1. gApprox: Mining Frequent Approximate Patterns from a Massive Network
Chen Cheny, Xifeng Yanz, Feida Zhuy, Jiawei Han
[ICDM 2007] reporter: Che-Wei, Liang 10/16 1

2. Outline Introduction Problem Formulation Algorithm Experiment
Pattern Space Exploration
Support Counting
Experiment
Conclusions 2

3. Introduction A set of graphs vs. a single network
Recently, a large number of graphs with massive sizes and complex structures in many applications.
Biological networks, social networks, Web.
demanding powerful data mining methods.
Now interested in patterns that frequently appear at many different places of a single network. 3

4. Introduction Protein-Protein Interaction (PPI) network
△= degree of approximation = 5 4

5. Two major complications
1. Mining frequent patterns in a single network
Partition it into regions
Each contains one occurrence of the pattern
2. Due to various inherent noise or data diversity, it is crucial to account for approximations so that all potentially interesting patterns can be captured. 5

6. Outline Introduction Problem Formulation Algorithm Experiment
Pattern Space Exploration
Support Counting
Experiment
Conclusion 6

7. Problem Formulation 7

8. Approximate Pattern Occurrences
Injective function m: Vp → VG mapping each vertex v Vp to m(v) VG
Quantify the degree of approximation m incurs
i.e., approximations can only happen within the matchable list. 8

9. Approximate Pattern Occurrences9

10. Approximate Pattern Occurrences10

11. Approximate Pattern Occurrences11

12. Pattern Support with Approximation12

13. Pattern Support with Approximation13

14. Pattern Support with Approximation14

15. Outline Introduction Problem Formulation Algorithm Experiment
Pattern Space Exploration
Support Counting
Experiment
Conclusion 15

16. Algorithm Two major issues: 1. Pattern Space Exploration
2. Support Counting
Enumerate approximate occurrences of each pattern in the network.
Decide the maximal number of disjoint occurrences. 16

17. Pattern Space Exploration
Decompose pattern space
Find all connected vertex sets in G that contain 1.
Remove 1 from G, and find all connected vertex sets in the new graph G’ that contain 2.
And so on so forth … 17

18. Pattern Space Exploration
Example: Generating all connected vertex sets starting from 1. Stage1. Start from 1 and mark 1.
Stage2. Expand from 1 to reach 2, 5, Mark 2, 5, 6. There are totally seven connected vertex sets in this stage {1,2}, {1,5}, {1,6}, {1,2,5}, {1,2,6}, {1,5,6}, {1,2,5,6}
Stage3. Taking each of the seven connected vertex sets in stage 2 as a starting point, continue expansion.
Stage4. Until there are no more unmarked vertices. 18

19. 19

20. 20

21. 21

22. Theorem 1 Explore() in Algorithm 1 is both complete and redundancy-free, i.e., given a network G (1) it only generates connected vertex sets in G. (2) it can generate all connected vertex sets in G. (3) it does not generate the same connected vertex set more than once. 22

23. Support Counting
A pattern P’s support is defined to be the maximal number of “disjoint” ones that can be chosen from P’s approximate occurrences in the network. — NP-Complete maximal independent set.
Use algorithm 2 can provide an upperbound. 23

24. Support Counting 24

25. gApprox
gApprox
Combine with pattern space exploration and support counting.
Conditional branch on the 3rd line of Algorithm 1’s DFS_horizontal() function. 25

26. Experiment 26

27. Conclusions
Give an approximation measure and show its impact on mining.
count a pattern’s support based on its approximate occurrences in the network.
The techniques is general
can be applied to networks from other domains.
Can be modified
to reach bigger, more interesting patterns even faster
with some sacrifice on the completeness of mining results. 27