Connected Substructure Similarity Search
Haichuan Shang
The University of New South Wales & NICTA, Australia
Joint Work:
Xuemin Lin (The University of New South Wales & NICTA, Australia)
Ying Zhang (The University of New South Wales, Australia)
Jeffrey Xu Yu (Chinese University of Hong Kong, China)
Wei Wang(The University of New South Wales & NICTA, Australia)
Outline
1. Motivation
2. Similarity Measure
3. Techniques
4. Experimental Study
5. Conclusion
Application
1. Chemistry
2. Bioinformatics
3. Software Engineering
4. Social Network
Chemical Compounds
Substructure Search
Substructure Similarity Search
Why Similarity Search?
Input Mistake
Exploration
......
Substructure Similarity Search
Why Similarity Search?
Input Mistake
Exploration
......
Existing Work
SIGMOD’05 Grafil
ICDE’06 Closure-tree
ICDE’07 GDIndex
VLDB’09 Comparing Stars
Graph Similarity
Subgraph Similarity
Similarity Measures
•Maximum Common
Subgraph (MCS)
(# of missing edges)
•Edit Distance.
•Variants.
No enforcement of
connectivity.
Graph Similarity
A New Similarity Measure.
Maximum Connected Common Subgraph – MCCS
(counting missing edges while retaining the
connectivity)
Graph Similarity
Maximum Connected Common Subgraph – MCCS: Given two
graphs g1 and g2, the maximum connected common subgraph of g1
and g2 is the largest connected subgraph of g1 which is subgraph
isomorphic to g2, denoted as mccs(g1, g2)
Graph Similarity
Maximum Connected Common Subgraph – MCCS: Given two
graphs g1 and g2, the maximum connected common subgraph of g1
and g2 is the largest connected subgraph of g1 which is subgraph
isomorphic to g2, denoted as mccs(g1, g2)
Subgraph Distance: Given a query graph q and a data graph g, the
Subgraph Distance is defined as,
dist(q, g) = |q| − |mccs(q, g)|
The graph size is defined as the number of edges.
(# of missing edges from the query)
Graph Similarity
Maximum Connected Common Subgraph – MCCS: Given two
graphs g1 and g2, the maximum connected common subgraph of g1
and g2 is the largest connected subgraph of g1 which is subgraph
isomorphic to g2, denoted as mccs(g1, g2)
Subgraph Distance: Given a query graph q and a data graph g, the
Subgraph Distance is defined as,
dist(q, g) = |q| − |mccs(q, g)|
The graph size is defined as the number of edges.
(# of missing edges from the query)
Substructure Similarity Search: Given a graph database D = {g1,
g2, ..., gn}, a query graph q, and a subgraph distance threshold ,
the substructure similarity search is to retrieve all the graphs gi ∈ D
with dist(q, gi) ≤ .
Framework
Feature-based exact subgraph search: overview

F

F

G

Q

G
Pruning: Q

Query

Feature(Index)

Query
Data
Data
Feature-based exact subgraph search: overview

F

F

G

Q

G
Pruning: Q

F

F

G

Q

G
Validation: Q

Query

Feature(Index)

Query
Data
Data
Similarity Search (triangular inequality)
dist(Q,F)+dist(F,D) ≥ dist(Q,D) ?
dist(Q,F)
Query
dist(F,D)
Feature(Index)
dist(Q,D)
Query
Data
Data
Similarity Search (triangular inequality)
dist(Q,F)+dist(F,D) ≥ dist(Q,D) ?
1
dist(Q,F)
Query
dist(F,D)
Feature(Index)
dist(Q,D)
Query
Data
Data
Similarity Search (triangular inequality)
dist(Q,F)+dist(F,D) ≥ dist(Q,D) ?
1
2
dist(Q,F)
Query
dist(F,D)
Feature(Index)
dist(Q,D)
Query
Data
Data
Similarity Search (triangular inequality)
dist(Q,F)+dist(F,D) ≥ dist(Q,D) – hold!
1
2
dist(Q,F)
Query
2
dist(F,D)
Feature(Index)
dist(Q,D)
Query
Data
Data
Triangular inequality: not always hold
dist(Q,F)+dist(F,D) ≥ dist(Q,D)
0
1
3
dist(Q,F)
Query
dist(F,D)
Feature(Index)
dist(Q,D)
Query
X
Data
Data
Triangular inequality: not always hold
dist(Q,F)+dist(F,D) ≥ dist(Q,D)
0
1
dist(Q,F)
Query
3
dist(F,D)
Feature(Index)
Data
dist(Q,D)
Query
X
Data
Connectivity Dominance
Connectivity Dominance: The connectivity of mccs(g1, g2)
dominates the connectivity of g2 if there is a subgraph isomorphic
mapping from mccs(g1, g2) to g2 such that if removing all the
edges from this mapping, then all the vertices in the embedding
mapping are disconnected. (i.e. The removing fully disconnected
g2 .)
Connectivity Dominance
Theorem. Given three graphs g1, g2, and g3, if the connectivity of
mccs(g1, g2) dominates g2 or the connectivity of mccs(g3, g2)
dominates g2, then dist(g1, g3) ≤ dist(g1, g2) + dist(g2, g3).
Connectivity Dominance
Theorem. Given three graphs g1, g2, and g3, if the connectivity of
mccs(g1, g2) dominates g2 or the connectivity of mccs(g3, g2)
dominates g2, then dist(g1, g3) ≤ dist(g1, g2) + dist(g2, g3).
Example 1
Example 2
g1=Query
g2=Feature(Index)
g3=Data
Connectivity Dominance
Theorem. Given three graphs g1, g2, and g3, if the connectivity of
mccs(g1, g2) dominates g2 or the connectivity of mccs(g3, g2)
dominates g2, then dist(g1, g3) ≤ dist(g1, g2) + dist(g2, g3).
Example 1
mccs(g1,g2) not dominate g2
mccs(g2,g3) dominates g2
Example 2
g1=Query
g2=Feature(Index)
g3=Data
Connectivity Dominance
Theorem. Given three graphs g1, g2, and g3, if the connectivity of
mccs(g1, g2) dominates g2 or the connectivity of mccs(g3, g2)
dominates g2, then dist(g1, g3) ≤ dist(g1, g2) + dist(g2, g3).
Example 1
mccs(g1,g2) not dominate g2
mccs(g2,g3) dominates g2
Example 2
mccs(g1,g2) not dominate g2
g1=Query
mccs(g2,g3) not dominate g2
g2=Feature(Index)
g3=Data
Connectivity Dominance
Theorem. Given three graphs g1, g2, and g3, if the connectivity of
mccs(g1, g2) dominates g2 or the connectivity of mccs(g3, g2)
dominates g2, then dist(g1, g3) ≤ dist(g1, g2) + dist(g2, g3).
Example 1
mccs(g1,g2) not dominate g2
mccs(g2,g3) dominates g2
Example 2
mccs(g1,g2) not dominate g2
mccs(g2,g3) not dominate g2
g1=Query
g2=Feature(Index)
g3=Data
Count # of disconnected components: Linear Algorithm
dist(Q,F)+dist(F,D) ≥ dist(Q,D)
Validation Rule 1:
dist(Q,F)+dist(F,D) ≤ 
=> dist(Q,D) ≤ 
mccs(Q, F) dominates F or mccs(F, D) dominates F
dist(Q,D)+dist(D,F) ≥ dist(Q,F)
Pruning Rule 1:
dist(Q,F)-dist(D,F)> 
mccs(D, F) dominates D
=> dist(Q,D)>

dist(F,Q)+dist(Q,D) ≥ dist(F,D)
Pruning Rule 2:
dist(F, D)-dist(F, Q)> 
mccs(F, Q) dominates Q
=> dist(Q,D)> 
Verification Algorithm
•
Basic idea:
1. enumerate sub-spanning tree of query graph such that the # of
missing edges ≤ ; try to terminate the algorithm as early as possible.
2. sharing the enumeration costs by two ways:
a. not enumerate every thing from scratch.
b. once enumerated, keep enumerated spanning trees.

•
Convert Query to QI-Sequence [VLDB08] to favour earlier
termination.
Prefix = Induced subgraph
1.1 Infrequent Label (in all data graphs) First
1.2 Higher Degree Vertex (in the query graph) First
1.3 Dense Induced Subgraph (in the query graph) First
Verification Algorithm
MCCS Detection Algorithm
1. Compute QI-Sequence
Verification Algorithm
MCCS Detection Algorithm
1. Compute QI-Sequence
2. DFS: Threshold based DFS Search(A-B-C Matched)
Verification Algorithm
Remove Edge B-D
MCCS Detection Algorithm
1. Compute QI-Sequence
2. DFS: Threshold based DFS Search(A-B-C Matched)
3. Generate new QI-Sequence from the existing one.
Verification Algorithm
Remove Edge B-E
MCCS Detection Algorithm
1. Compute QI-Sequence
2. DFS: Threshold based DFS Search(A-B-C Matched)
3. Generate new QI-Sequence from the existing one.
Verification Algorithm
Remove Edge B-F
MCCS Detection Algorithm
1. Compute QI-Sequence
2. DFS: Threshold based DFS Search(A-B-C Matched)
3. Generate new QI-Sequence from the existing one.
Verification Algorithm
Right Subtree
MCCS Detection Algorithm
1.Compute QI-Sequence
2.DFS: Threshold based DFS Search(A-B-C Matched)
3.Generate new QI-Sequence from the existing one.
4.DFS: Threshold based DFS Search (The second A-B Matched)
Verification Algorithm
Remove Edge B-C
MCCS Detection Algorithm
1.Compute QI-Sequence
2.DFS: Threshold based DFS Search(A-B-C Matched)
3.Generate new QI-Sequence from the existing one.
4.DFS: Threshold based DFS Search (The second A-B Matched)
5.Generate new QI-Sequence from the existing one.
Verification Algorithm
MCCS Detection Algorithm
1.Compute QI-Sequence
2.DFS: Threshold based DFS Search(A-B-C Matched)
3.Generate new QI-Sequence from the existing one.
4.DFS: Threshold based DFS Search (The second A-B Matched)
5.Generate new QI-Sequence from the existing one.
6.Terminate. (dist(q,g) ≤ 3)
Feature Selection
• Pruning Rule 1: mccs(D, F) dominates D
• Pruning Rule 2: mccs(F, Q) dominates Q
=>F should be dense.
=>Discriminative Frequent Induced Subgraph
• Validation Rule 1: mccs(F, D) dominates F or mccs(Q, F) dominates F
=>F nearly contains Q and F should be sparse.
=>Frequent Large Sparse Subgraphs
Algorithm: gSpan[ICDM02] with our on-the-fly feature selection.
Experiments
Settings
CPU
Intel Xeon 2.40GHz
Memory
4G
System
Debian Linux
Complier
GNU GCC
AIDS Antiviral dataset, a popular benchmark, 43k chemical bonds
Experiments
Experiments
Conclusion
Connected Substructure Similarity Search
1.Measure: Maximum Connected Common Subgraph – MCCS
2.Connectivity Dominance => Triangular inequality
3.MCCS Detection Algorithm
(Index, Filtering & Validation, Verification Techniques)
Future Work:
Large Graphs? New Measures?
Thanks