connected substructure similarity search haichuan shang the university of new south wales &...
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Connected Substructure Similarity Search
Haichuan ShangThe 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)
Application1. Chemistry
2. Bioinformatics
3. Software Engineering
4. Social Network
Chemical Compounds
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 SimilaritySubgraph Similarity Similarity Measures
•Maximum Common Subgraph (MCS) (# of missing edges)
•Edit Distance.
•Variants.
No enforcement ofconnectivity.
Graph Similarity
A New Similarity Measure.
Maximum Connected Common Subgraph – MCCS(counting missing edges while retaining the connectivity)
Graph SimilarityMaximum 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 SimilarityMaximum 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 SimilarityMaximum 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)
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) ≤ .
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)
Feature-based exact subgraph search: overview
Query Data
Query Feature(Index) Data
Q F F G Q G Pruning:
Query Data
Query Feature(Index) Data
Q F F G Q G Q F F G Q G
Feature-based exact subgraph search: overview
Pruning:Validation:
Similarity Search (triangular inequality)
dist(Q,F)+dist(F,D) ≥ dist(Q,D) ?
Query Data
dist(Q,D)
dist(Q,F)
dist(F,D)
Query Feature(Index) Data
Query Data
dist(Q,D)
dist(Q,F)
dist(F,D)
Query Feature(Index) Data
dist(Q,F)+dist(F,D) ≥ dist(Q,D) ?1
Similarity Search (triangular inequality)
Query Data
dist(Q,D)
dist(Q,F)
dist(F,D)
Query Feature(Index) Data
1 2
dist(Q,F)+dist(F,D) ≥ dist(Q,D) ?
Similarity Search (triangular inequality)
Query Data
dist(Q,D)
dist(Q,F)
dist(F,D)
Query Feature(Index) Data
1 2 2
dist(Q,F)+dist(F,D) ≥ dist(Q,D) – hold!
Similarity Search (triangular inequality)
dist(Q,F)
dist(F,D)
0 1 3
Query Feature(Index) Data
Query Data
dist(Q,D)
dist(Q,F)+dist(F,D) ≥ dist(Q,D)
X
Triangular inequality: not always hold
dist(Q,F)
dist(F,D)
0 1 3
Query Feature(Index) Data
Query Data
dist(Q,D)
Triangular inequality: not always hold
dist(Q,F)+dist(F,D) ≥ dist(Q,D)
X
Connectivity DominanceConnectivity 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 .)
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).
Connectivity Dominance
g1=Query g2=Feature(Index) g3=Data
Example 1
Example 2
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
g1=Query g2=Feature(Index) g3=Data
Example 1
Example 2
mccs(g2,g3) dominates g2
mccs(g1,g2) not dominate g2
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
g1=Query g2=Feature(Index) g3=Data
Example 1
Example 2
mccs(g2,g3) dominates g2
mccs(g1,g2) not dominate g2
mccs(g1,g2) not dominate g2 mccs(g2,g3) not dominate g2
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
g1=Query g2=Feature(Index) g3=Data
Example 1
Example 2
mccs(g2,g3) dominates g2
Count # of disconnected components: Linear Algorithm
mccs(g1,g2) not dominate g2
mccs(g1,g2) not dominate g2 mccs(g2,g3) not dominate g2
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)> => dist(Q,D)>mccs(D, F) dominates D
dist(F,Q)+dist(Q,D) ≥ dist(F,D)
Pruning Rule 2:dist(F, D)-dist(F, Q)> => dist(Q,D)>mccs(F, Q) dominates Q
• 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 subgraph1.1 Infrequent Label (in all data graphs) First1.2 Higher Degree Vertex (in the query graph) First1.3 Dense Induced Subgraph (in the query graph) First
Verification Algorithm
Verification Algorithm
MCCS Detection Algorithm
1.Compute QI-Sequence2.DFS: Threshold based DFS Search(A-B-C Matched)
Verification Algorithm
Remove Edge B-D
MCCS Detection Algorithm
1.Compute QI-Sequence2.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-Sequence2.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-Sequence2.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-Sequence2.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-Sequence2.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-Sequence2.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.
ExperimentsSettings
CPU Intel Xeon 2.40GHz
Memory 4G
System Debian Linux
Complier GNU GCC
AIDS Antiviral dataset, a popular benchmark, 43k chemical bonds