CMU SCS
Finding patterns in large, real
networks
Christos Faloutsos
CMU
CMU SCS
Thanks to
• Deepayan Chakrabarti (CMU/Yahoo)
• Michalis Faloutsos (UCR)
• Spiros Papadimitriou (CMU/IBM)
• George Siganos (UCR)
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Introduction
Internet Map
[lumeta.com]
Food Web
[Martinez ’91]
Protein Interactions
[genomebiology.com]
Graphs are everywhere!
Friendship Network
[Moody ’01]
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Outline
•
Laws
–
–
–
•
•
static
time-evolution laws
why so many power laws?
tools/results
future steps
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Motivating questions
• Q1: What do real graphs look like?
• Q2: How to generate realistic graphs?
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Virus propagation
• Q3: How do viruses/rumors propagate?
• Q4: Who is the best person/computer to
immunize against a virus?
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Graph clustering & mining
• Q5: which edges/nodes are ‘abnormal’?
• Q6: split a graph in k ‘natural’ communities
- but how to determine k?
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Outline
•
Laws
–
–
–
•
•
static
time-evolution laws
why so many power laws?
tools/results
future steps
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Topology
Q1: How does the Internet look like? Any
rules?
(Looks random – right?)
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Laws – degree distributions
• Q: avg degree is ~2 - what is the most
probable degree?
count
??
2
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degree
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Laws – degree distributions
• Q: avg degree is ~2 - what is the most
probable degree?
count
??
2
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count
degree
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degree
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CMU SCS
I.Power-law: outdegree O
Frequency
Exponent = slope
O = -2.15
-2.15
Nov’97
Outdegree
The plot is linear in log-log scale [FFF’99]
freq = degree (-2.15)
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II.Power-law: rank R
att.com
log(degree)
April’98
ibm.com
-0.82
R
log(rank)
Rank: nodes in decreasing degree order
• The plot is a line in log-log scale
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III.Power-law: eigen E
Eigenvalue
Exponent = slope
E = -0.48
Dec’98
Rank of decreasing eigenvalue
• Eigenvalues in decreasing order (first 20)
• [Mihail+, 02]: R = 2 * E
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IV. The Node Neighborhood
• N(h) = # of pairs of nodes within h hops
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IV. The Node Neighborhood
• Q: average degree = 3 - how many
neighbors should I expect within 1,2,… h
hops?
• Potential answer:
1 hop -> 3 neighbors
2 hops -> 3 * 3
…
h hops -> 3h
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IV. The Node Neighborhood
• Q: average degree = 3 - how many
neighbors should I expect within 1,2,… h
hops?
• Potential answer:
1 hop -> 3 neighbors
2 hops -> 3 * 3
WE HAVE DUPLICATES!
…
h hops -> 3h
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IV. The Node Neighborhood
• Q: average degree = 3 - how many
neighbors should I expect within 1,2,… h
hops?
• Potential answer:
1 hop -> 3 neighbors
2 hops -> 3 * 3
‘avg’ degree: meaningless!
…
h hops -> 3h
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IV. Power-law: hopplot H
# of Pairs
H = 4.86
Hops
Dec 98
H = 2.83
# of Pairs
Hops
Router level ’95
Pairs of nodes as a function of hops N(h)= hH
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Observation
• Q: Intuition behind ‘hop exponent’?
• A: ‘intrinsic=fractal dimensionality’ of the
network
...
N(h) ~ h1
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N(h) ~ h2
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Hop plots
• More on fractal/intrinsic dimensionalities:
very soon
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Any other ‘laws’?
Yes!
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Any other ‘laws’?
Yes!
• Small diameter
– six degrees of separation / ‘Kevin Bacon’
– small worlds [Watts and Strogatz]
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Any other ‘laws’?
• Bow-tie, for the web [Kumar+ ‘99]
• IN, SCC, OUT, ‘tendrils’
• disconnected components
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Any other ‘laws’?
• power-laws in communities (bi-partite cores)
[Kumar+, ‘99]
Log(count)
n:1
n:3
n:2
Log(m)
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2:3 core
(m:n core)
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More Power laws
• Also hold for other web graphs [Barabasi+,
‘99], [Kumar+, ‘99]
• citation graphs (see later)
• and many more
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Outline
•
Laws
–
–
–
•
•
static
time-evolution laws
why so many power laws?
tools/results
future steps
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How do graphs evolve?
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Time Evolution: rank R
Rank exponent
-0.5
0
200
400
600
800
-0.6
-0.7
Domain
level
-0.8
-0.9
-1
Instances in
time: Nov'97
- now ‘97
#days
since Nov.
The rank exponent has not changed! [Siganos+, ‘03]
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How do graphs evolve?
• degree-exponent seems constant - anything
else?
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Evolution of diameter?
• Prior analysis, on power-law-like graphs,
hints that
diameter ~ O(log(N)) or
diameter ~ O( log(log(N)))
• i.e.., slowly increasing with network size
• Q: What is happening, in reality?
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Evolution of diameter?
• Prior analysis, on power-law-like graphs,
hints that
diameter ~ O(log(N)) or
diameter ~ O( log(log(N)))
• i.e.., slowly increasing with network size
• Q: What is happening, in reality?
• A: It shrinks(!!), towards a constant value
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Shrinking diameter
[Leskovec+05a]
• Citations among physics
papers
• 11yrs; @ 2003:
– 29,555 papers
– 352,807 citations
• For each month M, create a
graph of all citations up to
month M
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Shrinking diameter
• Authors &
publications
• 1992
– 318 nodes
– 272 edges
• 2002
– 60,000 nodes
• 20,000 authors
• 38,000 papers
– 133,000 edges
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Shrinking diameter
• Patents & citations
• 1975
– 334,000 nodes
– 676,000 edges
• 1999
– 2.9 million nodes
– 16.5 million edges
• Each year is a
datapoint
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Shrinking diameter
• Autonomous
systems
• 1997
– 3,000 nodes
– 10,000 edges
• 2000
– 6,000 nodes
– 26,000 edges
• One graph per day
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Temporal evolution of graphs
• N(t) nodes; E(t) edges at time t
• suppose that
N(t+1) = 2 * N(t)
• Q: what is your guess for
E(t+1) =? 2 * E(t)
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Temporal evolution of graphs
• N(t) nodes; E(t) edges at time t
• suppose that
N(t+1) = 2 * N(t)
• Q: what is your guess for
E(t+1) =? 2 * E(t)
• A: over-doubled!
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Temporal evolution of graphs
• A: over-doubled - but obeying:
E(t) ~ N(t)a
for all t
where 1<a<2
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Temporal evolution of graphs
• A: over-doubled - but obeying:
E(t) ~ N(t)a
for all t
• Identically:
log(E(t)) / log(N(t)) = constant
for all t
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Densification Power Law
ArXiv: Physics papers
and their citations
E(t)
1.69
N(t)
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Densification Power Law
ArXiv: Physics papers
and their citations
E(t)
1
1.69
‘tree’
N(t)
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Densification Power Law
ArXiv: Physics papers
and their citations
‘clique’
E(t)
2
1.69
N(t)
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Densification Power Law
U.S. Patents, citing each
other
E(t)
1.66
N(t)
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Densification Power Law
Autonomous Systems
E(t)
1.18
N(t)
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Densification Power Law
ArXiv: authors & papers
E(t)
1.15
N(t)
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Summary of ‘laws’
•
•
•
•
•
Power laws for degree distributions
…………... for eigenvalues, bi-partite cores
Small & shrinking diameter (‘6 degrees’)
‘Bow-tie’ for web; ‘jelly-fish’ for internet
``Densification Power Law’’, over time
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Outline
•
Laws
–
–
–
static
time-evolution laws
why so many power laws?
•
•
•
•
•
other power laws
fractals & self-similarity
case study: Kronecker graphs
tools/results
future steps
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Power laws
• Many more power laws, in diverse settings:
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A famous power law: Zipf’s law
log(freq)
“a”
• Bible - rank vs
frequency (log-log)
“the”
log(rank)
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Power laws, cont’ed
• length of file transfers [Bestavros+]
• web hit counts [Huberman]
• magnitude of earthquakes (GuttenbergRichter law)
• sizes of lakes/islands (Korcak’s law)
• Income distribution (Pareto’s law)
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The Peer-to-Peer Topology
[Jovanovic+]
• Frequency versus degree
• Number of adjacent peers follows a power-law
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Click-stream data
Web Site Traffic
log(count)
u-id’s
url’s
Zipf
‘yahoo’
log(freq)
log(count)
‘super-surfer’
log(freq)
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Lotka’s law
(Lotka’s law of publication count); and
citation counts: (citeseer.nj.nec.com 6/2001)
log(count)
J. Ullman
log(#citations)
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Swedish sex-web
Nodes: people (Females; Males)
Links: sexual relationships
Albert Laszlo Barabasi
http://www.nd.edu/~networks/
Publication%20Categories/
04%20Talks/2005-norway3hours.ppt
4781 Swedes; 18-74;
59% response rate.
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Liljeros et al. Nature 2001
CMU SCS
Swedish sex-web
Nodes: people (Females; Males)
Links: sexual relationships
Albert Laszlo Barabasi
http://www.nd.edu/~networks/
Publication%20Categories/
04%20Talks/2005-norway3hours.ppt
4781 Swedes; 18-74;
59% response rate.
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Liljeros et al. Nature 2001
CMU SCS
Power laws
• Q: Why so many power laws?
• A1: self-similarity
• A2: ‘rich-get-richer’
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Digression: intro to fractals
• Fractals: sets of points that are self similar
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A famous fractal
e.g., Sierpinski triangle:
...
zero area;
infinite length!
dimensionality = ??
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A famous fractal
e.g., Sierpinski triangle:
...
zero area;
infinite length!
dimensionality = log(3)/log(2) = 1.58
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A famous fractal
equivalent graph:
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Intrinsic (‘fractal’) dimension
How to estimate it?
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Intrinsic (‘fractal’) dimension
• Q: fractal dimension
of a line?
• A: nn ( <= r ) ~ r^1
(‘power law’: y=x^a)
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• Q: fd of a plane?
• A: nn ( <= r ) ~ r^2
fd== slope of (log(nn) vs
log(r) )
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Sierpinsky triangle
== ‘correlation integral’
log(#pairs
within <=r )
= CDF of pairwise distances
1.58
log( r )
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Sierpinsky triangle
== ‘correlation integral’
log(#pairs
within <=r )
= CDF of pairwise distances
1.58
log( r )
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Line
== ‘correlation integral’
log(#pairs
within <=r )
= CDF of pairwise distances
1.58
...
1
log( r )
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2-d (Plane)
== ‘correlation integral’
log(#pairs
within <=r )
= CDF of pairwise distances
2
1.58
log( r )
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Recall: Hop Plot
• Internet routers: how many neighbors
within h hops? (= correlation integral!)
log(#pairs)
Reachability function:
number of neighbors
within r hops, vs r (loglog).
2.8
log(hops)
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Mbone routers, 1995
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Fractals and power laws
They are related concepts:
• fractals <=>
• self-similarity <=>
• scale-free <=>
• power laws ( y= xa )
– F = C r(-2)
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Fractals and power laws
• Power laws and fractals are closely related
• And fractals appear in MANY cases
–
–
–
–
–
–
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coast-lines: 1.1-1.5
brain-surface: 2.6
rain-patches: 1.3
tree-bark: ~2.1
stock prices / random walks: 1.5
... [see Mandelbrot; or Schroeder]
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Outline
•
Laws
–
–
–
static
time-evolution laws
why so many power laws?
•
•
•
•
•
other power laws
fractals & self-similarity
case study: Kronecker graph generator
tools/results
future steps
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Self-similarity at work
• Q2: How to generate realistic graphs?
• (Q2’) and how to grow it over time?
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Wish list for a generator:
• Power-law-tail in- and out-degrees
• Power-law-tail scree plots
• shrinking/constant diameter
• Densification Power Law
• communities-within-communities
Q: how to achieve all of them?
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Wish list for a generator:
• Power-law-tail in- and out-degrees
• Power-law-tail scree plots
• shrinking/constant diameter
• Densification Power Law
• communities-within-communities
Q: how to achieve all of them?
A: Kronecker matrix product [Leskovec+05b]
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Kronecker product
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Kronecker product
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Kronecker product
N
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N*N
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N**4
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Properties of Kronecker graphs:
•
•
•
•
•
Power-law-tail in- and out-degrees
Power-law-tail scree plots
constant diameter
perfect Densification Power Law
communities-within-communities
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Properties of Kronecker graphs:
• Power-law-tail in- and out-degrees
• Power-law-tail scree plots
• constant diameter
• perfect Densification Power Law
• communities-within-communities
and we can prove all of the above
(first and only generator that does that)
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Kronecker - ArXiv
real
(det.
Kronecker)
(stochastic)
Kronecker
Degree Scree Diameter D.P.L.
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Kronecker - patents
Degree
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Scree
Diameter
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D.P.L.
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Kronecker - A.S.
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Outline
•
Laws
–
–
–
•
static
time-evolution laws
why so many power laws?
tools/results
–
–
–
•
virus propagation
connection subgraphs
cross-associations
future steps
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Virus propagation
• Q3: How do viruses/rumors propagate?
• Q4: Who is the best person/computer to
immunize against a virus?
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The model: SIS
• ‘Flu’ like: Susceptible-Infected-Susceptible
• Virus ‘strength’ s= b/d
Healthy
Prob. d
N2
Prob. b
N1
N
Infected
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Epidemic threshold t
of a graph: the value of t, such that
if strength s = b / d < t
an epidemic can not happen
Thus,
• given a graph
• compute its epidemic threshold
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Epidemic threshold t
What should t depend on?
• avg. degree? and/or highest degree?
• and/or variance of degree?
• and/or third moment of degree?
• and/or diameter?
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Epidemic threshold
• [Theorem] We have no epidemic, if
β/δ <τ = 1/ λ1,A
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Epidemic threshold
• [Theorem] We have no epidemic, if
epidemic threshold
recovery prob.
β/δ <τ = 1/ λ1,A
attack prob.
largest eigenvalue
of adj. matrix A
Proof: [Wang+03]
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Experiments (Oregon)
Number of Infected Nodes
500
Oregon
β = 0.001
b/d > τ
(above threshold)
400
300
200
b/d = τ
(at the threshold)
100
0
0
250
500
750
Time
δ:
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0.05
0.06
1000
b/d < τ
(below threshold)
0.07
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Our result:
• Holds for any graph
• includes older results as special cases
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Virus propagation
• Q3: How do viruses/rumors propagate?
• Q4: Who is the best person/computer to
immunize against a virus?
• A4: the one that causes the highest decrease
in l1
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Outline
•
Laws
–
–
–
•
static
time-evolution laws
why so many power laws?
tools/results
–
–
–
•
virus propagation
connection subgraphs
cross-associations
future steps
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(A-A) Kidman-Diaz
• What are the best paths between ‘Kidman’
and ‘Diaz’?
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(A-A) Kidman-Diaz
• What are the best paths between ‘Kidman’
and ‘Diaz’? [KDD’04, w/
McCurley+Tomkins]
Strong, direct link
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Outline
•
Laws
–
–
–
•
static
time-evolution laws
why so many power laws?
tools/results
–
–
–
•
virus propagation
connection subgraphs
cross-associations
future steps
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Graph clustering & mining
• Q5: which edges/nodes are ‘abnormal’?
• Q6: split a graph in k ‘natural’ communities
- but how to determine k?
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Graph partitioning
• Documents x terms
• Customers x products
• Users x web-sites
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Graph partitioning
• Documents x terms
• Customers x products
• Users x web-sites
• Q: HOW MANY
PIECES?
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Graph partitioning
• Documents x terms
• Customers x products
• Users x web-sites
• Q: HOW MANY
PIECES?
• A: MDL/ compression
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Cross-associations
1x2
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2x2
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Cross-associations
2x3
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3x4
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Cross-associations
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Cross-associations
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Graph clustering & mining
• Q5: which edges/nodes are ‘abnormal’?
• Q6: split a graph in k ‘natural’ communities
- but how to determine k?
• A6: choose the k that leads to best overall
compression (= MDL = Minimum
Description Language)
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Cross-associations
missing edge
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outlier edge
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Outline
•
Laws
–
–
–
•
•
•
static
time-evolution laws
why so many power laws?
tools/results
other related projects
future steps
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chlorine concentrations
Motivation
Phase 1
: :
: :
Phase 2
Phase 3
sensors
: : near leak
: :
sensors
: : away : :
from leak
water distribution network
normal operation
May have hundreds of measurements, but
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it is unlikely they are completely unrelated!
CMU SCS
Motivation
chlorine concentrations
Phase 1
: :
: :
Phase 2
: :
: :
Phase 3
: :
: :
sensors
near leak
sensors
away
from leak
water distribution network
normal operation
major leak
May have hundreds of measurements, but
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it is unlikely they are completely unrelated!
CMU SCS
Motivation
chlorine concentrations
Phase 1
: :
Phase 1
: :
: :
k=1
: :
: :
: :
actual measurements
(n streams)
k hidden variable(s)
We would like to discover a few “hidden
(latent)
variables” that
summarize the key trends
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chlorine concentrations
Phase 1
: :
Phase 2
: :
Motivation
Phase 1
Phase 2
: :
k=2
: :
: :
: :
actual measurements
(n streams)
k hidden variable(s)
We would like to discover a few “hidden
(latent)
variables” that
summarize the key trends
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chlorine concentrations
Phase 1
: :
: :
Phase 2
: :
: :
Motivation
Phase 3
Phase 2
Phase 3
: :
: :
actual measurements
(n streams)
Phase 1
k=1
k hidden variable(s)
We would like to discover a few “hidden
(latent)
variables” that
summarize the key trends
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Stream mining
• Solution: SPIRIT [VLDB’05]
– incremental, on-line PCA
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Outline
•
•
•
•
Laws
tools/results
other related projects
future steps
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Future steps
•
•
•
•
connection sub-graphs on multi-graphs [Tong]
traffic matrix, evolving over time [Sun+]
influence propagation on blogs [Leskovec+]
automatic web site classification [Leskovec]
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Conclusions
• Power laws & self similarity
– excellent tools, for real datasets,
– where Gaussian, uniform etc assumptions fail
• Additional tools
– Kronecker, for graph generators
– eigenvalues for virus propagation
– MDL/compression for graph partitioning
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References
• [Aiello+, '00] William Aiello, Fan R. K. Chung, Linyuan
Lu: A random graph model for massive graphs. STOC
2000: 171-180
• [Albert+] Reka Albert, Hawoong Jeong, and Albert-Laszlo
Barabasi: Diameter of the World Wide Web, Nature 401 130131 (1999)
• [Barabasi, '03] Albert-Laszlo Barabasi Linked: How
Everything Is Connected to Everything Else and What It
Means (Plume, 2003)
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References, cont’d
• [Barabasi+, '99] Albert-Laszlo Barabasi and Reka Albert.
Emergence of scaling in random networks. Science,
286:509--512, 1999
• [Broder+, '00] Andrei Broder, Ravi Kumar, Farzin Maghoul,
Prabhakar Raghavan, Sridhar Rajagopalan, Raymie Stata,
Andrew Tomkins, and Janet Wiener. Graph structure in the
web, WWW, 2000
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References, cont’d
• [Chakrabarti+, ‘04] RMAT: A recursive graph generator, D.
Chakrabarti, Y. Zhan, C. Faloutsos, SIAM-DM 2004
• [Dill+, '01] Stephen Dill, Ravi Kumar, Kevin S. McCurley,
Sridhar Rajagopalan, D. Sivakumar, Andrew Tomkins: Selfsimilarity in the Web. VLDB 2001: 69-78
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References, cont’d
• [Fabrikant+, '02] A. Fabrikant, E. Koutsoupias, and C.H.
Papadimitriou. Heuristically Optimized Trade-offs: A New
Paradigm for Power Laws in the Internet. ICALP, Malaga,
Spain, July 2002
• [FFF, 99] M. Faloutsos, P. Faloutsos, and C. Faloutsos, "On
power-law relationships of the Internet topology," in
SIGCOMM, 1999.
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References, cont’d
• [Leskovec+05a] Jure Leskovec, Jon Kleinberg and Christos
Faloutsos Graphs over Time: Densification Laws,
Shrinking Diameters and Possible Explanations KDD
2005, Chicago, IL. (Best research paper award)
• [Leskovec+05b] Jure Leskovec, Deepayan Chakrabarti, Jon
Kleinberg and Christos Faloutsos, Realistic, Mathematically
Tractable Graph Generation and Evolution, Using
Kronecker Multiplication, ECML/PKDD 2005, Porto,
Portugal.
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References, cont’d
• [Jovanovic+, '01] M. Jovanovic, F.S. Annexstein, and K.A.
Berman. Modeling Peer-to-Peer Network Topologies
through "Small-World" Models and Power Laws. In
TELFOR, Belgrade, Yugoslavia, November, 2001
• [Kumar+ '99] Ravi Kumar, Prabhakar Raghavan, Sridhar
Rajagopalan, Andrew Tomkins: Extracting Large-Scale
Knowledge Bases from the Web. VLDB 1999: 639-650
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References, cont’d
• [Leland+, '94] W. E. Leland, M.S. Taqqu, W. Willinger,
D.V. Wilson, On the Self-Similar Nature of Ethernet
Traffic, IEEE Transactions on Networking, 2, 1, pp 1-15,
Feb. 1994.
• [Mihail+, '02] Milena Mihail, Christos H. Papadimitriou:
On the Eigenvalue Power Law. RANDOM 2002: 254-262
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References, cont’d
• [Milgram '67] Stanley Milgram: The Small World Problem,
Psychology Today 1(1), 60-67 (1967)
• [Montgomery+, ‘01] Alan L. Montgomery, Christos
Faloutsos: Identifying Web Browsing Trends and Patterns.
IEEE Computer 34(7): 94-95 (2001)
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References, cont’d
• [Palmer+, ‘01] Chris Palmer, Georgos Siganos, Michalis
Faloutsos, Christos Faloutsos and Phil Gibbons The
connectivity and fault-tolerance of the Internet topology
(NRDM 2001), Santa Barbara, CA, May 25, 2001
• [Pennock+, '02] David M. Pennock, Gary William Flake,
Steve Lawrence, Eric J. Glover, C. Lee Giles: Winners don't
take all: Characterizing the competition for links on the web
Proc. Natl. Acad. Sci. USA 99(8): 5207-5211 (2002)
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References, cont’d
• [Schroeder, ‘91] Manfred Schroeder Fractals, Chaos,
Power Laws: Minutes from an Infinite Paradise W H
Freeman & Co., 1991 (excellent book on fractals)
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References, cont’d
• [Siganos+, '03] G. Siganos, M. Faloutsos, P. Faloutsos, C.
Faloutsos Power-Laws and the AS-level Internet Topology,
Transactions on Networking, August 2003.
• [Wang+03] Yang Wang, Deepayan Chakrabarti, Chenxi
Wang and Christos Faloutsos: Epidemic Spreading in
Real Networks: an Eigenvalue Viewpoint, SRDS 2003,
Florence, Italy.
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Reference
• [Watts+ Strogatz, '98] D. J. Watts and S. H. Strogatz
Collective dynamics of 'small-world' networks, Nature,
393:440-442 (1998)
• [Watts, '03] Duncan J. Watts Six Degrees: The Science of a
Connected Age W.W. Norton & Company; (February 2003)
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Thank you!
www.cs.cmu.edu/~christos
www.db.cs.cmu.edu
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