1. Frequent Subgraph Discovery Michihiro Kuramochi and George Karypis ICDM 2001

2. Outline Introduction FSG algo. Canonical Labeling Conclusion

3. Introduction  Most of existing data mining algorithms assume that the data is represented via   Transactions (set of items)   Sequence of items or events  Multi-dimensional vectors   Time series  Scientific datasets with layers, hierarchy, geometry, and arbitrary relations can not be accurately modeled using such frameworks.  e.g. chemical compounds

4. Graphs can accurately model and represent scientific data sets Graphs are suitable for capturing arbitrary relations between the various elements

5. Example

6. FSG(Frequent Subgraph Discovery Algorithm) Level-by-level approach Incremental on the number of edges of the frequent subgraphs(like Apriori) Counting of frequent single and double edge subgraphs For finding frequent size k-subgraphs(k=3) Candidate generation Joining two size (k-1) subgraphs similar to each other Candidate pruning by downward closure property Frequency counting Check if a subgraph is contained in a transaction Repeat the steps for k = k + 1 Increase the size of subgraphs by one edge

7. FSG Approach: Candidate Generation Generate a size k-subgraph by merging two size (k-1) subgraphs

8. FSG Approach: Candidate Pruning Pruning of size k-candidates For all the (k-1)-subgraphs of a size k-canidate, check if downward closure property holds

9. FSG Approach: Frequency Counting For each subgraph to scan each one of the transaction graphs and determine if it is contained or not using subgraph isomorphism In Apriori,the frequency counting is performed substantially faster by building a hash-tree of candidate itemsets To scan each transaction to determine which of the itemsets in the hash-tree it supports

10. FSG Approach: Frequency Counting Keep track of the TID-List Perform subgraph isomorphism only on the intersection of the TID lists of the parent frequent subgraphs of size k-1

11. FSG Approach: Frequency Counting Perform only subgraph_isomorphism(c, T1)

12. Key to Performance: Canonical Labeling Given a graph, we want to find a unique order of vertices, by permuting rows and columns of its adjacency matrix

13. Canonical Labeling Find the vertex order so that the matrix becomes lexicographically the Largest when we compare in the column-wise way

14. Conclusion During both candidate generation and frequency counting, what FSG essentially does is to solve subgraph isomorphism Canonical Labeling can reduce our search space Using TID List Reduce the number of subgraph isomorphism checks Suitable for sparse graph transactions