Post on 19-May-2020
transcript
Faloutsos, Tong CIKM, 2008
1
SCS CMU
Large Graph Mining:Patterns, Tools and Case Studies
CIKM’08 Copyright: Faloutsos, Tong (2008)
Christos FaloutsosHanghang Tong
CMU
3-1
SCS CMU
Outline
• Part 1: Patterns• Part 2: Matrix and Tensor Tools• Part 3: Proximity
CIKM’08 Copyright: Faloutsos, Tong (2008)
• Part 3: Proximity• Part 4: Case Studies
3-2
SCS CMU
Part 3: Proximity on Graphs-Definitions, Fast Solutions, and Applications
Copyright: Faloutsos, Tong (2008) 3-3CIKM, 2008
SCS CMU
Joint work with
Christos Faloutsos (CMU)
Jia-Yu Pan(Google)
Yehuda Koren(AT&T Labs)
IBM
3-4
Spiros Papadimitriou
Tina Eliassi-Rad(LLNL)
Brian Gallagher(LLNL)
Kensuke Oonuma(Sony Corp.)
Yasushi Sakurai (NTT Labs)
Philip S. Yu Huiming Qu Hani Jamjoom
SCS CMU
Recap: Graphs are everywhere!
Copyright: Faloutsos, Tong (2008) 3-5CIKM, 2008
SCS CMU
Graph Mining: Big Picture
John
Smith
TomJones
Alan
Peter
Adam
+ Graph Level-Patterns-Laws-Generators
+ Subgraph Level- Community
Adam
Dan
Jack
Alice
Amy Anna
Beck
Tom
Cell PhoneMessageE-Mail
Community+ Node Level
-Association-Correlation-Causality-Proximity
Anna
We are here! Copyright: Faloutsos, Tong (2008) 3-6
Faloutsos, Tong CIKM, 2008
2
SCS CMU
Proximity on Graph: What?
A BH1 1
I J1
1 1
Copyright: Faloutsos, Tong (2008)
D1 1
EF
G1 11
a.k.a Relevance, Closeness, ‘Similarity’… 3-7
SCS CMU
Proximity on Graphs: Why?
• Link prediction [Liben-Nowell+], [Tong+]• Ranking [Haveliwala], [Chakrabarti+]• Email Management [Minkov+]• Image caption [Pan+]g p [ ]• Neighborhooh Formulation [Sun+]• Conn. subgraph [Faloutsos+], [Tong+], [Koren+]• Pattern match [Tong+]• Collaborative Filtering [Fouss+]• Many more…
Will return to this laterCopyright: Faloutsos, Tong (2008) 3-8CIKM, 2008
SCS CMU
0.2
0.25
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Link Prediction
Prox. Hist. for a set of deleted links
density
density
Prox (i j)+Prox (j i)
Prox. is effective to ‘deleted’ and absent edges!
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090
0.05
0.1
0.15
Prox (i j)+Prox (j i)
Q: How to predict the existence of the link?A: Proximity! [Liben-Nowell + 2003]
Prox. Hist. for a set of absent links
Copyright: Faloutsos, Tong (2008) 3-9CIKM, 2008
SCS CMU Neighborhood Search on graphs
ICDM
KDD
SDM
Philip S. Yu
IJCAI
AAAI M. Jordan
Ning Zhong
R. Ramakrishnan
… …
NIPS…
…
Conference Author
A: Proximity! [Sun+ ICDM2005]
Q: what is most related conference to ICDM?
Copyright: Faloutsos, Tong (2008) 3-10CIKM, 2008
SCS CMU
Image
Region Automatic Image Caption
Test Image
Sea Sun Sky Wave Cat Forest Tiger Grass
Keyword
Q: How to assign keywords to the test image?A: Proximity! [Pan+ 2004]
3-11Copyright: Faloutsos, Tong (2008)CIKM, 2008
SCS CMU
Center-Piece Subgraph(CePS)
B
CePS guy
Input Output
A C
Original Graph CePS
Q: How to find hub for the black nodes?A: Proximity! [Tong+ KDD 2006]
Copyright: Faloutsos, Tong (2008) 3-12CIKM, 2008
Faloutsos, Tong CIKM, 2008
3
SCS CMU
OutputInput
Data Graph
Query Graph
Best-Effort Pattern Match
Matching Subgraph
Copyright: Faloutsos, Tong (2008) 3-13
Q: How to find matching subgraph?A: Proximity![Tong+ KDD 2007 b]CIKM, 2008
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions
• Basic: RWR
• Variants
• Properties
• Applications• Conclusion
• Generalizations
Copyright: Faloutsos, Tong (2008) 3-14CIKM, 2008
SCS CMU
Why not shortest path?
A BD1 1
E F11 1
A BD1 1
Some ``bad’’ proximities
‘pizza delivery guy’ problem
A BD1 1
A BD1 1
EF
G1 11
‘multi-facet’ relationship
Copyright: Faloutsos, Tong (2008) 3-15CIKM, 2008
SCS CMU
A BD1 1
Why not max. netflow?Some ``bad’’ proximities
A BD1 11 E
No punishment on long paths
Copyright: Faloutsos, Tong (2008) 3-16CIKM, 2008
SCS CMU
What is a ``good’’ Proximity?
A BH1 1
I J1
1 1
D1 1
EF
G1 11
• Multiple Connections
• Quality of connection
•Direct & In-directed Conns
•Length, Degree, Weight…
…
Copyright: Faloutsos, Tong (2008) 3-17CIKM, 2008
SCS CMU
13
2
910
811
12
Random walk with restart
Copyright: Faloutsos, Tong (2008)
4
3
56
7
3-18CIKM, 2008
Faloutsos, Tong CIKM, 2008
4
SCS CMU
Random walk with restartNode 4
Node 1Node 2Node 3Node 4Node 5Node 6
0.130.100.130.220.130.05
13
2
910
811
120.13
0.10
0.13
0.08
0.04
0.02
0.04
0.03
Node 7Node 8Node 9Node 10Node 11Node 12
0.050.080.040.030.040.02
4
5 6
7
0.13
0.05
0.05
Ranking vector More red, more relevantNearby nodes, higher scores
4rCopyright: Faloutsos, Tong (2008) 3-19CIKM, 2008
SCS CMU
Why RWR is a good score?
W : adjacency matrix. c: damping factor
1( )Q I cW −= − =
,( , ) i jQ i j r∝
i
j
2c 3cQ c= ...W 2W 3W
all paths from ito j with length 1
all paths from ito j with length 2
all paths from ito j with length 3
Copyright: Faloutsos, Tong (2008) 3-20CIKM, 2008
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions
• Basic: RWR
• Variants
• Properties
• Applications• Conclusion
• Generalizations
Copyright: Faloutsos, Tong (2008) 3-21CIKM, 2008
SCS CMU
Variant: escape probability• Define Random Walk (RW) on the graph• Esc_Prob(CMU Napa)
– Prob (starting at CMU, reaches Napa before returning to CMU)
CMU Napathe remaining graph
Copyright: Faloutsos, Tong (2008) 22Esc_Prob = Pr (smile before cry)
SCS CMU
Other Variants
• Other measure by RWs– Community Time/Hitting Time [Fouss+]– SimRank [Jeh+]
• Equivalence of Random WalksAll are “related to” or “similar to”Equivalence of Random Walks– Electric Networks:
• EC [Doyle+]; SAEC[Faloutsos+]; CFEC[Koren+] – String Systems
• Katz [Katz], [Huang+], [Scholkopf+]• Matrix-Forest-based Alg [Chobotarev+]
All are “related to” or “similar to” random walk with restart!
Copyright: Faloutsos, Tong (2008) 3-23CIKM, 2008
SCS CMU
Chaptering different measurements
RWR
Esc_Prob+ Sink
Hitting Time/Commute Time
RegularizedUn-constrainedQuad Opt.
relax
4 ssp decides 1 esc_prob
KatzNormalize
+ Sink Commute Time
Effective Conductance
String System
Harmonic Func.ConstrainedQuad Opt.
Mathematic Tools
X out-degree
“voltage = position”
CIKM, 2008 Physical Models
Faloutsos, Tong CIKM, 2008
5
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions
• Basic: RWR
• Variants
• Properties
• Applications• Conclusion
• Generalizations
Copyright: Faloutsos, Tong (2008) 3-25CIKM, 2008
SCS CMU
Property: Monotonicity
A B
Candidate Graph
Original Graph
We want:
A B
Prox ( ) Prox ( )candi origianla b a b≤→ →A: degree preserving! [Koren+ KDD06][Tong+ KDD07a][Tong+ SDM08]
Copyright: Faloutsos, Tong (2008) 26CIKM, 2008
SCS CMU
Property: Asymmetry [Tong+ KDD07 a]
A B
1 1
1 A B
1 1
0.5
111
11
10.51
What is Prox from A to B?What is Prox from B to A?
What is Prox between A and B?
Copyright: Faloutsos, Tong (2008) 3-27CIKM, 2008
SCS CMU
Asymmetry in un-directed graphs• Hanghang’s # 1 Conf. is CIKM• The #1 author of CIKM is ...
So is love…
HanghangCIKM
Copyright: Faloutsos, Tong (2008) 3-28CIKM, 2008
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions
• Basic: RWR
• Variants
• PropertiesFast Solutions
• Applications• Conclusion
• Generalizations
Copyright: Faloutsos, Tong (2008) 3-29CIKM, 2008
SCS CMU
Group Proximity [Tong+ KDD07 a]
• Q: How close are Accountants to SECs?
Copyright: Faloutsos, Tong (2008) 3-30CIKM, 2008
• A: Prob (starting at any RED, reaches anyGREEN before touching any RED again)
Faloutsos, Tong CIKM, 2008
6
SCS CMU
Proximity on Attributed Graphs [Tong+ KDD07 b]
What is the proximity from node 7 to 10?If we know that…
Copyright: Faloutsos, Tong (2008) 3-31CIKM, 2008
SCS CMU
A: Augmented graphs [Tong+ KDD07 b]
Copyright: Faloutsos, Tong (2008) 3-32CIKM, 2008
SCS CMU
More on Generalizations
• Attributed on edges [Chakrabarti+ KDD 06]
• Proximity w/ Time– [Minkov+], [Tong+ SDM 2008], [Tong+ CIKM 2008]
• Proximity w/ Side Information [Tong+ 2008]
• …
Copyright: Faloutsos, Tong (2008) 3-33CIKM, 2008
SCS CMU
Summary of Proximity Definitions• Goal: Summarize multiple … relationship
• Solutions– Basic: Random Walk with Restart
• [Pan+ 2004][Sun+ 2006][Tong+ 2006][ ][ ][ g ]
– Properties: Asymmetry, monotonicity• [Koren+ 2006][Tong+ 2007] [Tong+ 2008]
– Variants: Esc_Prob and many others.• [Faloutsos+ 2004] [Koren+ 2006][Tong+ 2007]
– Generalizations: Group Prox, w/ Attr., w/ Time, w/ Side Information
• [Charkrabarti+ 2006][Tong+ 2007] [Tong+ 2008]Copyright: Faloutsos, Tong (2008) 3-34CIKM, 2008
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions• Applications• Conclusion
Copyright: Faloutsos, Tong (2008) 3-35CIKM, 2008
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions
• B_Lin: RWR
• BB_Lin: Skewed BGs
• Applications• Conclusion
Copyright: Faloutsos, Tong (2008) 3-36CIKM, 2008
• FastUpdate: Time-Evolving
Faloutsos, Tong CIKM, 2008
7
SCS CMU
Preliminary: Sherman–Morrison Lemma
Tv
uAA =If:
1 11 1 1
1( )1
TT
T
A u v AA A u v Av A u
− −− − −
−
⋅ ⋅= + ⋅ = −+ ⋅ ⋅
Then:
Copyright: Faloutsos, Tong (2008) 3-37CIKM, 2008
SCS CMU
SM: The block-form A
D
B
C1 1 1 1 1 1 1 1 1
1 1 1 1 1
( ) ( )( ) ( )
A B A A B D CA B CA A B D CA BC D D CA B CA D CA B
− − − − − − − − −
− − − − −
⎡ ⎤+ − − −⎡ ⎤= ⎢ ⎥⎢ ⎥ − − −⎢ ⎥⎣ ⎦ ⎣ ⎦
Or1 1 1 1 1 1
1 1 1 1 1
( ) ( )( ) ( )
A B A BD C A B D CA BC D D C A BD C D CA B
− − − − − −
− − − − −
⎡ ⎤− − −⎡ ⎤= ⎢ ⎥⎢ ⎥ − − −⎢ ⎥⎣ ⎦ ⎣ ⎦
And many other variants…Also known as Woodburg Identity
Or…
Copyright: Faloutsos, Tong (2008) 3-38CIKM, 2008
SCS CMU
SM Lemma: Applications
• RLS (Recursive least square)– and almost any algorithm in time series!
• Leave-one-out cross validation for LSR• Kalman filtering• Incremental matrix decomposition• …• … and all the fast sol.s we will introduce!
Copyright: Faloutsos, Tong (2008) 3-39CIKM, 2008
SCS CMU
Computing RWR
9 1012
0.13 0 1/3 1/3 1/3 0 0 0 0 0 0 0 00.10 1/3 0 1/3 0 0 0 0 1/4 0 0 0 0.13
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
01/3 1/3 0 1/3 0 0 0 0 0 0 0 0
⎛ ⎞⎜⎜⎜⎜
0.13 00.10 00.13 0
⎛ ⎞ ⎛ ⎞⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟
Ranking vector Starting vectorAdjacency matrix
(1 )i i ir cWr c e= + −Restart p
1
43
2
5 6
7
811
120.220.130.05
0.90.050.080.040.030.040.02
⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ = ×⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1/3 0 1/3 0 1/4 0 0 0 0 0 0 00 0 0 1/3 0 1/2 1/2 1/4 0 0 0 00 0 0 0 1/4 0 1/2 0 0 0 0 0 0 0 0 0 1/4 1/2 0 0 0 0 0 0 0 1/3 0 0 1/4 0 0 0 1/2 0 1/3 00 0 0 0 0 0 0 1/4 0 1/3 0 00 0 0 0 0 0 0 0 1/2 0 1/3 1/2 0 0 0 0 0 0 0 1/4 0 1/3 0 1/2 0 0 0 0 0 0 0 0 0 1/3 1/3 0
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎠
0.220.13 00.05 0
0.10.05 00.08 00.04 00.03 00.04 0
2 0
1
0.0
⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟+ ×⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
n x n n x 1n x 1
1
Copyright: Faloutsos, Tong (2008) 3-40CIKM, 2008
SCS CMU
0 1/3 1/3 1/3 0 0 0 0 0 0 0 01/3 0 1/3 0 0 0 0 1/4 0 0 0 01/3 1/3 0 1/3 0 0 0 0 0 0 0 01/3 0 1/3 0 1/4 0 0 0 0 0 0 00 0 0 1/3 0 1/2 1/2 1/4 0 0 0 00 0 0 0 1/4 0 1/2 0 0 0 0 0
000
00
1
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
Q: Given query i, how to solve it?
??
Query
0.9= ×0 0 0 0 1/4 0 1/2 0 0 0 0 0
0 0 0 0 1/4 1/2 0 0 0 0 0 0 0 1/3 0 0 1/4 0 0 0 1/2 0 1/3 00 0 0 0 0 0 0 1/4 0 1/3 0 00 0 0 0 0 0 0 0 1/2 0 1/3 1/2
0 0 0 0 0
00.1
0000
0 0 1/4 0 1/3 0 1/2 0 0 0 0 0 0 0 0 0 0 1/3 1/3 0 0
⎜ ⎟ ⎜ ⎟+ ×⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
??
Adjacency matrix Starting vectorRanking vectorRanking vector
Copyright: Faloutsos, Tong (2008) 3-41CIKM, 2008
SCS CMU
14
32
6
9 10
8 11120.13
0.10
0.13
0 05
0.08
0.04
0.02
0.04
0.03
OntheFly: 0 1/3 1/3 1/3 0 0 0 0 0 0 0 01/3 0 1/3 0 0 0 0 1/4 0 0 0 01/3 1/3 0 1/3 0 0 0 0 0 0 0 01/3 0 1/3 0 1/4
0.9= ×
0 0 0 0 0 0 00 0 0 1/3 0 1/2 1/2 1/4 0 0 0 00 0 0 0 1/4 0 1/2 0 0 0 0 0
0 0 0 0 1/4 1/2 0 0 0 0 0 0 0 1/3 0 0 1/4 0 0 0 1/2 0 1/3 00 0 0 0 0 0 0 1/4 0 1/3 0 0
000
00
0.1000
1
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
+ ×⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
000100000
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
0.130.100.130.220.130.050.050.080.04
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
1
43
2
6
9 10
8 11
12
5 6
70.13
0.05
0.05
0 0 0 0 0 0 0 0 1/2 0 1/3 1/2 0 0 0 0 0
00 0 1/4 0 1/3 0 1/2 0
0 0 0 0 0 0 0 0 0 1/3 1/3 0 0
⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
000
⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
0.030.040.02
⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
5 6
7
No pre-computation/ light storage
Slow on-line response O(mE)
ir ir
Copyright: Faloutsos, Tong (2008) 3-42CIKM, 2008
Faloutsos, Tong CIKM, 2008
8
SCS CMU
0.20 0.13 0.14 0.13 0.68 0.56 0.56 0.63 0.44 0.35 0.39 0.340.28 0.20 0.13 0.96 0.64 0.53 0.53 0.85 0.60 0.48 0.53 0.450.14 0.13 0.20 1.29 0.68 0.56 0.56 0.63 0.44 0.35 0.39 0.330.13 0.10 0.13 2.06 0.95 0.78 0.78 0.61 0.43 0.34 0.38 0.320.09 0.09 0.09 1.27 2.41 1.97 1.97 1.05 0.73 0.58 0.66 0.560.03 0.04 0.04 0.52 0.98 2.06 1.37 0.43 0.30 0.24 0.27 0.22
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
4
PreCompute1 2 3 4 5 6 7 8 9 10 11 12r r r r r r r r r r r r
1 32
6
9 10
8 11
12
R:0.03 0.04 0.04 0.52 0.98 1.37 2.06 0.43 0.30 0.24 0.27 0.220.08 0.11 0.04 0.82 1.05 0.86 0.86 2.13 1.49 1.19 1.33 1.130.03 0.04 0.03 0.28 0.36 0.30 0.30 0.74 1.78 1.00 0.76 0.790.04 0.04 0.04 0.34 0.44 0.36 0.36 0.89 1.50 2.45 1.54 1.800.04 0.05 0.04 0.38 0.49 0.40 0.40 1.00 1.14 1.54 2.28 1.720.02 0.03 0.02 0.21 0.28 0.22 0.22 0.56 0.79 1.20 1.14 2.05
⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
5 6
7
[Haveliwala+ 2002]c x Q
Q Copyright: Faloutsos, Tong (2008) 3-43CIKM, 2008
SCS CMU
2.20 1.28 1.43 1.29 0.68 0.56 0.56 0.63 0.44 0.35 0.39 0.341.28 2.02 1.28 0.96 0.64 0.53 0.53 0.85 0.60 0.48 0.53 0.451.43 1.28 2.20 1.29 0.68 0.56 0.56 0.63 0.44 0.35 0.39 0.331.29 0.96 1.29 2.06 0.95 0.78 0.78 0.61 0.43 0.34 0.38 0.320.91 0.86 0.91 1.27 2.41 1.97 1.97 1.05 0.73 0.58 0.66 0.560.37 0.35 0.37 0.52 0.98 2.06 1.37 0.43 0.30 0.24 0.27 0.22
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
PreCompute:
14
32
6
9 10
8 11120.13
0.10
0.13
0.08
0.04
0.02
0.04
0.03
0.37 0.35 0.37 0.52 0.98 1.37 2.06 0.43 0.30 0.24 0.27 0.220.84 1.14 0.84 0.82 1.05 0.86 0.86 2.13 1.49 1.19 1.33 1.130.29 0.40 0.29 0.28 0.36 0.30 0.30 0.74 1.78 1.00 0.76 0.790.35 0.48 0.35 0.34 0.44 0.36 0.36 0.89 1.50 2.45 1.54 1.800.39 0.53 0.39 0.38 0.49 0.40 0.40 1.00 1.14 1.54 2.28 1.720.22 0.30 0.22 0.21 0.28 0.22 0.22 0.56 0.79 1.20 1.14 2.05
⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
5 6
70.13
0.05
0.05
Fast on-line response
Heavy pre-computation/storage costO(n ) O(n )3 2 3-44
SCS CMU
Q: How to Balance?
On-line Off-line
Copyright: Faloutsos, Tong (2008) 3-45CIKM, 2008
SCS CMU
B_Lin: Basic Idea[Tong+ ICDM 2006]
1 32
9 10
8 11120.13
0.10
0.13
0.08
0.04
0.02
0 04
0.03
13
29 10
811
12
Find Community 14
32
5 6
7
9 10
8 1112
9 10
811
12
14
32
5 6
7
9 10
8 1112
13
2
45 6
70.13
0.05
0.05
0.044
5 6
7
Fix the remaining
Combine1
43
2
5 6
7
9 10
8 1112
5 6
7
14
32
5 6
7
9 10
8 1112
4
3-46
SCS CMU
+~~
B_Lin: details
W~
details
W 1: within community~
Cross-community
Copyright: Faloutsos, Tong (2008) 3-47CIKM, 2008
SCS CMU
B_Lin: details
W~I – c ~~ I – c – U S VW1~
-1 -1
details
Easy to be inverted LRA difference
SM Lemma!
Copyright: Faloutsos, Tong (2008) 3-48CIKM, 2008
Faloutsos, Tong CIKM, 2008
9
SCS CMU
B_Lin: summary
• Pre-Computational Stage• Q: • A: A few small, instead of ONE BIG, matrices inversions
Efficiently compute and store Q
• On-Line Stage• Q: Efficiently recover one column of Q• A: A few, instead of MANY, matrix-vector multiplications
Copyright: Faloutsos, Tong (2008) 3-49CIKM, 2008
SCS CMU
Query Time vs. Pre-Compute Time
Log Query Time
•Quality: 90%+ •On-line:
Log Pre-compute Time
•Up to 150x speedup•Pre-computation:
•Two orders saving
Copyright: Faloutsos, Tong (2008) 3-50CIKM, 2008
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions
• B_Lin: RWR
• BB_Lin: Skewed BGs
• Applications• Conclusion
Copyright: Faloutsos, Tong (2008) 3-51CIKM, 2008
• FastUpdate: Time-Evolving
SCS CMU
RWR on Bipartite Graph
n
authorsAuthor-Conf. Matrix
Observation: n >> m!
Examples: n
mConferences
p1. DBLP: 400k aus, 3.5k confs2. NetFlix: 2.7M usrs,18k mvs
Copyright: Faloutsos, Tong (2008) 3-52CIKM, 2008
SCS CMU
• Q: Given query i, how to solve it?
0 1/3 1/3 1/3 0 0 0 0 0 0 0 01/3 0 1/3 0 0 0 0 1/4 0 0 0 01/3 1/3 0 1/3 0 0 0 0 0 0 0 01/3 0 1/3 0 1/4 0 0 0 0 0 0 0
0001
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
RWR on Skewed bipartite graphs
… .. .
. .
m confs
1/3 0 1/3 0 1/4
0.9= ×
0 0 0 0 0 0 00 0 0 1/3 0 1/2 1/2 1/4 0 0 0 00 0 0 0 1/4 0 1/2 0 0 0 0 0
0 0 0 0 1/4 1/2 0 0 0 0 0 0 0 1/3 0 0 1/4 0 0 0 1/2 0 1/3 00 0 0 0 0 0 0 1/4 0 1/3 0 00 0 0 0 0 0 0 0 1/2 0 1/3 1/2
0 0 0 0 0
00
0.10000
0 0 1/4 0 1/3 0 1/2 0 0 0 0 0 0 0 0 0 0 1/3 1/3
1
0 0
⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
+ ×⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
??…. .. . . …. ...
0
0
n m
Ar
… .
. .. . …. .. . . …. ... Ac n aus
3-53
SCS CMU
• Step 1:
BB_Lin: Pre-Computation dd[Tong+ ICDM 06]
M = AcArX
2-step RWR for Conferences
m conferences
details
n authorsCopyright: Faloutsos, Tong (2008) 3-54CIKM, 2008
Faloutsos, Tong CIKM, 2008
10
SCS CMU
• Step 1:
• Step 2:
M = AcArX
1( 0 9 )I M −Λ = ×
2-step RWR for Conferences
All Conf-Conf Prox. Scores
m conferences
BB_Lin: Pre-Computation dd[Tong+ ICDM 06] details
• Step 2: ( 0.9 )I MΛ = − ×
n authorsCopyright: Faloutsos, Tong (2008) 3-55CIKM, 2008
SCS CMU
• Step 1:
• Step 2:
M = Ac ArX
1( 0 9 )I M −Λ = ×
2-step RWR for Conferences
All Conf-Conf Prox. Scores
BB_Lin: Pre-Computation dd[Tong+ ICDM 06] details
• Step 2:
• Cost:
• Examples – NetFlix: 1.5hr for pre-computation; – DBLP: 1 few minutes
( 0.9 )I MΛ = − ×3( )O m m E+ ⋅
Ac/ArE edges
m x m
SCS CMU
BB_Lin: On-Line Stage
(Base) Case 1: - Conf - Conf
authors
Conferences
Read out !
Ac/ArE edges 3-57CIKM, 2008
SCS CMU
BB_Lin: On-Line Stage
Case 2: - Au - Conf
authors
Conferences
Ac/ArE edges
1 matrix-vec!
3-58CIKM, 2008
SCS CMU
BB_Lin: On-Line Stage
Case 3: - Au - Au
authors
Conferences
Ac/ArE edges
2 matrix-vec!
3-59CIKM, 2008
SCS CMU
BB_Lin: Examples
Dataset Off-Line Cost On-Line CostDBLP a few minutes frac. of sec.
NetFlix 1.5 hours <0.01 sec.
400k authors x 3.5k conf.s 2.7m user
x 18k movies
Copyright: Faloutsos, Tong (2008) 3-60CIKM, 2008
Faloutsos, Tong CIKM, 2008
11
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions
• B_Lin: RWR
• BB_Lin: Skewed BGs
• Applications• Conclusion
Copyright: Faloutsos, Tong (2008) 3-61CIKM, 2008
• FastUpdate: Time-Evolving
SCS CMU
Challenges
• BB_Lin is good for skewed bipartite graphs– for NetFlix (2.7M nodes and 100M edges)– On-line cost for query: fraction of seconds
• w/ 1.5 hr pre-computation for m x m core matrix
• But…what if the graph is evolving over time– New edges/nodes arrive; edge weights increase…– On-line cost: 1.5hr itself becomes a part of this!
Copyright: Faloutsos, Tong (2008) 3-62CIKM, 2008
SCS CMU
1( 0.9 )I M −Λ = − × 3( )O m m E+ ⋅t=0
Q: How to update the core matrix?
t=11( 0.9 )I M −Λ = − ×
~ ~ 3( )O m m E+ ⋅
?Copyright: Faloutsos, Tong (2008) 3-63CIKM, 2008
SCS CMU
Update the core matrix• Step 1:
M = Ac ArX~ M= X+
details
• Step 2:1( 0.9 )I M −Λ = − ×~ ~ Rank 2 update
= + X
2( )O m 3( )O m m E+ ⋅ 3-64CIKM, 2008
SCS CMU
Update : General Case
M = AcArX~
n authors
m Conferences
details
• E’ edges changed• Involves n’ authors, m’ confs.
• Observationmin( ', ') 'n m E
Copyright: Faloutsos, Tong (2008) 3-65CIKM, 2008
SCS CMU
• Observation: – the rank of update is small!– Real Example (DBLP Post)
• 1258 time steps
Update : General Case
min( ', ') 'n m E n authors
m Conferences
details
• 1258 time steps• E’ up to ~20,000!• min(n’,m’) <=132
• Our Algorithm2(min( ', ') ')O n m m E+
3( )O m m E+ ⋅ 663-66
Faloutsos, Tong CIKM, 2008
12
SCS CMU
Fast-Single-Update
176x speeduplog(Time) (Seconds)
40x speedup
Datasets
Our method
Our method
Copyright: Faloutsos, Tong (2008)
SCS CMU
Fast-Batch-UpdateTime (Seconds)Time (Seconds)
Min (n’, m’)E’
15x speed-up on average!
Our method Our method
Copyright: Faloutsos, Tong (2008) 3-68CIKM, 2008
SCS CMU
More on “Fast Solutions”
• FastAllDAP– Simultaneously solve multiple linear systems– [Tong+ KDD 2007 a]
• MT3– Multiple-Resolution Analysis on Time [Tong+ CIKM 2008]
• Fast-ProSIN– On-Line response for users’ feedback [Tong+ ICDM 2008]
Copyright: Faloutsos, Tong (2008) 3-69CIKM, 2008
SCS CMU
Summary of Fast Solutions
• Goal: Efficiently Solve Linear System(s)
• Sols.– B_Lin: one large linear system [Tong+ ICDM06]
– BB_Lin: the intrinsic complexity is small [Tong+ ICDM06]
– FastUpdate: dynamic linear system [Tong+ SDM08]– FastAllDAP: multiple linear systems [Tong+ KDD07 a]– MT3: [Tong+ CIKM 2008]
– Fast-ProSIN: [Tong+ 2008]Copyright: Faloutsos, Tong (2008) 3-70CIKM, 2008
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions• Applications• Conclusion
Copyright: Faloutsos, Tong (2008) 3-71CIKM, 2008
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions
• Link Prediction & +
• Ranking Related Tasks• Applications• Conclusion
Copyright: Faloutsos, Tong (2008) 3-72CIKM, 2008
• Ranking Related Tasks
• User Specific Patterns
• Time Related Tasks
Faloutsos, Tong CIKM, 2008
13
SCS CMU
0.2
0.25
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18 Link Prediction:existence
Prox. Hist. for a set of deleted links
density
density
Prox (i j)+Prox (j i)
Prox. is effective to ‘deleted’ and absent edges!
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090
0.05
0.1
0.15
Prox (i j)+Prox (j i)
Q: How to predict the existence of the link?A: Proximity! [Liben-Nowell + 2003]
Prox. Hist. for a set of absent links
Copyright: Faloutsos, Tong (2008) 3-73CIKM, 2008
SCS CMU
Link Prediction: direction [Tong+ KDD 07 a]
• Q: Given the existence of the link, what is the direction of the link?
• A: Compare prox(i j) and prox(j i)>70%>70%
Prox (i j) - Prox (j i)
density
CIKM, 2008
SCS CMU
Beyond Link Prediction
• Collaborative Filtering [Fouss+]
• Name Disambiguation – [Minkov+ SIGIR 06]
• Anomaly Nodes/Edges – ‘a’ is abnormal if the neighborhood of ‘a’ is so different– [Sun+ ICDM 2005]
Copyright: Faloutsos, Tong (2008) 3-75CIKM, 2008
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions
• Link Prediction & +
• Ranking Related Tasks• Applications• Conclusion
Copyright: Faloutsos, Tong (2008) 3-76CIKM, 2008
• Ranking Related Tasks
• User Specific Patterns
• Time Related Tasks
SCS CMU Neighborhood Search on graphs
ICDM
KDD
SDM
Philip S. Yu
IJCAI
AAAI M. Jordan
Ning Zhong
R. Ramakrishnan
… …
NIPS…
…
Conference Author
A: Proximity! [Sun+ ICDM2005]
Q: what is most related conference to ICDM?
Copyright: Faloutsos, Tong (2008)CIKM, 2008
SCS CMU
NF: example
ICDM
KDD
SDM
PKDD
PAKDD
ICML0.009
0.011
0.0080.007
0.005
ICDM
ECML
CIKM
DMKD
SIGMOD
ICDE0.005
0.0050.004
0.004
0.004
Copyright: Faloutsos, Tong (2008) 3-78CIKM, 2008
Faloutsos, Tong CIKM, 2008
14
SCS CMU
gCaP: Automatic Image Caption• Q
…
Sea Sun Sky Wave{ } { }Cat Forest Grass Tiger
{?, ?, ?,}
?A: Proximity!
[Pan+ KDD2004]
Copyright: Faloutsos, Tong (2008) 3-79CIKM, 2008
SCS CMU
Image
Region
Test Image
Sea Sun Sky Wave Cat Forest Tiger Grass
Image
KeywordCopyright: Faloutsos, Tong (2008) 3-80
SCS CMU
Image
Region
Test Image
Sea Sun Sky Wave Cat Forest Tiger Grass
Image
KeywordCopyright: Faloutsos, Tong (2008) 3-81
SCS CMU
C-DEM: Multi-Modal Query System for DrosophilaEmbryo Databases [Fan+ VLDB 2008]
Copyright: Faloutsos, Tong (2008) 3-82CIKM, 2008
SCS CMUC-DEM (Screen-shot)
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions
• Link Prediction & +
• Ranking Related Tasks• Applications• Conclusion
Copyright: Faloutsos, Tong (2008) 3-84CIKM, 2008
• Ranking Related Tasks
• User Specific Patterns
• Time Related Tasks
Faloutsos, Tong CIKM, 2008
15
SCS CMU
Center-Piece Subgraph(CePS)
B
CePS guy
Input Output
A C
Original Graph CePS
Q: How to find hub for the black nodes?A: Proximity! [Tong+ KDD 2006]
Red: Max (Prox(A, Red) x Prox(B, Red) x Prox(C, Red))
SCS CMU
CePS: Example
R. Agrawal Jiawei Han
H.V. Jagadish
Laks V.S. Lakshmanan
Heikki
15 10 13
1 110
V. Vapnik M. Jordan
Mannila
Christos Faloutsos
Padhraic Smyth
Corinna Cortes
1 1
6
1 1
4 Daryl Pregibon
2
11 3
16
Copyright: Faloutsos, Tong (2008) 3-86CIKM, 2008
SCS CMU
K_SoftAnd: Relaxation of AND
Asking AND query? No Answer!
Disconnected Communities
Noise
Copyright: Faloutsos, Tong (2008) 3-87CIKM, 2008
SCS CMU
R. Agrawal Jiawei Han
H.V. Jagadish
Laks V.S. Lakshmanan
Umeshwar D l
1510 13
3 3
CePS: 2 SoftANDDB
V. Vapnik M. Jordan
Dayal
Bernhard Scholkopf
Peter L. Bartlett
Alex J. Smola
5 2 2
327
4
Stat.
SCS CMU
OutputInput
Data Graph
Query Graph
Best-Effort Pattern Match
Matching Subgraph
Q: How to find matching subgraph?A: Proximity![Tong+ KDD 2007 b]
CIKM, 2008
SCS CMU
G-Ray: How to?
matching nodematching node
matching node
details
matching node
Goodness = Prox (12, 4) x Prox (4, 12) xProx (7, 4) x Prox (4, 7) x
Prox (11, 7) x Prox (7, 11) xProx (12, 11) x Prox (11, 12)
Copyright: Faloutsos, Tong (2008) 3-90CIKM, 2008
Faloutsos, Tong CIKM, 2008
16
SCS CMU
Effectiveness: star-query
Query ResultCopyright: Faloutsos, Tong (2008) 3-91CIKM, 2008
SCS CMU
Effectiveness: line-query
Query
ResultCopyright: Faloutsos, Tong (2008) 3-92CIKM, 2008
SCS CMU
Query
Effectiveness: loop-query
Result
3-93CIKM, 2008
SCS CMU
Outline: Part 3
• Motivation• Definitions• Fast Solutions
• Link Prediction & +
• Ranking Related Tasks• Applications• Conclusion
Copyright: Faloutsos, Tong (2008) 3-94CIKM, 2008
• Ranking Related Tasks
• User Specific Patterns
• Time Related Tasks
SCS CMU
Challenge• Graphs are evolving over time!
–New nodes/edges show up; –Existing nodes/edges die out;
Ed i ht h–Edge weights change…
Q: How to generalize everything? A: Track Proximity! [Tong+ SDM 2008]
Copyright: Faloutsos, Tong (2008) 3-95CIKM, 2008
SCS CMU
pTrack/cTrack: Trend analysis on graph level
G.Hinton
T. SejnowskiRank of Influential-ness
M. Jordan
C. Koch
Year
Copyright: Faloutsos, Tong (2008) 3-96CIKM, 2008
Faloutsos, Tong CIKM, 2008
17
SCS CMU
pTrack: Problem Definition
• [Given]– (1) a large, skewed time-evolving bipartite
graphs, – (2) the query nodes of interest
3-97
• [Track]– (1) top-k most related nodes for each query
node at each time step t; – (2) the proximity score (or rank of proximity)
between any two query nodes at each time step t
Copyright: Faloutsos, Tong (2008)CIKM, 2008
SCS CMU
pTrack: Philip S. Yu’s Top-5 conferences up to each year
ICDEICDCS
SIGMETRICSPDISVLDB
CIKMICDCSICDE
SIGMETRICSICMCS
KDDSIGMOD
ICDMCIKM
ICDCS
ICDMKDDICDESDMVLDB
1992 1997 2002 2007
DatabasesPerformanceDistributed Sys.
DatabasesData Mining
DBLP: (Au. x Conf.)- 400k aus, - 3.5k confs - 20 yrs
Copyright: Faloutsos, Tong (2008) 3-98CIKM, 2008
SCS CMU
KDD’s Rank wrt. VLDB over years
Prox.Rank
Year
Data Mining and Databases are more and more relavant!
CIKM, 2008 3-99
SCS CMU
cTrack:10 most influential authors in NIPS community up to each year
T. Sejnowski
Author-paper bipartite graph from NIPS 1987-1999. 3k. 1740 papers, 2037 authors, spreading over 13 years
M. Jordan
3-100
SCS CMU
T3: Understand Time in Complex Context [Tong+ CIKM 2008]
Time Cluster, rep. entities: b7,b6, b8
Abnormal Time
Time Event Entityt1 e1 b1, b2
e2 b2, b3t2 e3 b2, b3
e b b
Input Output
rep. entities: b5,b4
Time Clusterrep. entities: b3, b2, b1
e4 b3, b4t3 e5 b4, b5t4 e6 b5, b6
e7 b6, b7t5 e8 b6, b7t6 e9 b7, b8
Copyright: Faloutsos, Tong (2008) 3-101CIKM, 2008
SCS CMU
T3: Time-to-Time Proximity Matrix
Time Event Entityt1 e1 b1, b2
e2 b2, b3t2 e3 b2, b3
e b be4 b3, b4t3 e5 b4, b5t4 e6 b5, b6
e7 b6, b7t5 e8 b6, b7t6 e9 b7, b8
Copyright: Faloutsos, Tong (2008) 3-102CIKM, 2008
Faloutsos, Tong CIKM, 2008
18
SCS CMU
More Applications• Clustering
– Proximity as input [Ding+ KDD 2007]• Email management [Minkov+ CEAS 06].
• Business Process Management [Qu+ 2008]
• ProSINProSIN– Listen to clients’ comments [Tong+ 2008]
• Ghost Edge– Within Network Classification [Gallagher & Tong+ KDD08 b]
• …
Copyright: Faloutsos, Tong (2008) 3-103
SCS CMU
gCap:
pTrack/cTrack:
LP: [Liben-Nowell+][Tong+ 2007]
NF:
CePS:
G-Ray:
T3:GhostEdge:
[Gallagher &
[Tong+ 2007]
[Tong+ 2006]
[Pan+ 2004]
[Pan+ 2005]
[Tong+ 2008]
FastAllDAP: [Tong+ 2007]
BB Lin: [Tong+ 2006]
FastUpdate: [Tong+ 2008]
ComputationsApplications
EfficientlySolve Linear
System(s)
MT3: [Tong+ 2008]Use Proximity as
Building block(gCap, CePS…)
Fast-ProSIN: [Tong+ 2008]
T3: & Tong+ 2008][Tong+ 2008]
A BH1 1
D1 1
EF
G1 11
I J11 1 RWR: [Pan+ 2004][Sun+ 2006][Tong+ 2006]
Variants: [Faloutsos+ 2004] [Koren+ 2006][Tong+ 2007]
Generalizations: [Charkrabarti+ 2006][Tong+ 2007, 2008]
Properties.: [Koren+ 2006]]Tong+ 2007, 2008]
Definitions
B_Lin: [Tong+ 2006]
BB_Lin: [Tong+ 2006]
ProximityOn
Graphs
Weighted Multiple
Relationship
System(s)
SCS CMU
Take-home messages• Proximity Definitions
– RWR– and a lot of variants
Computations• Computations– Sherman–Morrison Lemma– Fast Incremental Computation
• Applications– Proximity as a building block
Copyright: Faloutsos, Tong (2008) 3-105CIKM, 2008
SCS CMU
References• L. Page, S. Brin, R. Motwani, & T. Winograd. (1998), The
PageRank Citation Ranking: Bringing Order to the Web, Technical report, Stanford Library.
• T.H. Haveliwala. (2002) Topic-Sensitive PageRank. In WWW, 517-526, 2002
• J Y Pan H J Yang C Faloutsos & P Duygulu (2004) Automatic• J.Y. Pan, H.J. Yang, C. Faloutsos & P. Duygulu. (2004) Automatic multimedia cross-modal correlation discovery. In KDD, 653-658, 2004.
• C. Faloutsos, K. S. McCurley & A. Tomkins. (2002) Fast discovery of connection subgraphs. In KDD, 118-127, 2004.
• J. Sun, H. Qu, D. Chakrabarti & C. Faloutsos. (2005) Neighborhood Formation and Anomaly Detection in Bipartite Graphs. In ICDM, 418-425, 2005.
• W. Cohen. (2007) Graph Walks and Graphical Models. Draft.Copyright: Faloutsos, Tong (2008) 3-106CIKM, 2008
SCS CMU
References• P. Doyle & J. Snell. (1984) Random walks and electric
networks, volume 22. Mathematical Association America, New York.
• Y. Koren, S. C. North, and C. Volinsky. (2006) Measuring and extracting proximity in networks. In KDD, 245–255, 2006.
• A Agarwal S Chakrabarti & S Aggarwal (2006) Learning to• A. Agarwal, S. Chakrabarti & S. Aggarwal. (2006) Learning to rank networked entities. In KDD, 14-23, 2006.
• S. Chakrabarti. (2007) Dynamic personalized pagerank in entity-relation graphs. In WWW, 571-580, 2007.
• F. Fouss, A. Pirotte, J.-M. Renders, & M. Saerens. (2007) Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation. IEEE Trans. Knowl. Data Eng. 19(3), 355-369 2007.
Copyright: Faloutsos, Tong (2008) 3-107CIKM, 2008
SCS CMU
References• H. Tong & C. Faloutsos. (2006) Center-piece subgraphs:
problem definition and fast solutions. In KDD, 404-413, 2006.• H. Tong, C. Faloutsos, & J.Y. Pan. (2006) Fast Random Walk
with Restart and Its Applications. In ICDM, 613-622, 2006.• H. Tong, Y. Koren, & C. Faloutsos. (2007) Fast direction-
aware proximity for graph mining In KDD 747 756 2007aware proximity for graph mining. In KDD, 747-756, 2007.• H. Tong, B. Gallagher, C. Faloutsos, & T. Eliassi-Rad. (2007)
Fast best-effort pattern matching in large attributed graphs. In KDD, 737-746, 2007.
• H. Tong, S. Papadimitriou, P.S. Yu & C. Faloutsos. (2008) Proximity Tracking on Time-Evolving Bipartite Graphs. to appear in SDM 2008.
Copyright: Faloutsos, Tong (2008) 3-108CIKM, 2008
Faloutsos, Tong CIKM, 2008
19
SCS CMU
References• B. Gallagher, H. Tong, T. Eliassi-Rad, C. Faloutsos. Using
Ghost Edges for Classification in Sparsely Labeled Networks. KDD 2008
• H. Tong, Y. Sakurai, T. Eliassi-Rad, and C. Faloutsos. Fast Mining of Complex Time-Stamped Events CIKM 08
• H Tong H Qu and H Jamjoom Measuring Proximity on• H. Tong, H. Qu, and H. Jamjoom. Measuring Proximity on Graphs with Side Information. ICDM 2008
Copyright: Faloutsos, Tong (2008) 3-109CIKM, 2008