discovering roles and anomalies in graphs: theory and...
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Discovering Roles and Anomalies in Graphs:
Theory and Applications
Part 2: Patterns and Anomalies Tina Eliassi-Rad (Rutgers) Christos Faloutsos (CMU)
ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 2
OVERVIEW - high level:
ECML PKDD 2013 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 ECML PKDD 2013 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
ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 5
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?
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T. Eliassi-Rad & C. Faloutsos 6
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
ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 7
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…
ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 8
Graph mining • Are real graphs random?
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T. Eliassi-Rad & C. Faloutsos 9
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
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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]
Dynam
ic
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]
10 ECML PKDD 2013 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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Roadmap
• Patterns in graphs – Overview – Static graphs – Weighted graphs – Time-evolving graphs
• Anomaly Detection • Application: ebay fraud • Conclusions
T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 12
Solution# S.1
• Power law in the degree distribution [SIGCOMM99]
log(rank)
log(degree)
Internet Domains att.com
ibm.com
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T. Eliassi-Rad & C. Faloutsos 13
Solution# S.1
• Power law in the degree distribution [SIGCOMM99]
log(rank)
log(degree)
-0.82
att.com
ibm.com
Internet Domains
ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 14
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
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T. Eliassi-Rad & C. Faloutsos 15
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
ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 16
But: How about graphs from other domains?
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T. Eliassi-Rad & C. Faloutsos 17
More power laws: • web hit counts [w/ A. Montgomery]
Web Site Traffic
in-degree (log scale)
Count (log scale)
Zipf
users sites
``ebay’’
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T. Eliassi-Rad & C. Faloutsos 18
epinions.com • who-trusts-whom
[Richardson + Domingos, KDD 2001]
(out) degree
count
trusts-2000-people user
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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’
T. Eliassi-Rad & C. Faloutsos 19 ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 20
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
ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 21
Solution# S.3: Triangle ‘Laws’
• Real social networks have a lot of triangles
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T. Eliassi-Rad & C. Faloutsos 22
Solution# S.3: Triangle ‘Laws’
• Real social networks have a lot of triangles – Friends of friends are friends
• Any patterns?
ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 23
Triangle Law: #S.3 [Tsourakakis ICDM 2008]
ASN HEP-TH
Epinions X-axis: # of participating triangles Y: count (~ pdf)
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Triangle Law: #S.3 [Tsourakakis ICDM 2008]
ASN HEP-TH
Epinions X-axis: # of participating triangles Y: count (~ pdf)
T. Eliassi-Rad & C. Faloutsos 24 ECML PKDD 2013 Tutorial
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Triangle Law: #S.4 [Tsourakakis ICDM 2008]
SN Reuters
Epinions X-axis: degree Y-axis: mean # triangles n friends → ~n1.6 triangles
T. Eliassi-Rad & C. Faloutsos 25 ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 26
Triangle Law: Computations [Tsourakakis ICDM 2008]
Triangles are expensive to compute (3-way join; several approximation algorithms)
Q: Can we do this computation quickly?
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T. Eliassi-Rad & C. Faloutsos 27
Triangle Law: Computations [Tsourakakis ICDM 2008]
Triangle are expensive to compute (3-way join; several approx. algos)
Q: Can we do this computation quickly? A: Yes!
#triangles = 1/6 Sum ( λi3 )
(and, because of skewness (S2) , we only need the top few eigenvalues!)
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T. Eliassi-Rad & C. Faloutsos 28
Triangle Law: Computations [Tsourakakis ICDM 2008]
1000x+ speed-up, >90% accuracy ECML PKDD 2013 Tutorial
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Triangle Counting on Big Graphs Anomalous nodes in Twitter (~ 3 billion edges)
[U Kang, Brendan Meeder, +, PAKDD’11] T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial 29
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Triangle Counting on Big Graphs Anomalous nodes in Twitter (~ 3 billion edges)
[U Kang, Brendan Meeder, +, PAKDD’11] T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial 30
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Triangle Counting on Big Graphs Anomalous nodes in Twitter(~ 3 billion edges)
[U Kang, Brendan Meeder, +, PAKDD’11] T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial 31
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Triangle Counting on Big Graphs Anomalous nodes in Twitter (~ 3 billion edges)
[U Kang, Brendan Meeder, +, PAKDD’11] T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial 32
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Triangle Counting on Big Graphs Anomalous nodes in Twitter (~ 3 billion edges)
[U Kang, Brendan Meeder, +, PAKDD’11] T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial 33
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Triangle Counting on Big Graphs Q: How to compute # triangles in B-node
graph? (O(dmax2) )?
T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial 34
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Triangle Counting on Big Graphs Q: How to compute # triangles in B-node
graph? A: cubes of eigvals T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial 35
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36
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
T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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How about cliques? • Large Human Communication Networks
Patterns and a Utility-Driven Generator. Nan Du, Christos Faloutsos, Bai Wang, Leman Akoglu, KDD 2009
37 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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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
T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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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
T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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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
T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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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
T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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42
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
T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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T. Eliassi-Rad & C. Faloutsos 43
S5: Clique-Degree Power-Law • Outlier Detection
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T. Eliassi-Rad & C. Faloutsos 44
S5: Clique-Degree Power-Law • Outlier Detection
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T. Eliassi-Rad & C. Faloutsos 45
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
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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(N2) space and up to O(N3) time – prohibitive (N~1B)
• Our HADI: linear on E (~10B) – Near-linear scalability w.r.t. # machines – Several optimizations → 5x faster
T. Eliassi-Rad & C. Faloutsos 46 ECML PKDD 2013 Tutorial
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????
19+ [Barabasi+]
47 T. Eliassi-Rad & C. Faloutsos
Radius
Count
~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+]
48 T. Eliassi-Rad & C. Faloutsos
Radius
Count ??
~1999, ~1M nodes
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????
19+? [Barabasi+]
49 T. Eliassi-Rad & C. Faloutsos
Radius
Count
14 (dir.) ~7 (undir.)
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • Largest publicly available graph ever studied.
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YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • 7 degrees of separation (!) • Diameter: shrunk
????
19+? [Barabasi+]
50 T. Eliassi-Rad & C. Faloutsos
Radius
Count
14 (dir.) ~7 (undir.)
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YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Q: Shape?
????
51 T. Eliassi-Rad & C. Faloutsos
Radius
Count
~7 (undir.)
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52 T. Eliassi-Rad & C. Faloutsos
YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) • Effective diameter: surprisingly small • Multi-modality (?!)
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Radius Plot of GCC of YahooWeb Graph
53 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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54 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
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55 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
EN
~7
Conjecture: DE
BR
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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: probably mixture of cores
~7
Conjecture:
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T. Eliassi-Rad & C. Faloutsos 57
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
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S6: EigenSpokes EigenSpokes: Surprising Patterns and Scalable Community Chipping in Large Graphs. B. Aditya Prakash, Mukund Seshadri, Ashwin Sridharan, Sridhar Machiraju, and Christos Faloutsos: PAKDD 2010.
T. Eliassi-Rad & C. Faloutsos 58 ECML PKDD 2013 Tutorial
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EigenSpokes • Eigenvectors of adjacency matrix
§ equivalent to singular vectors (symmetric, undirected graph)
A = U�UT
59 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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EigenSpokes • Eigenvectors of adjacency matrix
§ equivalent to singular vectors (symmetric, undirected graph)
A = U�UT
�u1 �ui60 T. Eliassi-Rad & C. Faloutsos
N
N
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EigenSpokes • Eigenvectors of adjacency matrix
§ equivalent to singular vectors (symmetric, undirected graph)
A = U�UT
�u1 �ui61 T. Eliassi-Rad & C. Faloutsos
N
N
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EigenSpokes • Eigenvectors of adjacency matrix
§ equivalent to singular vectors (symmetric, undirected graph)
A = U�UT
�u1 �ui62 T. Eliassi-Rad & C. Faloutsos
N
N
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EigenSpokes • Eigenvectors of adjacency matrix
§ equivalent to singular vectors (symmetric, undirected graph)
A = U�UT
�u1 �ui63 T. Eliassi-Rad & C. Faloutsos
N
N
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EigenSpokes • EigenEigen (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 64
u1
u2
1st Principal component
2nd Principal component
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EigenSpokes • EigenEigen (EE)
Plot – Scatter plot of
scores of u1 vs u2 • One would expect
– Many points @ origin
– A few scattered ~randomly
65
u1
u2 90o
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EigenSpokes - pervasiveness • Present in mobile social graph
§ Across time and space • Patent citation graph
66 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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EigenSpokes - explanation
Near-cliques, or near-bipartite-cores, loosely connected
67 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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EigenSpokes - explanation
Near-cliques, or near-bipartite-cores, loosely connected
68 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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EigenSpokes - explanation
Near-cliques, or near-bipartite-cores, loosely connected
69 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 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
70 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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Bipartite Communities!
magnified bipartite community
patents from same inventor(s)
`cut-and-paste’ bibliography!
71 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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72
Roadmap
• Patterns in graphs – Overview – Static graphs – Weighted graphs – Time-evolving graphs
• Anomaly Detection • Application: ebay fraud • Conclusions
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73
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. KDD 2008
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74
Observation W.1: Fortification
Q: How do the weights of nodes relate to degree?
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75
Observation W.1: Fortification
More donors, more $ ?
$10
$5
‘Reagan’
‘Clinton’ $7
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Edges (# donors)
In-weights ($)
76
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
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77
Roadmap
• Patterns in graphs – Overview – Static graphs – Weighted graphs – Time-evolving graphs
• Anomaly Detection • Application: ebay fraud • Conclusions
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78
Problem: Time evolution • with Jure Leskovec
(CMU → Stanford)
• and Jon Kleinberg (Cornell – sabb. @ CMU)
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79
T.1 Evolution of the Diameter • Prior work on Power Law graphs hinted
at slowly growing diameter: – diameter ~ O(log N) – diameter ~ O(log log N)
• What is happening in real data?
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80
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
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81
T.1 Diameter – “Patents”
• Patent citation network
• 25 years of data • @1999
– 2.9 M nodes – 16.5 M edges
time [years]
diameter
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82
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)
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83
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”
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84
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 85
Roadmap
• Patterns in graphs – … – Time-evolving graphs
• T1: Shrinking diameter • T2: Densification • T3: Connected components • T4: Popularity over time • T5: Phone-call patterns
• …
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86
More on Time-evolving graphs
M. McGlohon, L. Akoglu, and C. Faloutsos. Weighted Graphs and Disconnected Components: Patterns and a Generator. KDD 2008
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Observation T.3: NLCC behavior Q: How do the NLCCs 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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88
Observation T.3: NLCC behavior Q: How do NLCCs 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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89
Observation T.3: NLCC behavior Q: How do NLCCs emerge and join with the
GCC? (“NLCC” = non-largest conn. components) – Do they continue to grow in size? – or do they shrink? – or stabilize?
YES
YES
YES
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90
Observation T.3: NLCC behavior • After the gelling point, the GCC takes off, but
the NLCCs remain ~constant (actually, oscillate).
IMDB
CC size
Time-stamp T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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91
Roadmap
• Patterns in graphs – … – Time-evolving graphs
• T1: Shrinking diameter • T2: Densification • T3: Connected components • T4: Popularity over time • T5: Phone-call patterns
• …
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92
Timing for Blogs
• with Mary McGlohon (CMU→Google) • Jure Leskovec (CMU→Stanford) • Natalie Glance (now at Google) • Matt Hurst (now at MSR) [SDM’07]
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93
T.4 : popularity over time
Post popularity drops-off – exponentially?
lag: days after post
# in links
1 2 3
@t
@t + lag T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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94
T.4 : popularity over time
Post popularity drops-off – exponentially? POWER LAW! Exponent?
# in links (log)
days after post (log)
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95
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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96
-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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97
Roadmap
• Patterns in graphs – … – Time-evolving graphs
• T1: Shrinking diameter • T2: Densification • T3: Connected components • T4: Popularity over time • T5: Phone-call patterns
• …
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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
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Probably, power law (?)
99
??
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No Power Law!
100 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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‘TLaC: Lazy Contractor’ • The longer a task (phone-call) has taken,
the even longer it will take
101
Odds ratio= Casualties(<x): Survivors(>=x) == power law
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102
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)
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Outliers:
103 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 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]
104 ECML PKDD 2013 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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105
Roadmap
• Patterns in graphs – Overview – Static graphs – Weighted graphs – Time-evolving graphs
• Anomaly Detection • Application: ebay fraud • Conclusions
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OddBall: Spotting Anomalies in Weighted Graphs
Leman Akoglu, Mary McGlohon,
Christos Faloutsos
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) • Compare with the rest of the population
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What is an egonet?
ego
108
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
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Near-Clique/Star
110 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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Near-Clique/Star
111 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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Near-Clique/Star
112 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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Andrew Lewis (director)
Near-Clique/Star
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Dominant Heavy Link
114 T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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115
Roadmap
• Patterns in graphs – overview – Static graphs – Weighted graphs – Time-evolving graphs
• Anomaly Detection • Application: ebay fraud • Conclusions
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NetProbe: The Problem Find bad sellers (fraudsters) on eBay who don’t deliver their (expensive) items
116
$$$
X T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial
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117
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
122
F A H Fraudster
Accomplice Honest
Darker means more likely
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NetProbe: Main Results
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124
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)
125
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
ECML PKDD 2013 Tutorial
127 T. Eliassi-Rad & C. Faloutsos Unifying Guilt-by-Association Approaches: Theorems and Fast Algorithms. Danai Koutra, et al PKDD’11
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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 algorithms’ to find such roles • Anomalies & patterns
– Static (power-laws, ‘six degrees’) – Weighted (super-linearity) – Time-evolving (densification, -1.5 exponent)
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Thank You!
Roles
Features
Anomalies
Patterns
= rare roles
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T. Eliassi-Rad & C. Faloutsos 131
Project info
Akoglu, Leman
Chau, Polo
Kang, U McGlohon, Mary
Tong, Hanghang
Prakash, Aditya
ECML PKDD 2013 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
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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
T. Eliassi-Rad & C. Faloutsos ECML PKDD 2013 Tutorial