gSpan algorithm
- 1. Gspan: Graph-based Substructure Pattern Mining Presented By: Sadik Mussah University of Vermont CS 332 – Data mining 1 - Algorithm -
- 2. Outlines • Background • Problem Definition • Authors Contribution • Concepts Behind Gspan • Experimental Result • Conclusion 2
- 3. Background • Frequent Subgraph Mining Is An Extension To Existing Frequent Pattern Mining Algorithms • A Major Challenge IsTo Count How Many Instances of patterns are in the Dataset • Counting Instances Might Be Easy For Sets, But Subtle For Graphs • Graph Isomorphism Problem 3
- 4. Background Theorem Given two graphs G and G’ (g prime), G isomorphic to G’ iff min(G) = min(G’) 04/12/16Sadik Mussah 4
- 5. Background 5 X W U Y V (a) X W U YV (b) Two Isomorphic graph (a) and (b) with their mapping function (c) Two Graphs Are Isomorphic If One Can Find A Mapping Of Nodes Of The First Graph To The Second Graph Such That Labels On Nodes And Edges Are Preserved. f(V1.1) = V2.2 f(V1.2) = V2.5 f(V1.3) = V2.3 f(V1.4) = V2.4 f(V1.5) = V2.1 (c) G1=(V1,E1,L1) G2=(V2,E2,L2) 1 2 3 4 5 1 2 3 4 5
- 6. Problem: Finding Frequent Subgraphs • Problem Setting: Similar To Finding Frequent Itemsets For Association Rule Discovery • Input: Database Of Graph Transactions • Undirected Simple Graph (No Multiples Edges) • Each Graph Transaction Has Labeled Edges/Vertices. • Transactions May Not Be Connected • Minimum Support Thresholds • Output: Frequent Subgraphs That Satisfy The Support Threshold, Where Each Frequent Subgraph Is Connected. 6
- 8. Authors Contribution • Representing Graphs As Strings (Like Treeminer) • No Candidate Generation! • “It Combines The Growing And Checking Of Frequent Subgraphs Into One Procedure,Thus Accelerates The Mining Process.” • Really Fast, Still A Standard Baseline System That Most Rivals Compare Their Systems To. 8
- 9. Concepts Behind Gspan • The Idea Is To Produces A Depth-first Search (DFS) Codes For Each Edge In Graphs • Edges Are Sorted According To Lexicographic Order Of Codes • Yan And Han Proved That Graph Isomororphism Can Be Tested For Two Graphs Annotated With DFS Codes • Starting With Small Graph Patterns Containing 1-edge, Patterns Are Expanded Systemically By The DFS Search • Employ Anti-monotonic Property Of Graph Frequency 9
- 10. Lexicographic Ordering In Graph • It Can Tell Us The Order Of Two Graphs. • The Design Can Help Us Build A Similar Hierarchy. • The Design Should Guarantee Easy-growing From One Level To The Lower Level And Easy-rolling-up From Low Level To Higher Level. • It May Be Difficult To Have Such Design That No Two Nodes In This Tree Are Same For Graph Case. • It Can Tell Us Whether The Graph Has Been Discovered. • And More,The Most Important, If A Graph Has Been Discovered, All Its Children Nodes In The Hierarchy Must Have Been Discovered. 10
- 11. Lexicographic Ordering in Graph11 ... ... ... 1-edge 2-edge ...3-edge ... ... ... ...
- 12. DFS Code And Minimum DFS Code • We Use A 5-tuple (Vi,Vj, L(vi), L(vj), L(vi,vj)) To Represent An Edge. (It May Be Redudant, But Much EasierTo Understand.) • Turn A Graph Into A SequenceWhose Basic Element Is 5-tuple. Form The Sequence In Such An Order: • To Extend One New Node,Add The Forward Edge That Connect One Node In The Old Graph With This New Node. • Add All Backward Edge That Connect This New Node To Other Nodes In The Old Graph • Repeat This Procedure. 12
- 13. DFS code 13 X Y X Z Z a a b b c d v0 v1 v2 v3 v4 X Y a e0: (0,1,x,y,a) X b e1: (1,2,y,x,b)a e2: (2,0,x,x,a) Z c e3: (2,3,x,z,c)b e4: (3,1,x,y,b) Z d e5: (1,4,x,z,d)
- 14. DFS Code And Minimum DFS Code 14 Depth First Tree And Forward/Backward Edge Set
- 15. Minimum DFS code 15 Each Graph may have lots of DFS code (why?): one smallest lexicographic one is its Minimum DFS Code Edge no. (B) (C) ( D) 0 (0,1,x,y,a) (0,1,y,x,a) (0,1,x,x,a) 1 (1,2,y,x,b) (1,2,x,x,a) (1,2,x,y,b) 2 (2,0,x,x,a) (2,0,x,y,b) (0,1,y,x,a) 3 (2,3,x,z,c) (2,3,x,z,c) (2,3,y,z,a) 4 (3,1,z,y,b) (3,0,z,y,b) (3,1,z,x,c) 5 (1,4,x,z,d) (0,4,y,z,d) (2,4,y,z,d)
- 16. Graph Parent And Its Children 16 X Y X Z Z a b c a Given a DFS code c0=(e0,e1,…,en) if c1=(e0,e1,…,en,ex) if c0<c1, then c0 is c1’s parent, c1 is c0’s child. ? ? ? ? ? ? ? ?
- 17. Theorem • 1. Given Two Graph G0 And G1, G0 Is Isomorphic To G1 Iff Min_dfs_code(g0)=min_dfs_code(g1). • 2. DFS CodeTree Covers All Graphs Although SomeTree Nodes May Represent The Same Graph • 3. Given A Node In DFS CodeTree, If Its DFS Code Is Not Its Minimum DFS Code, PruneThis Node And Its All DescendantsWon’t Change.“Covering”. 17
- 18. DFS Code Tree 18 ... ... ... 1-edge 2-edge ...3-edge ... ... ... ... pruned
- 19. FSG: two substructure patterns and their potential candidates. 04/12/16Sadik Mussah 19
- 20. 04/12/16SADIK MUSSAH 20 AGM: two substructures joined by two chains
- 21. Algorithm 21
- 22. Algorithm 22
- 27. Conclusion • No Candidate Generation And FalseTest • Space Saving From Depth First Search • Good Performance: Using “Memory Pool” And One Major Counting Improvement, It SeemsThe PerformanceWill Be Improved 5Times More. (But Need MoreTesting). 27
- 28. Questions Q1) What Two Major Costs From Apriori-like, Frequent Substructure Mining Algorithms Did Gspan Aim To Reduce/Avoid? Answer: 1)The Creation Of Size K+1 Candidate Subgraphs From Size K Frequent Subgraphs Is More Complicated And Costly The Standard Apriori Large Itemset Generation. 2) Pruning False Positives Is An Expensive Process. Subgraph Isomorphism Problem Is Np-complete. 28
- 29. Security Graph 3DVisualization • https://www.youtube.com/watch?v=JsEm-CDj4qM 04/12/16Sadik Mussah 29
- 30. Questions (cont.) • Q2) Which DFSTree Does The DFS Code Below BelongTo? 30
- 32. Questions • Q3) What Does Gspan CompareWhen Testing For Isomorphism Between Two Graphs,AndWhy? • Answer: Gspan Compares The Minimum Dfs Codes Of The Two Graphs. GivenTwo Graphs G And G’, G Is Isomorphic To G’ If Min(g)=min(g’).This Theorem Allows For A Simple String Comparison Of More Complicated Graphs. If Two Nodes Contain The Same Graph But Different Minimum DFS Codes,We Can Prune The Sub-branch Of The Rightmost Of The Two Nodes.This Greatly Decreases The Problem Size. 32
- 33. Questions? 33
Editor's Notes
- #4: Isomorphisim: The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. Which is MP - it is one of a very small number of problems belonging to NP neither known to be solvable in polynomial time nor NP-complete: