1. Mining Frequent Subgraphs
COMP Seminar
Spring 2011

2. FFSM: Fast Frequent Subgraph Mining -- An Overview:
How to solve graph isomorphism problem?
A Novel Graph Canonical Form: CAM
How to tackle subgraph isomorphism problem (NP-complete)?
Incrementally maintained embeddings
How to enumerate subgraphs:
An Efficient Data Structure: CAM Tree
Two Operations: CAM-join, CAM-extension.
FSG: level wise search gSpan: depth first search
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3. Adjacency Matrix
Every diagonal entry of adjacency matrix M corresponds to a distinct vertex in G and is filled with the label of this vertex.
Every off-diagonal entry in the lower triangle
part of M1 corresponds to a pair of vertices in G and is filled with the label of the edge between the two vertices and zero if there is no edge. p2 p5 a b d y x
(P) p1 p3 p4 c M1 y b d c x a M2 y b c d x a M3 d b x a y c
1for an undirected graph, the upper triangle is always a mirror of the lower triangle
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4. Code
A Code of n  n adjacency matrix M is defined as sequence of lower triangular entries (including the diagonal entries) in the order:
M1,1 M2,1 M2,2 … Mn,1 Mn,2 …Mn,n-1 Mn,n M1 y b d c x a a M3 d b x a y c
Code(M1): aybyxb0y0c00y0d < Code(M2): aybyxb00yd0y00c < Code(M3): bxby0d0y0cyy00a y b y x b y d y c M2
The Canonical Adjacency Matrix is the one produces the maximal code, using lexicographic order.
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5. MP Submatrix
For an m  m matrix A, an n  n matrix B is A’s maximal proper submatrix (MP Submatrix), iff B is obtained by removing the last none-zero entry from A. M6 y b d c x a M5 b y c x a M2 b y a M1 M3 M4 b y x a
We define a CAM is connected iff the corresponding graph is connected.
Theorem I: A CAM’s MP submatrix is CAM
Theorem II: A connected CAM’s MP submatrix is connected
Also explain, the symmetric property and we also remove the row if there is no edge entry left.
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6. CAM Tree: Subgraphs y x p2 p5 a b d (P) p1 p3 p4 c 11/30/2018 b c d ay a a a a d y b a c y b x b y y b b y b x b x x b b y c y d b x y a y a c b d y b a y a c b x d y b x a y a c b x d y b x a d y c b x
The first blink object can be obtained by “superimposing” two objects above it.
The second one can not. This is the motivation for suboptimal CAMs p2 p5 a b d y x
(P) p1 p3 p4 c y a c b x y a d b x y b d c a y b d c x a y b c d x a
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7. CAM Tree: Frequent Subgraphsx y a
= 2/3 p2 p5 a b d y x
(P) p1 p3 p4 c a b y x
(Q) q1 q3 q2 a b y
(S) s1 s3 s2
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8. How to Enumerate Nodes in a CAM Tree?
Two operations to explore CAM tree:
CAM-Join
CAM-Extension
Augmenting CAM tree with Suboptimal CAMs
Objectives:
no false dismissal
no redundancy
Plus: We want to this efficiently!
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9. Suboptimal Tree
We define a Suboptimal CAM as a matrix that its MP submatrix is a CAM. b c d a a b c y b b y b x b y d b x y a c d j e j e c y b x d j y a c b x d e j d y c b x p2 p5 a b d y x
(P) p1 p3 p4 c
May explain depth first search and using information from siblings.
We don’t show the biggest one which has five edges in it due to space limitation y b d c x a j
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10. Summary
Theorem:
For a graph G, let CK-1 (Ck) be set of the suboptimal CAMs of all size-(K-1) (K) subgraphs of G (K ≥ 2). Every member of set CK can be enumerated unambiguously either by joining two members of set CK-1 or by extending a member in CK-1.
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11. Experimental Study Predictive Toxicology Evaluation Competition (PTE)
Contains: 337 compounds
Each graph contains 27 nodes and 27 edges on average
NIH DTP Anti-Viral Screen Test (DTP CA/CM)
Chemicals are classified to be Confirmed Active (CA), Confirmed Moderate Active (CM) and Confirmed Inactive (CI).
We formed a dataset contains CA (423) and CM (1083).
Each graph contains 25 nodes and 27 edges on average
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12. Performance (PTE) Support Threshold (%) Support Threshold (%)
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13. Performance (DTP CACM)
Support Threshold (%)
Support Threshold (%)
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