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Mining Frequent Subgraphs

Mining Frequent Subgraphs. COMP 790-90 Seminar Spring 2011. 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

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Mining Frequent Subgraphs

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  1. Mining Frequent Subgraphs COMP 790-90 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.

  3. a b y b x b y x b y 0 d 0 y c 0 y 0 c 0 0 0 y 0 d y y 0 0 a M3 M1 p2 p5 a y c b y y b p1 y x b x a y 0 0 y d y d b 0 y 0 c 0 p4 p3 M2 (P) 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. 1for an undirected graph, the upper triangle is always a mirror of the lower triangle

  4. b x b y 0 d 0 y 0 c y y 0 0 a a y b M3 y x b 0 y c 0 0 0 y 0 d M1 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 Code(M1): aybyxb0y0c00y0d < Code(M2): aybyxb00yd0y00c < Code(M3): bxby0d0y0cyy00a a y b y x b 0 0 y d 0 y 0 c 0 M2 • TheCanonical Adjacency Matrix is the one produces the maximal code, using lexicographic order.

  5. a a M1 a y b y b y x b a a 0 y c y x b y b y b a 0 0 y c 0 0 y 0 d y 0 b y x b y b 0 M5 M2 M3 M4 M6 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. • 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

  6. b b a y d x b y b y 0 c a a a y b y b a a a a a a b y b 0 y d y 0 b y y y y b b b b y y b b x b y x b y 0 c 0 y y y x x x 0 b b b b y 0 x 0 b b 0 y 0 d 0 0 0 0 y y y y 0 0 0 0 c d c c 0 0 y y 0 0 d d a a y b y b 0 x b 0 x b p2 p5 y 0 0 y c 0 0 y d c b y p1 a a a x a y y y b b b y 0 0 y x x 0 b b b y d b 0 0 0 y y y 0 0 0 c c d p4 0 0 0 0 0 0 y y y 0 0 0 d c d p3 (P) CAM Tree: Subgraphs b d c a b b y c x b a b a a y y b b y b x b 0 y c y 0 d 0 0 x x b b

  7. a b a b y b x b a a y b y b 0 x b y 0 b a y b y x b p2 p5 s1 q1 y c b y y y b b s2 p1 q2 x a a a x y y y y d b b b p4 s3 q3 p3 (S) (P) (Q) CAM Tree: Frequent Subgraphs = 2/3

  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!

  9. a b y b y c y x b j e e j j e e a a b b b b y b x b y b x b x b x b 0 y c 0 y d 0 y d y 0 c y 0 d 0 y c j j e e j j a a a a b b y b y b y b y b x b x b y x b y x b y 0 c y 0 d y x b y x b p2 p5 y 0 0 y c 0 0 y d 0 y 0 d 0 y 0 c 0 y 0 c 0 y 0 d c b y j j p1 a a x a y y b y b y y x b y x b d b 0 y 0 c 0 0 y d p4 p3 0 0 y 0 d 0 0 y 0 c (P) Suboptimal Tree We define a Suboptimal CAM as a matrix that its MP submatrix is a CAM. d b c a b b a y d x b y b

  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.

  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

  12. Performance (PTE) Support Threshold (%) Support Threshold (%)

  13. Performance (DTP CACM) Support Threshold (%) Support Threshold (%)

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