Proximity on Graphs - Carnegie Mellon School of Computer...

H. Tong 15-826

Guest-lecture 1

SCS CMU

Proximity on Graphs-Definitions, Fast Solutions, and Applications

Hanghang [email protected]

SCS CMU

-----

Graphs are everywhere!

Internet Map [Koren 2009] Food Web [2007] Terrorist Network

2

p [ ] [ ]

Protein Network [Salthe 2004]

Social Network [Newman 2005]

Web Graph

[Krebs 2002]

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! 826 guest lecture 3

H. Tong 15-826

Guest-lecture 2

SCS CMU

Proximity on Graph: What?

a.k.a Relevance, Closeness, ‘Similarity’…826 guest lecture 4

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 later826 guest lecture 5

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


Q: How to predict the existence of the link?A: Proximity! [Liben-Nowell + 2003]

Prox. Hist. for a set of absent links

826 guest lecture 6

H. Tong 15-826

Guest-lecture 3

SCS CMU Neighborhood Search on graphs

… …

…

…

Conference Author

A: Proximity! [Sun+ ICDM2005]

Q: what is most related conference to ICDM?

826 guest lecture 7

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]

826 guest lecture 8

SCS CMU

A A

Center-Piece Subgraph Discovery

Input: original graph Output: CePS

CePS Node

3-9

B C B C

Q: How to find hub for the black nodes?A: Proximity! [Tong+ KDD 2006]

H. Tong 15-826

Guest-lecture 4

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]826 guest lecture

SCS CMU

Outline

• Motivation• Definitions• Fast Solutions

• Basic: RWR

• Variants

• Properties

• Applications• Conclusion

• Generalizations

826 guest lecture 11

SCS CMU

Why not shortest path?Some ``bad’’ proximities

‘pizza delivery guy’ problem

A BD1 1

‘multi-facet’ relationship


H. Tong 15-826

Guest-lecture 5

SCS CMU

Why not max. netflow?Some ``bad’’ proximities

No punishment on long paths


SCS CMU

Random Walk with Restart Node 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

4r14

SCS CMU

RWR: Think of it as Wine Spill

15

1. Spill a drop of wine on cloth 2. Spread/diffuse to the neighborhood

H. Tong 15-826

Guest-lecture 6

SCS CMU

13

2

910

811

12

RWR: Wine Spill on a Graph

16

4

3

56

7

11

wine spill on cloth RWR on a graph

Query

Same Diffusion Eq.

SCS CMU

13

2

910

811

12

Random Walk with Restart

4

3

56

7

17Same Diffusion Eq.

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




H. Tong 15-826

Guest-lecture 7

SCS CMU

Why RWR is A Good Score?

Prox (A B) =A B

…

High proximity many, short, heavy-weighted paths

Prox (A, B) = Score (Red Path) + Score (Green Path) + Score (Blue Path) +Score (Purple Path) +

…

19

SCS CMU

Outline


• Basic: RWR

• Variants

• Properties


• Generalizations


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

Esc_Prob = Pr (smile before cry) 21

H. Tong 15-826

Guest-lecture 8

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!


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”

Physical Models

SCS CMU

Outline


• Basic: RWR

• Variants

• Properties


• Generalizations


H. Tong 15-826

Guest-lecture 9

SCS CMU

Property: Monotonicity

We want: Prox ( ) Prox ( )candi origianla b a b≤→ →A: degree preserving! [Koren+ KDD06][Tong+ KDD07a][Tong+ SDM08]

826 guest lecture25

SCS CMU

Property: Asymmetry [Tong+ KDD07 a]

What is Prox from A to B?What is Prox from B to A?

What is Prox between A and B?


SCS CMU

Asymmetry in un-directed graphs• Hanghang’s # 1 Conf. is CIKM• The #1 author of CIKM is ...

So is love…

HanghangCIKM

826 guest lecture27

H. Tong 15-826

Guest-lecture 10

SCS CMU

Outline


• Basic: RWR

• Variants

• Properties


• Generalizations


SCS CMU

Group Proximity [Tong+ KDD07 a]

• Q: How close are Accountants to SECs?

826 guest lecture

• A: Prob (starting at any RED, reaches anyGREEN before touching any RED again)

29

SCS CMU

Proximity on Attributed Graphs [Tong+ KDD07 b]

What is the proximity from node 7 to 10?If we know that…

826 guest lecture30

H. Tong 15-826

Guest-lecture 11

SCS CMU

A: Augmented graphs [Tong+ KDD07 b]


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]

• …


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]826 guest lecture

33

H. Tong 15-826

Guest-lecture 12

SCS CMU

Outline

• Motivation• Definitions• Fast Solutions• Applications• Conclusion


SCS CMU

Outline


• B_Lin: RWR

• BB_Lin: Skewed BGs


826 guest lecture

• FastUpdate: Time-Evolving

35

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:


H. Tong 15-826

Guest-lecture 13

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…


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!


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


H. Tong 15-826

Guest-lecture 14

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


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

0.300.30.10.30000

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

0.120.180.120.350.030.070.070.070

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

0.190.090.190.180.180.040.040.060.02

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

0.140.130.140.260.100.060.060.080.01

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

0.160.100.160.210.150.050.050.070.02

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

0.130.100.130.220.130.050.050.080.04

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ 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

0 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

⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

000

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

0.030.040.02

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

5 6

7

000

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

000

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

00.020

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

0.010.010

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

0.010.020.01

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

0.030.040.02

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

No pre-computation/ light storage

Slow on-line response O(mE)

ir ir


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

14

32

6

9 10

8 11120.13

0.10

0.13

0 05

0.08

0.04

0.02

0.04

0.03

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

70.13

0.05

0.05

5 6

7

[Haveliwala+ 2002]c x Q

Q 826 guest lecture

42

H. Tong 15-826

Guest-lecture 15

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

1

43

29 10

8 11

12

0.130.100.130.220.130.050 05

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟0.37 0.35 0.37 0.52 0.98 1.37 2.06 0.43 0.30 0.24 0.27 0.22

0.84 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

5 6

7

Fast on-line response

Heavy pre-computation/storage costO(n ) O(n )

0.050.080.040.030.040.02

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

3 2 43

SCS CMU

Q: How to Balance?

On-line Off-line


SCS CMU

B_Lin: Pre-Compute[Tong+ ICDM 2006]

10

Find Communities

10 1010

Compute & Store Within-Communities Scores

Q33Q11

1

4

3

2

5 6

7

9 10

811

12

5 6

7

9 10

811

12

1

4

3

2

45

1

4

3

2

5 6

7

9 10

811

12

5 6

7

9 10

811

12

1

4

3

2

Q22

Q11

H. Tong 15-826

Guest-lecture 16

SCS CMU

B_Lin: On-Line[Tong+ ICDM 2006]

13

29 10

811

120.130.10

0.13

0.08

0.04

0.02

0 04

0.03

13

2

9 10

811

12

Find Communities

9 10

811

12

1

43

2

13

2

1

43

2

4

5 6

70.13

0.05

0.05

0.044

5 6

7

Estimate the impact of the bridges

Combine1

43

2

5 6

7

9 10

811

12

5 6

7

1

43

2

5 6

7

9 10

811

12

4

46

SCS CMU

B_Lin: details

+W =

W1,1

W2,2

W2,2

+~~

W1: within community Cross community 47

W1,1

W2,2

W2,2

U

S V

SCS CMU

B_Lin: details

-1 -1

W1,1

W2,2

W2,2

U

S V

WI – c ~~ I – c – cUSVW1

Easy to be inverted LRA difference

48

H. Tong 15-826

Guest-lecture 17

SCS CMU

B_Lin: details

-1 -1

W1,1

W2,2

W2,2

U

S V

WI – c ~~ I – c – cUSVW1

Easy to be inverted LRA difference

Sherman–Morrison Lemma: If B=A-USVThen B-1 = A-1 + A-1U(S-1-VA-1U)-1VA-1

A USV

SCS CMU

B_Lin: Pre-Compute Stage• Q: Efficiently compute and store Q• A: A few small, instead of ONE BIG, inverted

matrices Proximities within communities Effect of bridges

50Footnote: Q1=(I-cW1)-1

U

VQ1,1

Q2,2

Qk,k

>Q1,1

Q2,2

Qk,k

. . .Q1=

SCS CMU

B_Lin: On-Line Stage• Q: Efficiently recover one column of Q• A: A few, instead of MANY, matrix-vector

multiplicationsriei

51

U

Vii

>

Pre-computed Query Result51

H. Tong 15-826

Guest-lecture 18

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


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Outline


• B_Lin: RWR



826 guest lecture


53

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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


H. Tong 15-826

Guest-lecture 19

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• 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


SCS CMU

• Step 1:

• Step 2:

BB_Lin: Pre-Computation dd[Tong+ ICDM 06]

M = AcArX

1( 0 9 )I M −Λ = ×

2-step RWR for Conferences

All Conf-Conf Prox. Scores

m conferences

• Step 2:

• Cost:

• Examples – NetFlix: 1.5hr for pre-computation; – DBLP: 1 few minutes

( 0.9 )I MΛ = − ×3( )O m m E+ ⋅

n authors826 guest lecture 56

SCS CMU

• Step 1:

• Step 2:

M = AcArX

1( 0 9 )I M −Λ = ×



m conferences

BB_Lin: Pre-Computation dd[Tong+ ICDM 06] details

• Step 2: ( 0.9 )I MΛ = − ×

n authors826 guest lecture 57

H. Tong 15-826

Guest-lecture 20

SCS CMU

• Step 1:

• Step 2:

M = Ac ArX

1( 0 9 )I M −Λ = ×



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

826 guest lecture58

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BB_Lin: On-Line Stage

(Base) Case 1: - Conf - Conf

authors

Conferences

Read out !

Ac/ArE edges826 guest lecture 59

SCS CMU


Case 2: - Au - Conf

authors

Conferences

Ac/ArE edges

1 matrix-vec!


H. Tong 15-826

Guest-lecture 21

SCS CMU


Case 3: - Au - Au

authors

Conferences

Ac/ArE edges

2 matrix-vec!


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


SCS CMU

Outline


• B_Lin: RWR



826 guest lecture


63

H. Tong 15-826

Guest-lecture 22

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!


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+ ⋅

?826 guest lecture 65

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+ ⋅826 guest lecture 66

H. Tong 15-826

Guest-lecture 23

SCS CMU

Update : General Case

M = AcArX~

n authors

m Conferences

details

• E’ edges changed• Involves n’ authors, m’ confs.

• Observationmin( ', ') 'n m E


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+ ⋅826 guest lecture 68

SCS CMU

176xlog(Time) (Seconds)

Speed Comparison

6969

40x

Data Sets

Our

DBLP Netflix

Our

H. Tong 15-826

Guest-lecture 24

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]


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]826 guest lecture 71

SCS CMU

Outline

• Motivation• Definitions• Fast Solutions• Applications• Conclusion


H. Tong 15-826

Guest-lecture 25

SCS CMU

Outline


• Link Prediction & +

• Ranking Related Tasks• Applications• Conclusion

826 guest lecture

• Ranking Related Tasks

• User Specific Patterns

• Time Related Tasks

73

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i

i i

i

i i

Link Prediction: existence

826 guest lecture

i

i

i

i

Prox (deleted) >> Prox (absent) !Footnote: - Red pair: ``deleted’’; - Blue pair: ``absent’’

74

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. 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


Q: How to predict the existence of the link?A: Proximity! [Liben-Nowell + 2003]

Prox. Hist. for a set of absent links


H. Tong 15-826

Guest-lecture 26

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%

i

j

i

i

i

>70%

Prox (i j) - Prox (j i)

density


SCS CMU

Beyond Link Prediction

• Collaborative Filtering [Fouss+]

• Name Disambiguation H. Tong

C. Faloutsos

J. Pan

P. S. Yu

…

– [Minkov+ SIGIR 06]

• Anomaly Nodes/Edges – ‘a’ is abnormal if the neighborhood of ‘a’ is so different– [Sun+ ICDM 2005]

826 guest lecture

Hanghang TongM.-J. Li

…

77

SCS CMU

Outline




826 guest lecture




78

H. Tong 15-826

Guest-lecture 27

SCS CMU Neighborhood Search on graphs

… …

…

…

Conference Author

A: Proximity! [Sun+ ICDM2005]

Q: what is most related conference to ICDM?


SCS CMU

NF: example


SCS CMU

gCaP: Automatic Image Caption• Q

…

Sea Sun Sky Wave{ } { }Cat Forest Grass Tiger

{?, ?, ?,}

?A: Proximity!

[Pan+ KDD2004]


H. Tong 15-826

Guest-lecture 28

SCS CMU

Image

Region

Test Image


Image

Keyword826 guest lecture 82

SCS CMU

Image

Region

Test Image


Image

Keyword

{Grass, Forest, Cat, Tiger}

83

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C-DEM: Multi-Modal Query System for DrosophilaEmbryo Databases [Guo+ VLDB 2008]


H. Tong 15-826

Guest-lecture 29

SCS CMU

Outline




826 guest lecture




85

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Center-Piece Subgraph Discovery [Tong+ KDD 06]

A

• Given: a graph W, and a query set • Find: the most central node (wrt the query

set)

Q: Who is the most central node wrt the black nodes?

(e.g., master-mind criminal, common advisor/collaborator, etc)

B C

86

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A A

Center-Piece Subgraph Discovery [Tong+ KDD 06]

Input: original graph Output: CePS

CePS Node

3-87

B C B C

Q: How to find hub for nodes A, B, C?Our Solution: Max (Prox(A, Red) x Prox(B, Red) x Prox(C, Red))

H. Tong 15-826

Guest-lecture 30

SCS CMU

?

CePS: Example (AND Query)

88

DBLP co-authorship network: -400,000 authors, 2,000,000 edges

Code at: http://www.cs.cmu.edu/~htong/soft.htm

SCS CMU

CePS: Example (AND Query)

10

10

1 2

1

89

DBLP co-authorship network: -400,000 authors, 2,000,000 edges

Code at: http://www.cs.cmu.edu/~htong/soft.htm

1

1 1

4 6

SCS CMU

K_SoftAnd: Relaxation of AND

Asking AND query? No Answer!

Disconnected Communities

Noise


H. Tong 15-826

Guest-lecture 31

SCS CMU

CePS: 2 SoftANDDB

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]

826 guest lecture

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) xProx (11, 7) x Prox (7, 11) xProx (12, 11) x Prox (11, 12)


H. Tong 15-826

Guest-lecture 32

SCS CMU

Effectiveness: star-query Databases

Query Result826 guest lecture

Bio-medicalIntelligent Agent

94

SCS CMU

Effectiveness: line-queryDatabases Learning Bio-medicalTheory

Query

Result826 guest lecture 95

SCS CMU

Outline




826 guest lecture




96

H. Tong 15-826

Guest-lecture 33

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]


SCS CMU

pTrack/cTrack: Trend analysis on graph level

G.Hinton

T. SejnowskiRank of Influential-ness

M. Jordan

C. Koch

Year


SCS CMU

pTrack: Problem Definition

• [Given]– (1) a large, skewed time-evolving bipartite

graphs, – (2) the query nodes of interest

• [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


H. Tong 15-826

Guest-lecture 34

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


SCS CMU

KDD’s Rank wrt. VLDB over years

Prox.Rank

Year

Data Mining and Databases are getting closer & closer


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

102

H. Tong 15-826

Guest-lecture 35

SCS CMU

T3/MT3: How to understand time?


SCS CMU

A Motivating Example: InputsTime Event(e.g., Session) EntityOct. 26 Link Analysis Tom, Bob

Clustering Bob, AlanOct. 27 Classification Bob, Alan

Anomaly Detection Alan, BeckOct. 28 Party Beck, DanOct. 29 Web Search Dan, Jack

Advertising Jack, PeterOct. 30 Enterprise Search Jack, PeterOct. 31 Q & A Peter, Smith

SCS CMU

Time Cluster, rep. entities: b7,b6, b8A Motivating Example: Outputs

JackOct. 29

Oct. 30Oct. 30

Oct 28

Time ClusterRep. Entities:

``Jack’’, ``Peter’’, ``Smith’’

Abnormal TimeOct. 28

Oct. 26

Oct. 27

Rep. Entities:``Beck’’ ``Dan’’

Time ClusterRep. Entities:

``Tom’’, ``Bob’’,``Alan’’

105

H. Tong 15-826

Guest-lecture 36

SCS CMU

Problem Definitions: (How to Understand Time in such complex context)

• Given datasets collected at different time stamps;

• Find– Q1: Time Cluster – Q2: Abnormal Time stamp– Q3: Interpretations– Q4: Right time granularity


SCS CMU

T3: Basic IdeaOct. 26, 2008

Oct. 27, 2008

Link Analysis

Clustering

Classification

Anomaly Dect.

Tom

Bob

AlanTT

Oct. 28, 2008

Oct. 29, 2008

Oct. 30, 2008

Oct. 31, 2008

Party

Web Search

Advertising

En. Search

Q & A

Beck

Dan

Jack

Peter

Smith

TP

826 guest lecture107

SCS CMU

Data Sets• CIKM: from cikm proceedings

• Time: Publication years (1993-2007, 15)• Event: Paper (952)• Entities: Authors (1895) & Sessions

(279)• Attribute: Keyword (158)


H. Tong 15-826

Guest-lecture 37

SCS CMU

T3 on `CIKM’ Data Set Rep. Authors Rep. Keywords

James. P. CallanW. Bruce CroftJames AllanPhilip S. Yu

George KarypisCharles Clarke

WebCluster

ClassificationXML

LanguageStream

Rep. Authors Rep. KeywordsElke Rundensteiner

Daniel MirankerAndreas Henrich

Il-Yeol SongScott B Huffman

Robert J. Hall

KnowledgeSystem

UnstructuredRule

Object-orientedDeductive 109

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]

• …


SCS CMU

gCap:

pTrack/cTrack:

LP: [Liben-Nowell+][Tong+ 2007]

NF:

CePS:

G-Ray:

T3:GhostEdge:

[Gallagher & 2008

[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]

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)

H. Tong 15-826

Guest-lecture 38

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


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.826 guest lecture 115

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.


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Guest-lecture 39

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.


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


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