Discovering Roles and Anomalies in Graphs: Theory and...
Transcript of Discovering Roles and Anomalies in Graphs: Theory and...
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Discovering Roles and Anomalies in Graphs: Theory
and Applications
Part 2: patterns, anomalies and applications
Tina Eliassi-Rad (Rutgers) Christos Faloutsos (CMU)
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T. Eliassi-Rad & C. Faloutsos 2
OVERVIEW - high level:
SDM'12 Tutorial
Roles
Features
Anomalies
Patterns
= rare roles
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T. Eliassi-Rad & C. Faloutsos 3
Resource:
Open source system for mining huge graphs: PEGASUS project (PEta GrAph mining
System) • www.cs.cmu.edu/~pegasus • code and papers SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 4
Roadmap
• Patterns in graphs – overview – Static graphs – Weighted graphs – Time-evolving graphs
• Anomaly Detection • Application: ebay fraud • Conclusions
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 5
Graphs - why should we care?
Internet Map [lumeta.com]
Food Web [Martinez ’91]
Friendship Network [Moody ’01]
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 6
Graphs - why should we care? • IR: bi-partite graphs (doc-terms)
• web: hyper-text graph
• ... and more:
D1
DN
T1
TM
... ...
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 7
Graphs - why should we care? • ‘viral’ marketing • web-log (‘blog’) news propagation • computer network security: email/IP traffic
and anomaly detection • ....
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 8
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?
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 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?
– To spot anomalies (rarities), we have to discover patterns
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 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
– Large datasets reveal patterns/anomalies that may be invisible otherwise…
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 11
Graph mining • Are real graphs random?
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 12
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
SDM'12 Tutorial
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Real Graph Patterns unweighted weighted static
P01. Power-law degree distribution [Faloutsos et. al.`99, Kleinberg et. al.`99, Chakrabarti et. al. `04, Newman`04] P02. Triangle Power Law [Tsourakakis `08] P03. Eigenvalue Power Law [Siganos et. al. `03] P04. Community structure [Flake et. al.`02, Girvan and Newman `02] P05. Clique Power Laws [Du et. al. ‘09]
P12. Snapshot Power Law [McGlohon et. al. `08]
dynamic
P06. Densification Power Law [Leskovec et. al.`05] P07. Small and shrinking diameter [Albert and Barabási `99, Leskovec et. al. ‘05, McGlohon et. al. ‘08] P08. Gelling point [McGlohon et. al. `08] P09. Constant size 2nd and 3rd connected components [McGlohon et. al. `08] P10. Principal Eigenvalue Power Law [Akoglu et. al. `08] P11. Bursty/self-similar edge/weight additions [Gomez and Santonja `98, Gribble et. al. `98, Crovella and Bestavros `99, McGlohon et .al. `08]
P13. Weight Power Law [McGlohon et. al. `08] P14. Skewed call duration distributions [Vaz de Melo et. al. `10]
13 SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos RTG: A Recursive Realistic Graph Generator using Random Typing Leman Akoglu and Christos Faloutsos. ECML PKDD’09.
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T. Eliassi-Rad & C. Faloutsos 14
Roadmap
• Patterns in graphs – overview – Static graphs – Weighted graphs – Time-evolving graphs
• Anomaly Detection • Application: ebay fraud • Conclusions
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 15
Solution# S.1
• Power law in the degree distribution [SIGCOMM99]
log(rank)
log(degree)
internet domains
att.com
ibm.com
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 16
Solution# S.1
• Power law in the degree distribution [SIGCOMM99]
log(rank)
log(degree)
-0.82
internet domains
att.com
ibm.com
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 17
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
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 18
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
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 19
But: How about graphs from other domains?
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 20
More power laws: • web hit counts [w/ A. Montgomery]
Web Site Traffic
in-degree (log scale)
Count (log scale)
Zipf
users sites
``ebay’’
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 21
epinions.com • who-trusts-whom
[Richardson + Domingos, KDD 2001]
(out) degree
count
trusts-2000-people user
SDM'12 Tutorial
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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’ SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 22
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T. Eliassi-Rad & C. Faloutsos 23
Roadmap
• Patterns in graphs – overview – Static graphs
• S1: Degree, S2: eigenvalues • S3-4: Triangles, S5: cliques • Radius plot • Other observations (‘eigenSpokes’)
– Weighted graphs – Time-evolving graphs
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 24
Solution# S.3: Triangle ‘Laws’
• Real social networks have a lot of triangles
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 25
Solution# S.3: Triangle ‘Laws’
• Real social networks have a lot of triangles – Friends of friends are friends
• Any patterns?
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 26
Triangle Law: #S.3 [Tsourakakis ICDM 2008]
ASN HEP-TH
Epinions X-axis: # of participating triangles Y: count (~ pdf)
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 27
Triangle Law: #S.3 [Tsourakakis ICDM 2008]
ASN HEP-TH
Epinions
SDM'12 Tutorial
X-axis: # of participating triangles Y: count (~ pdf)
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T. Eliassi-Rad & C. Faloutsos 28
Triangle Law: #S.4 [Tsourakakis ICDM 2008]
SN Reuters
Epinions X-axis: degree Y-axis: mean # triangles n friends -> ~n1.6 triangles
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 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?
details
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 30
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
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 31
Triangle Law: Computations [Tsourakakis ICDM 2008]
1000x+ speed-up, >90% accuracy
details
SDM'12 Tutorial
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Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges)
[U Kang, Brendan Meeder, +, PAKDD’11] 32 SDM'12 Tutorial 32 T. Eliassi-Rad & C. Faloutsos
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Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges)
[U Kang, Brendan Meeder, +, PAKDD’11] 33 SDM'12 Tutorial 33 T. Eliassi-Rad & C. Faloutsos
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Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges)
[U Kang, Brendan Meeder, +, PAKDD’11] 34 SDM'12 Tutorial 34 T. Eliassi-Rad & C. Faloutsos
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Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges)
[U Kang, Brendan Meeder, +, PAKDD’11] 35 SDM'12 Tutorial 35 T. Eliassi-Rad & C. Faloutsos
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Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges)
[U Kang, Brendan Meeder, +, PAKDD’11] 36 SDM'12 Tutorial 36 T. Eliassi-Rad & C. Faloutsos
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Triangle counting for large graphs? Q: How to compute # triangles in B-node
graph? (O(dmax ** 2) )? 37 SDM'12 Tutorial 37 T. Eliassi-Rad & C. Faloutsos
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Triangle counting for large graphs? Q: How to compute # triangles in B-node
graph? (O(dmax ** 2) )? A: cubes of eigvals 38 SDM'12 Tutorial 38 T. Eliassi-Rad & C. Faloutsos
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T. Eliassi-Rad & C. Faloutsos 39
Roadmap
• Patterns in graphs – overview – Static graphs
• S1: Degree, S2: eigenvalues • S3-4: Triangles, S5: cliques • Radius plot • Other observations (‘eigenSpokes’)
– Weighted graphs – Time-evolving graphs
SDM'12 Tutorial
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How about cliques?
SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 40
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Large Human Communication Networks Patterns and a Utility-Driven Generator
Nan Du, Christos Faloutsos, Bai Wang, Leman Akoglu KDD 2009
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SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 42
Cliques • Clique is a complete subgraph. • If a clique can not be
contained by any larger clique, it is called the maximal clique.
2 0
1 3
4
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SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 43
Clique • Clique is a complete subgraph. • If a clique can not be
contained by any larger clique, it is called the maximal clique.
2 0
1 3
4
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SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 44
Clique • Clique is a complete subgraph. • If a clique can not be
contained by any larger clique, it is called the maximal clique.
2 0
1 3
4
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SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 45
Clique • Clique is a complete subgraph. • If a clique can not be
contained by any larger clique, it is called the maximal clique.
• {0,1,2}, {0,1,3}, {1,2,3} {2,3,4}, {0,1,2,3} are cliques;
• {0,1,2,3} and {2,3,4} are the maximal cliques.
2 0
1 3
4
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SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 46
S5: Clique-Degree Power-Law • Power law:
α∝idg iavC d
1 8 2 2 is the power law exponent
[ . , . ] for S1~S3αα ∈
More friends, even more social circles !
# maximal cliques of node i
degree of node i
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SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 47
S5: Clique-Degree Power-Law • Outlier Detection
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SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 48
S5: Clique-Degree Power-Law • Outlier Detection
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T. Eliassi-Rad & C. Faloutsos 49
Roadmap
• Patterns in graphs – overview – Static graphs
• S1: Degree, S2: eigenvalues • S3-4: Triangles, S5: cliques • Radius plot • Other observations (‘eigenSpokes’)
– Weighted graphs – Time-evolving graphs
SDM'12 Tutorial
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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
T. Eliassi-Rad & C. Faloutsos 50 SDM'12 Tutorial
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????
19+ [Barabasi+]
51 T. Eliassi-Rad & C. Faloutsos
Radius
Count
SDM'12 Tutorial
~1999, ~1M nodes
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YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • Largest publicly available graph ever studied.
????
19+ [Barabasi+]
52 T. Eliassi-Rad & C. Faloutsos
Radius
Count
SDM'12 Tutorial
??
~1999, ~1M nodes
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YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • Largest publicly available graph ever studied.
????
19+? [Barabasi+]
53 T. Eliassi-Rad & C. Faloutsos
Radius
Count
SDM'12 Tutorial
14 (dir.) ~7 (undir.)
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YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • 7 degrees of separation (!) • Diameter: shrunk
????
19+? [Barabasi+]
54 T. Eliassi-Rad & C. Faloutsos
Radius
Count
SDM'12 Tutorial
14 (dir.) ~7 (undir.)
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YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Q: Shape?
????
55 T. Eliassi-Rad & C. Faloutsos
Radius
Count
SDM'12 Tutorial
~7 (undir.)
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56 T. Eliassi-Rad & C. Faloutsos
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • effective diameter: surprisingly small. • Multi-modality (?!)
SDM'12 Tutorial
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Radius Plot of GCC of YahooWeb.
57 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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58 T. Eliassi-Rad & C. Faloutsos
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • effective diameter: surprisingly small. • Multi-modality: probably mixture of cores .
SDM'12 Tutorial
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59 T. Eliassi-Rad & C. Faloutsos
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • effective diameter: surprisingly small. • Multi-modality: probably mixture of cores .
SDM'12 Tutorial
EN
~7
Conjecture: DE
BR
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60 T. Eliassi-Rad & C. Faloutsos
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • effective diameter: surprisingly small. • Multi-modality: probably mixture of cores .
SDM'12 Tutorial
~7
Conjecture:
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T. Eliassi-Rad & C. Faloutsos 61
Roadmap
• Patterns in graphs – overview – Static graphs
• S1: Degree, S2: eigenvalues • S3-4: Triangles, S5: cliques • Radius plot • Other observations (‘eigenSpokes’)
– Weighted graphs – Time-evolving graphs
SDM'12 Tutorial
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S6: 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.
T. Eliassi-Rad & C. Faloutsos 62 SDM'12 Tutorial
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EigenSpokes • Eigenvectors of adjacency matrix
§ equivalent to singular vectors (symmetric, undirected graph)
A = U�UT
63 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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EigenSpokes • Eigenvectors of adjacency matrix
§ equivalent to singular vectors (symmetric, undirected graph)
A = U�UT
�u1 �ui64 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
N
N
details
![Page 65: Discovering Roles and Anomalies in Graphs: Theory and ...eliassi.org/sdm12tut/sdm12tut_roles_part2_final.pdfReal Graph Patterns unweighted weighted static Newman `02] P01.Power-law](https://reader034.fdocuments.in/reader034/viewer/2022042414/5f2ed255fb59005ae1731d56/html5/thumbnails/65.jpg)
EigenSpokes • Eigenvectors of adjacency matrix
§ equivalent to singular vectors (symmetric, undirected graph)
A = U�UT
�u1 �ui65 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
N
N
details
![Page 66: Discovering Roles and Anomalies in Graphs: Theory and ...eliassi.org/sdm12tut/sdm12tut_roles_part2_final.pdfReal Graph Patterns unweighted weighted static Newman `02] P01.Power-law](https://reader034.fdocuments.in/reader034/viewer/2022042414/5f2ed255fb59005ae1731d56/html5/thumbnails/66.jpg)
EigenSpokes • Eigenvectors of adjacency matrix
§ equivalent to singular vectors (symmetric, undirected graph)
A = U�UT
�u1 �ui66 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
N
N
details
![Page 67: Discovering Roles and Anomalies in Graphs: Theory and ...eliassi.org/sdm12tut/sdm12tut_roles_part2_final.pdfReal Graph Patterns unweighted weighted static Newman `02] P01.Power-law](https://reader034.fdocuments.in/reader034/viewer/2022042414/5f2ed255fb59005ae1731d56/html5/thumbnails/67.jpg)
EigenSpokes • Eigenvectors of adjacency matrix
§ equivalent to singular vectors (symmetric, undirected graph)
A = U�UT
�u1 �ui67 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
N
N
details
![Page 68: Discovering Roles and Anomalies in Graphs: Theory and ...eliassi.org/sdm12tut/sdm12tut_roles_part2_final.pdfReal Graph Patterns unweighted weighted static Newman `02] P01.Power-law](https://reader034.fdocuments.in/reader034/viewer/2022042414/5f2ed255fb59005ae1731d56/html5/thumbnails/68.jpg)
EigenSpokes • EE plot: • Scatter plot of
scores of u1 vs u2 • One would expect
– Many points @ origin
– A few scattered ~randomly
T. Eliassi-Rad & C. Faloutsos 68
u1
u2
SDM'12 Tutorial
1st Principal component
2nd Principal component
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EigenSpokes • EE plot: • Scatter plot of
scores of u1 vs u2 • One would expect
– Many points @ origin
– A few scattered ~randomly
T. Eliassi-Rad & C. Faloutsos 69
u1
u2 90o
SDM'12 Tutorial
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EigenSpokes - pervasiveness • Present in mobile social graph
§ across time and space • Patent citation graph
70 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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EigenSpokes - explanation
Near-cliques, or near-bipartite-cores, loosely connected
71 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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EigenSpokes - explanation
Near-cliques, or near-bipartite-cores, loosely connected
72 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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EigenSpokes - explanation
Near-cliques, or near-bipartite-cores, loosely connected
73 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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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
74 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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Bipartite Communities!
magnified bipartite community
patents from same inventor(s)
`cut-and-paste’ bibliography!
75 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 76
Roadmap
• Patterns in graphs – overview – Static graphs – Weighted graphs – Time-evolving graphs
• Anomaly Detection • Application: ebay fraud • Conclusions
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 77
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
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 78
Observation W.1: Fortification Q: How do the weights of nodes relate to degree?
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 79
Observation W.1: Fortification
More donors, more $ ?
$10
$5
SDM'12 Tutorial
‘Reagan’
‘Clinton’ $7
![Page 80: Discovering Roles and Anomalies in Graphs: Theory and ...eliassi.org/sdm12tut/sdm12tut_roles_part2_final.pdfReal Graph Patterns unweighted weighted static Newman `02] P01.Power-law](https://reader034.fdocuments.in/reader034/viewer/2022042414/5f2ed255fb59005ae1731d56/html5/thumbnails/80.jpg)
Edges (# donors)
In-weights ($)
T. Eliassi-Rad & C. Faloutsos 80
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
SDM'12 Tutorial
![Page 81: Discovering Roles and Anomalies in Graphs: Theory and ...eliassi.org/sdm12tut/sdm12tut_roles_part2_final.pdfReal Graph Patterns unweighted weighted static Newman `02] P01.Power-law](https://reader034.fdocuments.in/reader034/viewer/2022042414/5f2ed255fb59005ae1731d56/html5/thumbnails/81.jpg)
T. Eliassi-Rad & C. Faloutsos 81
Roadmap
• Patterns in graphs – overview – Static graphs – Weighted graphs – Time-evolving graphs
• Anomaly Detection • Application: ebay fraud • Conclusions
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 82
Problem: Time evolution • with Jure Leskovec (CMU ->
Stanford)
• and Jon Kleinberg (Cornell – sabb. @ CMU)
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 83
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?
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 84
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
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 85
T.1 Diameter – “Patents”
• Patent citation network
• 25 years of data • @1999
– 2.9 M nodes – 16.5 M edges
time [years]
diameter
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 86
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)
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 87
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’’
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 88
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
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T. Eliassi-Rad & C. Faloutsos 89
Roadmap
• Patterns in graphs – … – Time-evolving graphs
• T1: shrinking diameter; • T2: densification • T3: connected components • T4: popularity over time • T5: phonecall patterns
• …
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 90
More on Time-evolving graphs
M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008
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T. Eliassi-Rad & C. Faloutsos 91
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?
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T. Eliassi-Rad & C. Faloutsos 92
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?
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 93
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?
SDM'12 Tutorial
YES YES
YES
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T. Eliassi-Rad & C. Faloutsos 94
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 SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 95
(Computation – scalability?) • Q: How to handle billion node graphs? • A: hadoop + ‘Pegasus’
– Most operations -> matrix-vector multiplications
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Generalized Iterated Matrix Vector Multiplication (GIMV)
T. Eliassi-Rad & C. Faloutsos 96
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).
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Generalized Iterated Matrix Vector Multiplication (GIMV)
T. Eliassi-Rad & C. Faloutsos 97
• PageRank • proximity (RWR) • Diameter • Connected components • (eigenvectors, • Belief Prop. • … )
Matrix – vector Multiplication
(iterated)
SDM'12 Tutorial
details
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98
Example: GIM-V At Work • Connected Components – 4 observations:
Size
Count
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99
Example: GIM-V At Work • Connected Components
Size
Count
T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
1) 10K x larger than next
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100
Example: GIM-V At Work • Connected Components
Size
Count
T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
2) ~0.7B singleton nodes
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101
Example: GIM-V At Work • Connected Components
Size
Count
T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
3) SLOPE!
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102
Example: GIM-V At Work • Connected Components
Size
Count 300-size
cmpt X 500. Why? 1100-size cmpt
X 65. Why?
T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
4) Spikes!
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103
Example: GIM-V At Work • Connected Components
Size
Count
suspicious financial-advice sites
(not existing now)
T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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104
GIM-V At Work • Connected Components over Time • LinkedIn: 7.5M nodes and 58M edges
Stable tail slope after the gelling point
T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 105
Roadmap
• Patterns in graphs – … – Time-evolving graphs
• T1: shrinking diameter; • T2: densification • T3: connected components • T4: popularity over time • T5: phonecall patterns
• …
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 106
Timing for Blogs
• with Mary McGlohon (CMU->Google) • Jure Leskovec (CMU->Stanford) • Natalie Glance (now at Google) • Mat Hurst (now at MSR) [SDM’07]
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 107
T.4 : popularity over time
Post popularity drops-off – exponentially?
lag: days after post
# in links
1 2 3
@t
@t + lag
SDM'12 Tutorial
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T. Eliassi-Rad & C. Faloutsos 108
T.4 : popularity over time
Post popularity drops-off – exponentially? POWER LAW! Exponent?
# in links (log)
days after post (log)
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T. Eliassi-Rad & C. Faloutsos 109
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)
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SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 110
-1.5 slope
J. G. Oliveira & A.-L. Barabási Human Dynamics: The Correspondence Patterns of Darwin and Einstein. Nature 437, 1251 (2005) . [PDF]
Log # days to respond
Log Prob() -1.5
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T. Eliassi-Rad & C. Faloutsos 111
Roadmap
• Patterns in graphs – … – Time-evolving graphs
• T1: shrinking diameter; • T2: densification • T3: connected components • T4: popularity over time • T5: phonecall patterns
• …
SDM'12 Tutorial
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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
SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 112
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Probably, power law (?)
SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 113
??
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No Power Law!
SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 114
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‘TLaC: Lazy Contractor’ • The longer a task (phonecall) has taken, • The even longer it will take
SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 115
Odds ratio= Casualties(<x): Survivors(>=x) == power law
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116
Data Description
n Data from a private mobile operator of a large city n 4 months of data n 3.1 million users n more than 1 billion phone records
n Over 96% of ‘talkative’ users obeyed a TLAC distribution (‘talkative’: >30 calls)
SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos
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Outliers:
SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos 117
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Real Graph Patterns unweighted weighted static
P01. Power-law degree distribution [Faloutsos et. al.`99, Kleinberg et. al.`99, Chakrabarti et. al. `04, Newman`04] P02. Triangle Power Law [Tsourakakis `08] P03. Eigenvalue Power Law [Siganos et. al. `03] P04. Community structure [Flake et. al.`02, Girvan and Newman `02] P05. Clique Power Laws [Du et. al. ‘09]
P12. Snapshot Power Law [McGlohon et. al. `08]
dynamic
P06. Densification Power Law [Leskovec et. al.`05] P07. Small and shrinking diameter [Albert and Barabási `99, Leskovec et. al. ‘05, McGlohon et. al. ‘08] P08. Gelling point [McGlohon et. al. `08] P09. Constant size 2nd and 3rd connected components [McGlohon et. al. `08] P10. Principal Eigenvalue Power Law [Akoglu et. al. `08] P11. Bursty/self-similar edge/weight additions [Gomez and Santonja `98, Gribble et. al. `98, Crovella and Bestavros `99, McGlohon et .al. `08]
P13. Weight Power Law [McGlohon et. al. `08] P14. Skewed call duration distributions [Vaz de Melo et. al. `10]
118 SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos RTG: A Recursive Realistic Graph Generator using Random Typing Leman Akoglu and Christos Faloutsos. ECML PKDD’09.
✓ ✓ ✓
✓ ✓
✓ ✓
✓
✓
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T. Eliassi-Rad & C. Faloutsos 119
Roadmap
• Patterns in graphs – overview – Static graphs – Weighted graphs – Time-evolving graphs
• Anomaly Detection • Application: ebay fraud • Conclusions
SDM'12 Tutorial
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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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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
T. Eliassi-Rad & C. Faloutsos 121 SDM'12 Tutorial
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What is an egonet?
ego
122
egonet
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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
123 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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Near-Clique/Star
124 SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos
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Near-Clique/Star
125 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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Near-Clique/Star
126 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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Andrew Lewis (director)
Near-Clique/Star
127 T. Eliassi-Rad & C. Faloutsos SDM'12 Tutorial
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Dominant Heavy Link
128 SDM'12 Tutorial 128 T. Eliassi-Rad & C. Faloutsos
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T. Eliassi-Rad & C. Faloutsos 129
Roadmap
• Patterns in graphs – overview – Static graphs – Weighted graphs – Time-evolving graphs
• Anomaly Detection • Application: ebay fraud • Conclusions
SDM'12 Tutorial
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NetProbe: The Problem Find bad sellers (fraudsters) on eBay who don’t deliver their (expensive) items
130 SDM'12 Tutorial T. Eliassi-Rad & C. Faloutsos
$$$
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E-bay Fraud detection
w/ Polo Chau & Shashank Pandit, CMU [www’07]
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E-bay Fraud detection
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E-bay Fraud detection
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E-bay Fraud detection - NetProbe
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NetProbe: Key Ideas • Fraudsters fabricate their reputation by
“trading” with their accomplices • Transactions form near bipartite cores • How to detect them?
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NetProbe: Key Ideas Use ‘Belief Propagation’ and ~heterophily
136
F A H Fraudster
Accomplice Honest
Darker means more likely
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NetProbe: Main Results
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T. Eliassi-Rad & C. Faloutsos 138
Roadmap
• Patterns in graphs • Anomaly Detection • Application: ebay fraud
– How-to: Belief Propagation • Conclusions
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Guilt-by-Association Techniques
Given: • graph and • few labeled nodes
Find: class (red/green) for rest nodes Assuming: network effects (homophily/ heterophily, etc)
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details
red green
F
H A
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Correspondence of Methods
Random Walk with Restarts (RWR) Google Semi-supervised Learning (SSL) Belief Propagation (BP) Bayesian
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Correspondence of Methods
Random Walk with Restarts (RWR) ≈ Semi-supervised Learning (SSL) ≈ Belief Propagation (BP)
Method Matrix unknown known RWR [I – c AD-1] × x = (1-c)y SSL [I + a(D - A)] × x = y
FABP [I + a D - c’A] × bh = φh
0 1 0 1 0 1 0 1 0
? 0 1 1
SDM'12 Tutorial 141 T. Eliassi-Rad & C. Faloutsos Unifying Guilt-by-Association Approaches: Theorems and Fast Algorithms. Danai Koutra, et al PKDD’11
details
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T. Eliassi-Rad & C. Faloutsos 142
Roadmap
• Patterns in graphs • Anomaly Detection • Application: ebay fraud • Conclusions
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Overall conclusions • Roles:
– Past work in social networks (‘regular’, ‘structural’ etc)
– Scalable algo’s to find such roles • Anomalies & patterns
– Static (power-laws, ‘six degrees’) – Weighted (super-linearity) – Time-evolving (densification, -1.5 exponent)
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T. Eliassi-Rad & C. Faloutsos 144
OVERALL CONCLUSIONS – high level:
SDM'12 Tutorial
Roles
Features
Anomalies
Patterns
= rare roles
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T. Eliassi-Rad & C. Faloutsos 145
OVERALL CONCLUSIONS – high level
• BIG DATA: -> roles/patterns/outliers that are invisible otherwise
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T. Eliassi-Rad & C. Faloutsos 146
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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T. Eliassi-Rad & C. Faloutsos 147
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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T. Eliassi-Rad & C. Faloutsos 148
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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T. Eliassi-Rad & C. Faloutsos 149
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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T. Eliassi-Rad & C. Faloutsos 150
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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T. Eliassi-Rad & C. Faloutsos 151
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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T. Eliassi-Rad & C. Faloutsos 153
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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T. Eliassi-Rad & C. Faloutsos 154
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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T. Eliassi-Rad & C. Faloutsos 155
Project info
Akoglu, Leman
Chau, Polo
Kang, U McGlohon, Mary
Tong, Hanghang
Prakash, Aditya
SDM'12 Tutorial
Thanks to: NSF IIS-0705359, IIS-0534205, CTA-INARC; ADAMS-DARPA; Yahoo (M45), LLNL, IBM, SPRINT, Google, INTEL, HP, iLab
www.cs.cmu.edu/~pegasus
Koutra, Danai