Mining Graph Patterns Efficiently via Randomized Summaries Chen Chen, Cindy X. Lin, Matt Fredrikson, Mihai Christodorescu, Xifeng Yan, Jiawei Han VLDB’09

Outline Motivation Preliminaries SUMMARIZE-MINE FRAMEWORK SUMMARIZE-MINE FRAMEWORK Bounding the False Negative Rate Bounding the False Negative Rate Experiments Experiments Conclusion Conclusion

Motivation Graphs Pattern Mining are heavily needed in many real applications, such as bioinformatics, hyperlinked webs and social network analysis. Unfortunately, due to the fundamental role subgraph isomorphism plays in existing methods, they may all enter into a pitfall when the cost to enumerate a huge set of isomorphic embeddings blows up, especially in large graphs with few identical labels.

Motivation Consider possible ways to reduce the number of embeddings. In particular, since in real applications, many embeddings overlap substantially, we explore the possibility of somehow “merging” these embeddings to significantly reduce the overall cardinality.

Preliminaries

SUMMARIZE-MINE FRAMEWORK Summarization: For raw database with frequency threshold min_sup, we bind vertices with identical labels into a single node and collapse the network correspondingly into a smaller summarized version. This step generalizes our view on the data to a higher level. Mining: Apply any state-of-art frequent subgraph mining algorithm on the summarized database D’ = {S 1, S 2,..., S n } with a slightly lowered support threshold min sup’, which generates the pattern set FP(D’). Verification: Check patterns in FP(D’) against the original database D, remove those p ∈ FP(D’) whose support in D is less than min sup and transform the result collection into R’ Iteration: Repeat steps 1, 2 and 3 for t times, and combine the results from each iteration. Let R’ 1,R’ 2,...,R’ t be the patterns obtained from different iterations, the final result is R’ = R’ 1 ∪ R’ 2 ∪ … ∪ R’ t. This step is to guarantee that the overall probability of missing any frequent pattern is bounded. Deal with false positive and false negative. Raw DB Summarized DB

Summarization: For raw database with frequency threshold min_sup, we bind vertices with identical labels into a single node and collapse the network correspondingly into a smaller summarized version. This step generalizes our view on the data to a higher level. Mining: Apply any state-of-art frequent subgraph mining algorithm on the summarized database D’ = {S 1, S 2,..., S n } with a slightly lowered support threshold min sup’, which generates the pattern set FP(D’). Verification: Check patterns in FP(D’) against the original database D, remove those p ∈ FP(D’) whose support in D is less than min sup and transform the result collection into R’ Iteration: Repeat steps 1, 2 and 3 for t times, and combine the results from each iteration. Let R’ 1,R’ 2,...,R’ t be the patterns obtained from different iterations, the final result is R’ = R’ 1 ∪ R’ 2 ∪ … ∪ R’ t. This step is to guarantee that the overall probability of missing any frequent pattern is bounded. Deal with false positive and false negative. SUMMARIZE-MINE FRAMEWORK Raw DB Summarized DB

Take gSpan as the skeleton of mining algorithm Each labeled graph pattern can be transformed into a sequential representation called DFS code With a defined lexicographical order on the DFS code space, all subgraph patterns can be organized into a tree structure, where  1. patterns with k edges are put on the k th level  2. a preorder traversal of this tree would generate the DFS codes of all possible patterns in the lexicographical order SUMMARIZE-MINE FRAMEWORK

According to DFS lexicographic order, SUMMARIZE-MINE FRAMEWORK

Summarization: For raw database with frequency threshold min_sup, we bind vertices with identical labels into a single node and collapse the network correspondingly into a smaller summarized version. This step generalizes our view on the data to a higher level. Mining: Apply any state-of-art frequent subgraph mining algorithm on the summarized database D’ = {S 1, S 2,..., S n } with a slightly lowered support threshold min sup’, which generates the pattern set FP(D’). Verification: Check patterns in FP(D’) against the original database D, remove those p ∈ FP(D’) whose support in D is less than min sup and transform the result collection into R’ Iteration: Repeat steps 1, 2 and 3 for t times, and combine the results from each iteration. Let R’ 1,R’ 2,...,R’ t be the patterns obtained from different iterations, the final result is R’ = R’ 1 ∪ R’ 2 ∪ … ∪ R’ t. This step is to guarantee that the overall probability of missing any frequent pattern is bounded. Deal with false positive and false negative. Raw DB Summarized DB

SUMMARIZE-MINE FRAMEWORK Reduce false positives  Technique 1: Bottom-up sup(p 1 ) > sup(p 2 ) >min_sup  Technique 2: Top-down min_sup > sup(p 1 ) > sup(p 2 ) It is guaranteed that there is no false positives. False Embeddings  False Positives

SUMMARIZE-MINE FRAMEWORK Summarization: For raw database with frequency threshold min_sup, we bind vertices with identical labels into a single node and collapse the network correspondingly into a smaller summarized version. This step generalizes our view on the data to a higher level. Mining: Apply any state-of-art frequent subgraph mining algorithm on the summarized database D’ = {S 1, S 2,..., S n } with a slightly lowered support threshold min sup’, which generates the pattern set FP(D’). Verification: Check patterns in FP(D’) against the original database D, remove those p ∈ FP(D’) whose support in D is less than min sup and transform the result collection into R’ Iteration: Repeat steps 1, 2 and 3 for t times, and combine the results from each iteration. Let R’ 1,R’ 2,...,R’ t be the patterns obtained from different iterations, the final result is R’ = R’ 1 ∪ R’ 2 ∪ … ∪ R’ t. This step is to guarantee that the overall probability of missing any frequent pattern is bounded. Deal with false positive and false negative. Raw DB Summarized DB

SUMMARIZE-MINE FRAMEWORK

Summarization: For raw database with frequency threshold min_sup, we bind vertices with identical labels into a single node and collapse the network correspondingly into a smaller summarized version. This step generalizes our view on the data to a higher level. Mining: Apply any state-of-art frequent subgraph mining algorithm on the summarized database D’ = {S 1, S 2,..., S n } with a slightly lowered support threshold min sup’, which generates the pattern set FP(D’). Verification: Check patterns in FP(D’) against the original database D, remove those p ∈ FP(D’) whose support in D is less than min sup and transform the result collection into R’ Iteration: Repeat steps 1, 2 and 3 for t times, and combine the results from each iteration. Let R’ 1,R’ 2,...,R’ t be the patterns obtained from different iterations, the final result is R’ = R’ 1 ∪ R’ 2 ∪ … ∪ R’ t. This step is to guarantee that the overall probability of missing any frequent pattern is bounded. Deal with false positive and false negative. Raw DB Summarized DB

Bounding the False Negative Rate Miss Embeddings  False Negatives q(p) The probability that all m j vertices with label l j are assigned to x j different groups (and thus f continues to exist) is Multiplying the probabilities for all L labels

Bounding the False Negative Rate

The false negative rate after t iterations is (1−P) t. To make (1−P) t less than some small  Technique 1: For raw database with frequency threshold min_sup, we adopt a lower frequency threshold min_sup’ for summarized database.  Technique 2: Iterate the mining steps for t times and combine the results generated in each time. It is NOT guaranteed that there is no false negaitives, but the possibility is bounded by

Experiments

Experiments

Conclusion Isomorphism test on small graphs is much more easier. Each graph does iteration t times to reduce the false negative rate, t = ?