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

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Lady Gaga is the centre of Twitterverse

Dalai Lama

Lady Gaga

Anderson Cooper

Queen Rania of Jordan

Arnold Schwarzenegger

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6 degrees of separation

• Stanley Milgram: famous chain letter experiment in 1967

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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

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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, 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

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6 Dimensions of Separation

OSN Dimension

YouTube 6Twitter 4Flickr 4

Cyworld 7

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

• 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)|

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Applications of Cops and Robbers

• moving target search– 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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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.

C

C

C

R

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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

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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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

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AM8204 – Topics in Discrete Mathematics

• Winter 2012• 6 weeks each: complex networks, graph

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

me)

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Graphs at Ryerson (G@R)