Master’s Seminar
November 2012
Modelling, Mining, and
Searching Networks
Anthony Bonato
Ryerson University
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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:
 deg(x)  2 | E(G) |
xV(G)
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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
Nuit
Blanche
City of
Toronto
Four
Seasons
Hotel
Frommer’s
Greenland
Tourism
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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
b
Ni,n  i n, some b  2
(Broder et al, 01)
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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 nodes
11000 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
Arnold
Schwarzenegger
Queen Rania
of Jordan
Anderson
Cooper
Lady Gaga
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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
Twitter
Flickr
Cyworld
6
4
4
7
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Cops and Robbers
C
C
R
C
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Cops and Robbers
C
C
R
C
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Cops and Robbers
C
R
C
C
cop number c(G) ≤ 3
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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
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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 variant
The robber fights back!
• robber can attack neighbouring cop
C
C
R
C
• 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.
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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
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Good guys vs bad guys games in graphs
bad
good
slow
slow
medium
fast
helicopter
eternal
security
traps, tandem-win
medium
robot vacuum
Cops and Robbers
edge searching
fast
cleaning
distance k Cops
and Robbers
Cops and Robbers The Angel
on disjoint edge
and Devil
sets
seepage
Helicopter Cops
and Robbers,
Marshals, The
Angel and Devil,
Firefighter
helicopter
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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 postdocs
• 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)
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