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Complex networks and their models

Anthony BonatoRyerson University

Graduate SeminarOctober 2011

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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)) = (V,E) consists of a nonempty set of vertices or nodes V, and a set of edges E, which is a symmetric binary relation on V (usually irreflexive)

nodes edges• 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 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 inPPI networks are vital for normalcellular 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 Twitter

• Java et al. (2009)– 6 degrees of

separation in Twitter

• other researchers found similar results in Facebook, LinkedIn, …

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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, 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

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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

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Applications of Cops and Robbers

• moving target search– missile-defense– gaming

• counter-terrorism– intercepting messages or agents

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The robber fights back! (Haidar’s thesis)

• 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

– 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)

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• preprints, reprints, contact:Google: “Anthony Bonato”

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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

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Graphs at Ryerson (G@R)