Modelling, Mining, and Searching Networks
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Transcript of Modelling, Mining, and Searching Networks
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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
nodes edges• 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 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 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
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R
Cops and Robbers
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C
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Cops and Robbers
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C
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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
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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)