mining billion-node graphs: patterns and toolschristos/talks/11-facebook/foils/...cmu scs c....
TRANSCRIPT
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CMU SCS
Mining Billion-node Graphs: Patterns and Tools
Christos Faloutsos CMU
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CMU SCS
Thank you! • Rong Yan
• Junfeng Pan
• Adam Ward
• Jordan MacDonald
FB'11 C. Faloutsos (CMU) 2
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CMU SCS
C. Faloutsos (CMU) 3
Our goal:
Open source system for mining huge graphs:
PEGASUS project (PEta GrAph mining System)
• www.cs.cmu.edu/~pegasus • code and papers
FB'11
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CMU SCS
C. Faloutsos (CMU) 4
Outline
• Introduction – Motivation • Problem#1: Patterns in graphs • Problem#2: Tools • Problem#3: Scalability • Conclusions
FB'11
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CMU SCS
C. Faloutsos (CMU) 5
Graphs - why should we care?
Internet Map [lumeta.com]
Food Web [Martinez ’91]
Friendship Network [Moody ’01]
FB'11
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CMU SCS
C. Faloutsos (CMU) 6
Graphs - why should we care? • IR: bi-partite graphs (doc-terms)
• web: hyper-text graph
• ... and more:
D1
DN
T1
TM
... ...
FB'11
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CMU SCS
C. Faloutsos (CMU) 7
Graphs - why should we care? • ‘viral’ marketing • web-log (‘blog’) news propagation • computer network security: email/IP traffic
and anomaly detection • ....
FB'11
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CMU SCS
C. Faloutsos (CMU) 8
Outline
• Introduction – Motivation • Problem#1: Patterns in graphs
– Static graphs – Weighted graphs – Time evolving graphs
• Problem#2: Tools • Problem#3: Scalability • Conclusions
FB'11
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CMU SCS
C. Faloutsos (CMU) 9
Problem #1 - network and graph mining
• What does the Internet look like? • What does FaceBook look like?
• What is ‘normal’/‘abnormal’? • which patterns/laws hold?
FB'11
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CMU SCS
C. Faloutsos (CMU) 10
Problem #1 - network and graph mining
• What does the Internet look like? • What does FaceBook look like?
• What is ‘normal’/‘abnormal’? • which patterns/laws hold?
– To spot anomalies (rarities), we have to discover patterns
FB'11
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CMU SCS
C. Faloutsos (CMU) 11
Problem #1 - network and graph mining
• What does the Internet look like? • What does FaceBook look like?
• What is ‘normal’/‘abnormal’? • which patterns/laws hold?
– To spot anomalies (rarities), we have to discover patterns
– Large datasets reveal patterns/anomalies that may be invisible otherwise…
FB'11
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CMU SCS
C. Faloutsos (CMU) 12
Graph mining • Are real graphs random?
FB'11
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CMU SCS
C. Faloutsos (CMU) 13
Laws and patterns • Are real graphs random? • A: NO!!
– Diameter – in- and out- degree distributions – other (surprising) patterns
• So, let’s look at the data
FB'11
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CMU SCS
C. Faloutsos (CMU) 14
Solution# S.1 • Power law in the degree distribution
[SIGCOMM99]
log(rank)
log(degree)
internet domains
att.com
ibm.com
FB'11
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CMU SCS
C. Faloutsos (CMU) 15
Solution# S.1 • Power law in the degree distribution
[SIGCOMM99]
log(rank)
log(degree)
-0.82
internet domains
att.com
ibm.com
FB'11
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CMU SCS
C. Faloutsos (CMU) 16
Solution# S.2: Eigen Exponent E
• A2: power law in the eigenvalues of the adjacency matrix
E = -0.48
Exponent = slope
Eigenvalue
Rank of decreasing eigenvalue
May 2001
FB'11
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CMU SCS
C. Faloutsos (CMU) 17
Solution# S.2: Eigen Exponent E
• [Mihail, Papadimitriou ’02]: slope is ½ of rank exponent
E = -0.48
Exponent = slope
Eigenvalue
Rank of decreasing eigenvalue
May 2001
FB'11
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CMU SCS
C. Faloutsos (CMU) 18
But: How about graphs from other domains?
FB'11
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CMU SCS
C. Faloutsos (CMU) 19
More power laws: • web hit counts [w/ A. Montgomery]
Web Site Traffic
in-degree (log scale)
Count (log scale)
Zipf
users sites
``ebay’’
FB'11
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CMU SCS
C. Faloutsos (CMU) 20
epinions.com • who-trusts-whom
[Richardson + Domingos, KDD 2001]
(out) degree
count
trusts-2000-people user
FB'11
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CMU SCS
And numerous more • # of sexual contacts • Income [Pareto] –’80-20 distribution’ • Duration of downloads [Bestavros+] • Duration of UNIX jobs (‘mice and
elephants’) • Size of files of a user • … • ‘Black swans’ FB'11 C. Faloutsos (CMU) 21
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CMU SCS
C. Faloutsos (CMU) 22
Outline
• Introduction – Motivation • Problem#1: Patterns in graphs
– Static graphs • degree, diameter, eigen, • triangles • cliques
– Weighted graphs – Time evolving graphs
• Problem#2: Tools FB'11
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CMU SCS
C. Faloutsos (CMU) 23
Solution# S.3: Triangle ‘Laws’
• Real social networks have a lot of triangles
FB'11
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CMU SCS
C. Faloutsos (CMU) 24
Solution# S.3: Triangle ‘Laws’
• Real social networks have a lot of triangles – Friends of friends are friends
• Any patterns?
FB'11
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CMU SCS
C. Faloutsos (CMU) 25
Triangle Law: #S.3 [Tsourakakis ICDM 2008]
ASN HEP-TH
Epinions X-axis: # of participating triangles Y: count (~ pdf)
FB'11
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CMU SCS
C. Faloutsos (CMU) 26
Triangle Law: #S.3 [Tsourakakis ICDM 2008]
ASN HEP-TH
Epinions
FB'11
X-axis: # of participating triangles Y: count (~ pdf)
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CMU SCS
C. Faloutsos (CMU) 27
Triangle Law: #S.4 [Tsourakakis ICDM 2008]
SN Reuters
Epinions X-axis: degree Y-axis: mean # triangles n friends -> ~n1.6 triangles
FB'11
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CMU SCS
C. Faloutsos (CMU) 28
Triangle Law: Computations [Tsourakakis ICDM 2008]
But: triangles are expensive to compute (3-way join; several approx. algos)
Q: Can we do that quickly?
details
FB'11
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CMU SCS
C. Faloutsos (CMU) 29
Triangle Law: Computations [Tsourakakis ICDM 2008]
But: triangles are expensive to compute (3-way join; several approx. algos)
Q: Can we do that quickly? A: Yes!
#triangles = 1/6 Sum ( λi3 )
(and, because of skewness (S2) , we only need the top few eigenvalues!
details
FB'11
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CMU SCS
C. Faloutsos (CMU) 30
Triangle Law: Computations [Tsourakakis ICDM 2008]
1000x+ speed-up, >90% accuracy
details
FB'11
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CMU SCS
Triangle counting for large graphs?
Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11]
31 FB'11 31 C. Faloutsos (CMU)
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CMU SCS
Triangle counting for large graphs?
Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11]
32 FB'11 32 C. Faloutsos (CMU)
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CMU SCS
EigenSpokes B. Aditya Prakash, Mukund Seshadri, Ashwin
Sridharan, Sridhar Machiraju and Christos Faloutsos: EigenSpokes: Surprising Patterns and Scalable Community Chipping in Large Graphs, PAKDD 2010, Hyderabad, India, 21-24 June 2010.
C. Faloutsos (CMU) 33 FB'11
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CMU SCS
EigenSpokes • Eigenvectors of adjacency matrix
equivalent to singular vectors (symmetric, undirected graph)
34 C. Faloutsos (CMU) FB'11
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CMU SCS
EigenSpokes • Eigenvectors of adjacency matrix
equivalent to singular vectors (symmetric, undirected graph)
35 C. Faloutsos (CMU) FB'11
N
N
details
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CMU SCS
EigenSpokes • Eigenvectors of adjacency matrix
equivalent to singular vectors (symmetric, undirected graph)
36 C. Faloutsos (CMU) FB'11
N
N
details
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CMU SCS
EigenSpokes • Eigenvectors of adjacency matrix
equivalent to singular vectors (symmetric, undirected graph)
37 C. Faloutsos (CMU) FB'11
N
N
details
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CMU SCS
EigenSpokes • Eigenvectors of adjacency matrix
equivalent to singular vectors (symmetric, undirected graph)
38 C. Faloutsos (CMU) FB'11
N
N
details
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CMU SCS
EigenSpokes • EE plot: • Scatter plot of
scores of u1 vs u2 • One would expect
– Many points @ origin
– A few scattered ~randomly
C. Faloutsos (CMU) 39
u1
u2
FB'11
1st Principal component
2nd Principal component
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CMU SCS
EigenSpokes • EE plot: • Scatter plot of
scores of u1 vs u2 • One would expect
– Many points @ origin
– A few scattered ~randomly
C. Faloutsos (CMU) 40
u1
u2 90o
FB'11
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CMU SCS
EigenSpokes - pervasiveness • Present in mobile social graph
across time and space
• Patent citation graph
41 C. Faloutsos (CMU) FB'11
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CMU SCS
EigenSpokes - explanation
Near-cliques, or near-bipartite-cores, loosely connected
42 C. Faloutsos (CMU) FB'11
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CMU SCS
EigenSpokes - explanation
Near-cliques, or near-bipartite-cores, loosely connected
43 C. Faloutsos (CMU) FB'11
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CMU SCS
EigenSpokes - explanation
Near-cliques, or near-bipartite-cores, loosely connected
44 C. Faloutsos (CMU) FB'11
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CMU SCS
EigenSpokes - explanation
Near-cliques, or near-bipartite-cores, loosely connected
So what? Extract nodes with high
scores high connectivity Good “communities”
spy plot of top 20 nodes
45 C. Faloutsos (CMU) FB'11
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CMU SCS
Bipartite Communities!
magnified bipartite community
patents from same inventor(s)
`cut-and-paste’ bibliography!
46 C. Faloutsos (CMU) FB'11
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CMU SCS
C. Faloutsos (CMU) 47
Outline
• Introduction – Motivation • Problem#1: Patterns in graphs
– Static graphs • degree, diameter, eigen, • triangles • cliques
– Weighted graphs – Time evolving graphs
• Problem#2: Tools FB'11
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CMU SCS
C. Faloutsos (CMU) 48
Observations on weighted graphs?
• A: yes - even more ‘laws’!
M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008
FB'11
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CMU SCS
C. Faloutsos (CMU) 49
Observation W.1: Fortification Q: How do the weights of nodes relate to degree?
FB'11
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CMU SCS
C. Faloutsos (CMU) 50
Observation W.1: Fortification
More donors, more $ ?
$10
$5
FB'11
‘Reagan’
‘Clinton’ $7
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CMU SCS
Edges (# donors)
In-weights ($)
C. Faloutsos (CMU) 51
Observation W.1: fortification: Snapshot Power Law
• Weight: super-linear on in-degree • exponent ‘iw’: 1.01 < iw < 1.26
Orgs-Candidates
e.g. John Kerry, $10M received, from 1K donors
More donors, even more $
$10
$5
FB'11
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CMU SCS
C. Faloutsos (CMU) 52
Outline
• Introduction – Motivation • Problem#1: Patterns in graphs
– Static graphs – Weighted graphs – Time evolving graphs
• Problem#2: Tools • …
FB'11
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CMU SCS
C. Faloutsos (CMU) 53
Problem: Time evolution • with Jure Leskovec (CMU ->
Stanford)
• and Jon Kleinberg (Cornell – sabb. @ CMU)
FB'11
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CMU SCS
C. Faloutsos (CMU) 54
T.1 Evolution of the Diameter • Prior work on Power Law graphs hints
at slowly growing diameter: – diameter ~ O(log N) – diameter ~ O(log log N)
• What is happening in real data?
FB'11
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CMU SCS
C. Faloutsos (CMU) 55
T.1 Evolution of the Diameter • Prior work on Power Law graphs hints
at slowly growing diameter: – diameter ~ O(log N) – diameter ~ O(log log N)
• What is happening in real data? • Diameter shrinks over time
FB'11
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CMU SCS
C. Faloutsos (CMU) 56
T.1 Diameter – “Patents”
• Patent citation network
• 25 years of data • @1999
– 2.9 M nodes – 16.5 M edges
time [years]
diameter
FB'11
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CMU SCS
C. Faloutsos (CMU) 57
T.2 Temporal Evolution of the Graphs
• N(t) … nodes at time t • 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)
FB'11
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CMU SCS
C. Faloutsos (CMU) 58
T.2 Temporal Evolution of the Graphs
• N(t) … nodes at time t • 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!
– But obeying the ``Densification Power Law’’ FB'11
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CMU SCS
C. Faloutsos (CMU) 59
T.2 Densification – Patent Citations
• Citations among patents granted
• @1999 – 2.9 M nodes – 16.5 M edges
• Each year is a datapoint
N(t)
E(t)
1.66
FB'11
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CMU SCS
C. Faloutsos (CMU) 60
Outline
• Introduction – Motivation • Problem#1: Patterns in graphs
– Static graphs – Weighted graphs – Time evolving graphs
• Problem#2: Tools • …
FB'11
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CMU SCS
C. Faloutsos (CMU) 61
More on Time-evolving graphs
M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008
FB'11
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CMU SCS
C. Faloutsos (CMU) 62
Observation T.3: NLCC behavior Q: How do NLCC’s emerge and join with
the GCC?
(``NLCC’’ = non-largest conn. components) – Do they continue to grow in size? – or do they shrink? – or stabilize?
FB'11
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CMU SCS
C. Faloutsos (CMU) 63
Observation T.3: NLCC behavior Q: How do NLCC’s emerge and join with
the GCC?
(``NLCC’’ = non-largest conn. components) – Do they continue to grow in size? – or do they shrink? – or stabilize?
FB'11
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CMU SCS
C. Faloutsos (CMU) 64
Observation T.3: NLCC behavior Q: How do NLCC’s emerge and join with
the GCC?
(``NLCC’’ = non-largest conn. components) – Do they continue to grow in size? – or do they shrink? – or stabilize?
FB'11
YES YES
YES
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CMU SCS
C. Faloutsos (CMU) 65
Observation T.3: NLCC behavior • After the gelling point, the GCC takes off, but
NLCC’s remain ~constant (actually, oscillate).
IMDB
CC size
Time-stamp FB'11
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CMU SCS
C. Faloutsos (CMU) 66
Timing for Blogs
• with Mary McGlohon (CMU->Google) • Jure Leskovec (CMU->Stanford) • Natalie Glance (now at Google) • Mat Hurst (now at MSR) [SDM’07]
FB'11
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CMU SCS
C. Faloutsos (CMU) 67
T.4 : popularity over time
Post popularity drops-off – exponentially?
lag: days after post
# in links
1 2 3
@t
@t + lag
FB'11
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CMU SCS
C. Faloutsos (CMU) 68
T.4 : popularity over time
Post popularity drops-off – exponentially? POWER LAW! Exponent?
# in links (log)
days after post (log)
FB'11
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CMU SCS
C. Faloutsos (CMU) 69
T.4 : popularity over time
Post popularity drops-off – exponentially? POWER LAW! Exponent? -1.6 • close to -1.5: Barabasi’s stack model • and like the zero-crossings of a random walk
# in links (log) -1.6
days after post (log)
FB'11
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CMU SCS
FB'11 C. Faloutsos (CMU) 70
-1.5 slope J. G. Oliveira & A.-L. Barabási Human Dynamics: The
Correspondence Patterns of Darwin and Einstein. Nature 437, 1251 (2005) . [PDF]
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CMU SCS
T.5: duration of phonecalls Surprising Patterns for the Call
Duration Distribution of Mobile Phone Users
Pedro O. S. Vaz de Melo, Leman Akoglu, Christos Faloutsos, Antonio A. F. Loureiro
PKDD 2010 FB'11 C. Faloutsos (CMU) 71
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CMU SCS
Probably, power law (?)
FB'11 C. Faloutsos (CMU) 72
??
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CMU SCS
No Power Law!
FB'11 C. Faloutsos (CMU) 73
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CMU SCS
‘TLaC: Lazy Contractor’ • The longer a task (phonecall) has taken, • The even longer it will take
FB'11 C. Faloutsos (CMU) 74
Odds ratio=
Casualties(<x): Survivors(>=x)
== power law
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CMU SCS
75
Data Description
Data from a private mobile operator of a large city 4 months of data 3.1 million users more than 1 billion phone records
Over 96% of ‘talkative’ users obeyed a TLAC distribution (‘talkative’: >30 calls)
FB'11 C. Faloutsos (CMU)
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CMU SCS
C. Faloutsos (CMU) 76
Outline
• Introduction – Motivation • Problem#1: Patterns in graphs • Problem#2: Tools
– OddBall (anomaly detection) – Belief Propagation – Immunization
• Problem#3: Scalability • Conclusions
FB'11
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CMU SCS
OddBall: Spotting Anomalies in Weighted Graphs
Leman Akoglu, Mary McGlohon, Christos Faloutsos
Carnegie Mellon University School of Computer Science
PAKDD 2010, Hyderabad, India
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CMU SCS
Main idea For each node, • extract ‘ego-net’ (=1-step-away neighbors) • Extract features (#edges, total weight, etc
etc) • Compare with the rest of the population
C. Faloutsos (CMU) 78 FB'11
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CMU SCS What is an egonet?
ego
79
egonet
C. Faloutsos (CMU) FB'11
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CMU SCS
Selected Features Ni: number of neighbors (degree) of ego i Ei: number of edges in egonet i Wi: total weight of egonet i λw,i: principal eigenvalue of the weighted
adjacency matrix of egonet I
80 C. Faloutsos (CMU) FB'11
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CMU SCS Near-Clique/Star
81 FB'11 C. Faloutsos (CMU)
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CMU SCS Near-Clique/Star
82 C. Faloutsos (CMU) FB'11
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CMU SCS Near-Clique/Star
83 C. Faloutsos (CMU) FB'11
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CMU SCS
Andrew Lewis (director)
Near-Clique/Star
84 C. Faloutsos (CMU) FB'11
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CMU SCS
C. Faloutsos (CMU) 85
Outline
• Introduction – Motivation • Problem#1: Patterns in graphs • Problem#2: Tools
– OddBall (anomaly detection) – Belief Propagation – Immunization
• Problem#3: Scalability • Conclusions
FB'11
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CMU SCS
FB'11 C. Faloutsos (CMU) 86
E-bay Fraud detection
w/ Polo Chau & Shashank Pandit, CMU [www’07]
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CMU SCS
FB'11 C. Faloutsos (CMU) 87
E-bay Fraud detection
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CMU SCS
FB'11 C. Faloutsos (CMU) 88
E-bay Fraud detection
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CMU SCS
FB'11 C. Faloutsos (CMU) 89
E-bay Fraud detection - NetProbe
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CMU SCS
Popular press
And less desirable attention: • E-mail from ‘Belgium police’ (‘copy of
your code?’) FB'11 C. Faloutsos (CMU) 90
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CMU SCS
C. Faloutsos (CMU) 91
Outline
• Introduction – Motivation • Problem#1: Patterns in graphs • Problem#2: Tools
– OddBall (anomaly detection) – Belief propagation – Immunization
• Problem#3: Scalability -PEGASUS • Conclusions
FB'11
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CMU SCS Immunization and epidemic
thresholds • Q1: which nodes to immunize? • Q2: will a virus vanish, or will it create an
epidemic?
FB'11 C. Faloutsos (CMU) 92
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CMU SCS
Q1: Immunization: • Given
• a network, • k vaccines, and • the virus details
• Which nodes to immunize?
FB'11 93 C. Faloutsos (CMU)
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CMU SCS
Q1: Immunization: • Given
• a network, • k vaccines, and • the virus details
• Which nodes to immunize?
FB'11 94 C. Faloutsos (CMU)
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CMU SCS
Q1: Immunization: • Given
• a network, • k vaccines, and • the virus details
• Which nodes to immunize?
FB'11 95 C. Faloutsos (CMU)
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CMU SCS
Q1: Immunization: • Given
• a network, • k vaccines, and • the virus details
• Which nodes to immunize?
A: immunize the ones that maximally raise the `epidemic threshold’ [Tong+, ICDM’10]
FB'11 96 C. Faloutsos (CMU)
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CMU SCS
Q2: will a virus take over? • Flu-like virus (no immunity, ‘SIS’) • Mumps (life-time immunity, ‘SIR’) • Pertussis (finite-length immunity, ‘SIRS’)
FB'11 C. Faloutsos (CMU) 97
β: attack prob δ: heal prob
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CMU SCS
Q2: will a virus take over? • Flu-like virus (no immunity, ‘SIS’) • Mumps (life-time immunity, ‘SIR’) • Pertussis (finite-length immunity, ‘SIRS’)
FB'11 C. Faloutsos (CMU) 98
β: attack prob δ: heal prob
Α: depends on connectivity (avg degree? Max degree? variance? Something else?
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CMU SCS
FB'11 C. Faloutsos (CMU) 99
Epidemic threshold τ
What should τ 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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CMU SCS
FB'11 C. Faloutsos (CMU) 100
Epidemic threshold
• [Theorem] We have no epidemic, if
β/δ <τ = 1/ λ1,A
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CMU SCS
FB'11 C. Faloutsos (CMU) 101
Epidemic threshold
• [Theorem] We have no epidemic, if
β/δ <τ = 1/ λ1,A
largest eigenvalue of adj. matrix A
attack prob.
recovery prob. epidemic threshold
Proof: [Wang+03] (for SIS=flu only)
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CMU SCS
A2: will a virus take over? • For all typical virus propagation models (flu,
mumps, pertussis, HIV, etc) • The only connectivity measure that matters, is
1/λ1 the first eigenvalue of the adj. matrix [Prakash+, ‘10, arxiv]
FB'11 C. Faloutsos (CMU) 102
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CMU SCS
A2: will a virus take over?
FB'11 C. Faloutsos (CMU) 103
Fraction of infected
Time ticks
Below: exp. extinction
Above: take-over
Graph: Portland, OR 31M links 1.5M nodes
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CMU SCS
C. Faloutsos (CMU) 104
Outline
• Introduction – Motivation • Problem#1: Patterns in graphs • Problem#2: Tools
– OddBall (anomaly detection) – Belief propagation – Immunization
• Problem#3: Scalability -PEGASUS • Conclusions
FB'11
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CMU SCS
FB'11 C. Faloutsos (CMU) 105
Scalability • Google: > 450,000 processors in clusters of ~2000
processors each [Barroso, Dean, Hölzle, “Web Search for a Planet: The Google Cluster Architecture” IEEE Micro 2003]
• Yahoo: 5Pb of data [Fayyad, KDD’07] • Problem: machine failures, on a daily basis • How to parallelize data mining tasks, then? • A: map/reduce – hadoop (open-source clone)
http://hadoop.apache.org/
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CMU SCS
C. Faloutsos (CMU) 106
Centralized Hadoop/PEGASUS
Degree Distr. old old
Pagerank old old
Diameter/ANF old HERE
Conn. Comp old HERE
Triangles done HERE
Visualization started
Outline – Algorithms & results
FB'11
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CMU SCS
HADI for diameter estimation • Radius Plots for Mining Tera-byte Scale
Graphs U Kang, Charalampos Tsourakakis, Ana Paula Appel, Christos Faloutsos, Jure Leskovec, SDM’10
• Naively: diameter needs O(N**2) space and up to O(N**3) time – prohibitive (N~1B)
• Our HADI: linear on E (~10B) – Near-linear scalability wrt # machines – Several optimizations -> 5x faster
C. Faloutsos (CMU) 107 FB'11
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CMU SCS
????
19+ [Barabasi+]
108 C. Faloutsos (CMU)
Radius
Count
FB'11
~1999, ~1M nodes
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CMU SCS
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • Largest publicly available graph ever studied.
????
19+ [Barabasi+]
109 C. Faloutsos (CMU)
Radius
Count
FB'11
??
~1999, ~1M nodes
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CMU SCS
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • Largest publicly available graph ever studied.
????
19+? [Barabasi+]
110 C. Faloutsos (CMU)
Radius
Count
FB'11
14 (dir.) ~7 (undir.)
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CMU SCS
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • 7 degrees of separation (!) • Diameter: shrunk
????
19+? [Barabasi+]
111 C. Faloutsos (CMU)
Radius
Count
FB'11
14 (dir.) ~7 (undir.)
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CMU SCS
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Q: Shape?
????
112 C. Faloutsos (CMU)
Radius
Count
FB'11
~7 (undir.)
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CMU SCS
113 C. Faloutsos (CMU)
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • effective diameter: surprisingly small. • Multi-modality (?!)
FB'11
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CMU SCS
Radius Plot of GCC of YahooWeb.
114 C. Faloutsos (CMU) FB'11
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CMU SCS
115 C. Faloutsos (CMU)
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • effective diameter: surprisingly small. • Multi-modality: probably mixture of cores .
FB'11
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CMU SCS
116 C. Faloutsos (CMU)
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • effective diameter: surprisingly small. • Multi-modality: probably mixture of cores .
FB'11
EN
~7
Conjecture: DE
BR
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CMU SCS
117 C. Faloutsos (CMU)
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • effective diameter: surprisingly small. • Multi-modality: probably mixture of cores .
FB'11
~7
Conjecture:
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CMU SCS
Running time - Kronecker and Erdos-Renyi Graphs with billions edges.
details
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CMU SCS
C. Faloutsos (CMU) 119
Centralized Hadoop/PEGASUS
Degree Distr. old old
Pagerank old old
Diameter/ANF old HERE
Conn. Comp old HERE
Triangles HERE
Visualization started
Outline – Algorithms & results
FB'11
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CMU SCS Generalized Iterated Matrix
Vector Multiplication (GIMV)
C. Faloutsos (CMU) 120
PEGASUS: A Peta-Scale Graph Mining System - Implementation and Observations. U Kang, Charalampos E. Tsourakakis, and Christos Faloutsos. (ICDM) 2009, Miami, Florida, USA. Best Application Paper (runner-up).
FB'11
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CMU SCS Generalized Iterated Matrix
Vector Multiplication (GIMV)
C. Faloutsos (CMU) 121
• PageRank • proximity (RWR) • Diameter • Connected components • (eigenvectors, • Belief Prop. • … )
Matrix – vector Multiplication
(iterated)
FB'11
details
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CMU SCS
122
Example: GIM-V At Work • Connected Components – 4 observations:
Size
Count
C. Faloutsos (CMU) FB'11
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CMU SCS
123
Example: GIM-V At Work • Connected Components
Size
Count
C. Faloutsos (CMU) FB'11
1) 10K x larger than next
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CMU SCS
124
Example: GIM-V At Work • Connected Components
Size
Count
C. Faloutsos (CMU) FB'11
2) ~0.7B singleton nodes
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CMU SCS
125
Example: GIM-V At Work • Connected Components
Size
Count
C. Faloutsos (CMU) FB'11
3) SLOPE!
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CMU SCS
126
Example: GIM-V At Work • Connected Components
Size
Count 300-size
cmpt X 500. Why? 1100-size cmpt
X 65. Why?
C. Faloutsos (CMU) FB'11
4) Spikes!
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CMU SCS
127
Example: GIM-V At Work • Connected Components
Size
Count
suspicious financial-advice sites
(not existing now)
C. Faloutsos (CMU) FB'11
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CMU SCS
128
GIM-V At Work • Connected Components over Time • LinkedIn: 7.5M nodes and 58M edges
Stable tail slope after the gelling point
C. Faloutsos (CMU) FB'11
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CMU SCS
C. Faloutsos (CMU) 129
Outline
• Introduction – Motivation • Problem#1: Patterns in graphs • Problem#2: Tools • Problem#3: Scalability • Conclusions
FB'11
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CMU SCS
C. Faloutsos (CMU) 130
OVERALL CONCLUSIONS – low level:
• Several new patterns (fortification, triangle-laws, conn. components, etc)
• New tools: – anomaly detection (OddBall), belief
propagation, immunization
• Scalability: PEGASUS / hadoop
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OVERALL CONCLUSIONS – high level
• BIG DATA: Large datasets reveal patterns/outliers that are invisible otherwise
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References • Leman Akoglu, Christos Faloutsos: RTG: A Recursive
Realistic Graph Generator Using Random Typing. ECML/PKDD (1) 2009: 13-28
• Deepayan Chakrabarti, Christos Faloutsos: Graph mining: Laws, generators, and algorithms. ACM Comput. Surv. 38(1): (2006)
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References • Deepayan Chakrabarti, Yang Wang, Chenxi Wang,
Jure Leskovec, Christos Faloutsos: Epidemic thresholds in real networks. ACM Trans. Inf. Syst. Secur. 10(4): (2008)
• Deepayan Chakrabarti, Jure Leskovec, Christos Faloutsos, Samuel Madden, Carlos Guestrin, Michalis Faloutsos: Information Survival Threshold in Sensor and P2P Networks. INFOCOM 2007: 1316-1324
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References • Christos Faloutsos, Tamara G. Kolda, Jimeng Sun:
Mining large graphs and streams using matrix and tensor tools. Tutorial, SIGMOD Conference 2007: 1174
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References • T. G. Kolda and J. Sun. Scalable Tensor
Decompositions for Multi-aspect Data Mining. In: ICDM 2008, pp. 363-372, December 2008.
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References • Jure Leskovec, Jon Kleinberg and Christos Faloutsos
Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations, KDD 2005 (Best Research paper award).
• Jure Leskovec, Deepayan Chakrabarti, Jon M. Kleinberg, Christos Faloutsos: Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication. PKDD 2005: 133-145
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References • Jimeng Sun, Yinglian Xie, Hui Zhang, Christos
Faloutsos. Less is More: Compact Matrix Decomposition for Large Sparse Graphs, SDM, Minneapolis, Minnesota, Apr 2007.
• Jimeng Sun, Spiros Papadimitriou, Philip S. Yu, and Christos Faloutsos, GraphScope: Parameter-free Mining of Large Time-evolving Graphs ACM SIGKDD Conference, San Jose, CA, August 2007
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References • Jimeng Sun, Dacheng Tao, Christos
Faloutsos: Beyond streams and graphs: dynamic tensor analysis. KDD 2006: 374-383
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References • Hanghang Tong, Christos Faloutsos, and
Jia-Yu Pan, Fast Random Walk with Restart and Its Applications, ICDM 2006, Hong Kong.
• Hanghang Tong, Christos Faloutsos, Center-Piece Subgraphs: Problem Definition and Fast Solutions, KDD 2006, Philadelphia, PA
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References • Hanghang Tong, Christos Faloutsos, Brian
Gallagher, Tina Eliassi-Rad: Fast best-effort pattern matching in large attributed graphs. KDD 2007: 737-746
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Project info
Akoglu, Leman
Chau, Polo
Kang, U McGlohon, Mary
Tong, Hanghang
Prakash, Aditya
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Thanks to: NSF IIS-0705359, IIS-0534205, CTA-INARC; Yahoo (M45), LLNL, IBM, SPRINT, Google, INTEL, HP, iLab
www.cs.cmu.edu/~pegasus
Out, next year
Koutra, Danae
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OVERALL CONCLUSIONS – high level
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Graphs/ Social net
Cyber-security
Fraud detection
Data center monitoring
Environmental data monitoring
Health db
Big data / analytics
Databases, Map/reduce