CMU SCS
Large Graph Mining:
Power Tools and a Practitioner’s guide
Task 4: Center-piece Subgraphs
Faloutsos, Miller and Tsourakakis
CMU
KDD'09
Faloutsos, Miller, Tsourakakis
P5-1
CMU SCS
Outline
•
•
•
•
•
•
•
•
Introduction – Motivation
Task 1: Node importance
Task 2: Community detection
Task 3: Recommendations
Task 4: Connection sub-graphs
Task 5: Mining graphs over time
…
Conclusions
KDD'09
Faloutsos, Miller, Tsourakakis
P5-2
CMU SCS
Detailed outline
• Problem definition
• Solution
• Results
H. Tong
& C. Faloutsos Center-piece
subgraphs: problem
KDD'09
Faloutsos, Miller, Tsourakakis
definition and fast solutions. In KDD, 404-413, 2006.
P5-3
CMU SCS
Center-Piece Subgraph(Ceps)
B
• Given Q query nodes
• Find Center-piece (  b )
• Input of Ceps
– Q Query nodes
– Budget b
– k softAnd number
C
A
B B
• App.
–
–
–
–
KDD'09
Social Network
Law Inforcement
Gene Network
…
Faloutsos, Miller, Tsourakakis
AA
C
C
P5-4
CMU SCS
Challenges in Ceps
• Q1: How to measure importance?
• (Q2: How to extract connection subgraph?
• Q3: How to do it efficiently?)
KDD'09
Faloutsos, Miller, Tsourakakis
P5-5
CMU SCS
Challenges in Ceps
• Q1: How to measure importance?
• A: “proximity” – but how to combine
scores?
• (Q2: How to extract connection subgraph?
• Q3: How to do it efficiently?)
KDD'09
Faloutsos, Miller, Tsourakakis
P5-6
CMU SCS
AND: Combine Scores
• Q: How to
combine scores?
KDD'09
Faloutsos, Miller, Tsourakakis
P5-7
CMU SCS
AND: Combine Scores
• Q: How to
combine scores?
• A: Multiply
• …= prob. 3
random particles
coincide on node j
KDD'09
Faloutsos, Miller, Tsourakakis
P5-8
CMU SCS
K_SoftAnd: Relaxation of AND
Noise
Disconnected
Communities
What if AND query  No Answer?
KDD'09
Faloutsos, Miller, Tsourakakis
P5-9
CMU SCS
K_SoftAnd: Combine Scores
Generalization –
SoftAND:
We want nodes close
to k of Q (k<Q)
query nodes.
Q: How to do that?
KDD'09
Faloutsos, Miller, Tsourakakis
P5-10
CMU SCS
K_SoftAnd: Combine Scores
Generalization –
softAND:
We want nodes close
to k of Q (k<Q)
query nodes.
Q: How to do that?
A: Prob(at least kout-of-Q will meet
each other at j)
KDD'09
Faloutsos, Miller, Tsourakakis
P5-11
CMU SCS
AND query vs. K_SoftAnd
query
0.0103
0.4505
5
5
x 1e-4
0.0046
0.1010
0.1010
0.0710
11
0.0046
0.0019
11
12
4
0.0046
0.1010
0.1010
10
0.2267
0.4505
0.0710
0.0710
2
6
10
0.0024
13
3
0.1010
0.1010
7
1
9
8
0.4505
0.0103
And Query
KDD'09
0.0046
13
3
1
12
4
0.0019
0.0019
2
6
0.0046
0.0046
7
9
8
0.0103
2_SoftAnd Query
Faloutsos, Miller, Tsourakakis
P5-12
CMU SCS
1_SoftAnd query = OR query
0.0103
5
0.1617
0.1617
0.1387
11
12
4
0.1617
0.1617
10
0.0849
13
3
1
0.0103
KDD'09
0.1387
0.1387
2
6
0.1617
0.1617
7
9
8
0.0103
Faloutsos, Miller, Tsourakakis
P5-13
CMU SCS
Detailed outline
• Problem definition
• Solution
• Results
KDD'09
Faloutsos, Miller, Tsourakakis
P5-14
CMU SCS
Case Study: AND query
H.V.
Jagadish
15
Laks V.S.
Lakshmanan
10
R. Agrawal
Jiawei Han
10
1
2
Heikki
Mannila
Christos
Faloutsos
KDD'09
1
Corinna
Cortes
1
6
Padhraic
Smyth
1
1
V. Vapnik
4
13
3
Daryl
Faloutsos, Miller, Tsourakakis
Pregibon
1
1
M. Jordan
6
P5-15
CMU SCS
Case Study: AND query
H.V.
Jagadish
15
Laks V.S.
Lakshmanan
10
R. Agrawal
Jiawei Han
10
1
2
Heikki
Mannila
Christos
Faloutsos
KDD'09
1
Corinna
Cortes
1
6
Padhraic
Smyth
1
1
V. Vapnik
4
13
3
Daryl
Faloutsos, Miller, Tsourakakis
Pregibon
1
1
M. Jordan
6
P5-16
CMU SCS
H.V.
Jagadish
15
10
Laks V.S.
Lakshmanan
database
13
R. Agrawal
Jiawei Han
Umeshwar
Dayal
3
Bernhard
Scholkopf
5
V. Vapnik
27
4
KDD'09
2
2_SoftAnd query
3
Peter L.
Bartlett
3
Statistic
2
M. Jordan
Alex J.
Smola
Faloutsos, Miller, Tsourakakis
P5-17
CMU SCS
Conclusions
Proximity (e.g., w/ RWR) helps answer
‘AND’ and ‘k_softAnd’ queries
KDD'09
Faloutsos, Miller, Tsourakakis
P5-18