Transcript LLNL Graph X-Ray: Fast Best-Effort Pattern Matching in Large Attributed Graphs Hanghang Tong, Brian Gallagher, Christos Faloutsos, Tina Eliassi-Rad 8/13/2007 KDD 2007, San Jose.

LLNL
Graph X-Ray:
Fast Best-Effort Pattern Matching
in Large Attributed Graphs
Hanghang Tong, Brian Gallagher,
Christos Faloutsos, Tina Eliassi-Rad
8/13/2007
KDD 2007, San Jose
Input
Output
Query Graph
CEO
SEC
Matching Subgraph
Accountant
Manager
Attributed Data Graph
2
Terminology: ``Conform’’
Matching Subgraph conforms Query Graph
3
Terminology: ``Interception’’
matching node
Intermediate node
matching node
matching node
matching node
Matching Subgraph
Query Graph
Path 12-13-4 is an Interception
4
Terminology: ``Instantiate’’
Matching Subgraph Ht
Query Graph Hq
Node 11 instantiates SEC node
Ht instantiates Hq
5
Roadmap
• Introduction
– Problem Definition
– Motivations
• How to: Graph X-Ray
• Experimental Results
• Conclusion
6
Motivation: Why Not SQL?
• Case 1: Exact match does not exist
– Q: How to find approximate answer?
• Case 2: Too many exact matches
– Q: How to rank them?
7
Motivation: Why Not SQL? (Cont.)
• Case 3: Exact match might be not the best answer
– ``Find CEO who has heavy contact with Accountant’’
• Q: how to find right?
2
3
4
11
4
12
1
...
99
Exact match
1 direct connection
Inexact match
Many indirect connections
8
Motivation: Efficiency
• Why Not Subgraph Isomorphism?
– Polynomial for fixed # of pattern query
• Q1: How to scale up linearly?
• Q2: … and with a small slope?
9
Wish List
• Effectiveness
– Both exact match & inexact Match
– Ranking among multiple results
– ``Best’’ answer (proximity-based)
• Efficiency
– Scale linearly
– Scale with small scope
G-Ray meets all!
10
Roadmap
• Introduction
– Problem Definition
– Motivations
• How to: Graph X-Ray
• Experimental Results
• Conclusion
11
Preliminary: Center-Piece Subgraph [Tong+]
Q
B
B
A
C
Original Graph
CePS
meta
Black:
queryis
nodes
A
C
=CePS( A , B , C )
opt. in G-Ray!
12
Preliminary: Augmented Graph
• Data nodes
8
4
– 1,…13
7
• Attribute nodes
13
3
11
–a
12
1
5
9
6
Footnote
2
10
Aug. Graph is crucial for computation!
13
G-Ray: quick overview (for loop
)
Step 1: SF
Step 7: BR
Step 2: NE
Step 3: BR
Step 6: NE
Step 4: NE
Step 5: BR
SF: Seed-Finder
NE: Neighborhood
-Expander
BR: Bridge
14
Step 8: BR
Seed-Finder (
)
• Q: How to instantiate SEC node?
• A:
11
=CePS(
)
8
4
7
Footnote
`11’ is close to some un-known
data nodes for `CEO’ `Account.’
and `Manager’
13
3
11
12
1
5
9
6
2
15
10
Neighborhood-Expander (
)
• Q: How to instantiate CEO node?
– Step 1  Step 2?
• A:
12
=CePS(
)
11
8
4
• Footnote:
7
– Step 3  Step 4?
7
=CePS(
11
)
=CePS(
7
11
12
1
5
– Step 5  Step 6?
4
13
3
9
12
)
6
2
16
10
Bridge (
)
Step 6: NE
• Q:
?
Step 7: BR
• A: Prim-like Alg.
– To maximize
– Should block node 11 and 7
• Footnote
– Connection subgraph, or one single path?
17
Roadmap
• Introduction
– Problem Definition
– Motivation
• How to: Graph X-Ray
• Experimental Results
• Conclusion
18
Experimental Results
• Datasets
– DBLP
– Node: author (315k)
– Edge: co-authorship (1,800k)
– Attribute: conference & year (13k)
• KDD-2001, SIGMOD…
19
Effectiveness: star-query
Query
Result
20
Effectiveness: line-query
Query
Result
21
Effectiveness:
loop-query
Query
Result
22
Efficiency
Response Time
80
Average Response Time (Seconds)
70
Fast FSGM
Iterative method
60
•Scale linearly
•Small slope
•3-5 Seconds
50
40
30
20
10
0
# of Edges
0
0.2
0.4
0.6
0.8
1
1.2
# of Edges
1.4
1.6
1.8
2
6
x 10
~2 M edges
23
Roadmap
• Introduction
– Problem Definition
– Motivation
• How to: Graph X-Ray
• Experimental Results
• Conclusion
24
Conclusion
• Graph X-Ray (G-Ray)
– Best effort pattern match
• in large attributed graphs
– Scale linearly
• with small slope
• More details in Poster Session
– Monday (tonight)
– board number 8
25
4
8
7
13
3
12
11
13
12
1
4
5
11
9
7
2
6
10
G-Ray
X-Ray
Thank you!
www.cs.cmu.edu/~htong
26
Backup-slides
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Proximity on Graph
10
a.k.a relevance, closeness
9
12
2
8
1
11
3
4
6
• Multi-faceted
• Punish long path
• Edge weight
5
7
How to:
---- random walk with restart
28
Random walk with restart
0.04
9
0.10
Node 4
12
2
0.13
1
0.03
10
0.08
3
0.02
8
0.13
11
0.04
4
0.13
6
5
7
0.05
Node 1
Node 2
Node 3
Node 4
Node 5
Node 6
Node 7
Node 8
Node 9
Node 10
Node 11
Node 12
0.13
0.10
0.13
0.22
0.13
0.05
0.05
0.08
0.04
0.03
0.04
0.02
0.05
Nearby nodes, higher scores
More red, more relevant
Ranking vector
r4
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How to rank the results
• Our goodness function
– Measure the proximity between any two matching
nodes if they are required to be connected. (two-way)
– Multiply them together
• In G-Ray, we approximately optimize this
goodness functions
• If we have multiple matching subgraphs, we can
rank them according to this goodness functions
30
How to rank the results
matching node
matching node
matching node
matching node
Goodness = Prox (12, 4) x Prox (4, 12) x
Prox (7, 4) x Prox (4, 7) x
Prox (11, 7) x Prox (7, 11) x
Prox (12, 11) x Prox (11, 12) 31