COM (Co-Occurrence Miner):Graph Classification Based on

Pattern Co-occurrence

Ning Jin, Calvin Young, Wei WangUniversity of North Carolina at

Chapel Hill11/04/2009

What Are Graphs?

Graph: • a set of nodes connected by a set of edges• nodes and edges can have labels• edges can have directions

1 2

1

2

Graph Classification: Example

Negative set:

Positive set:

Graph Classification: Example

Negative set:

Positive set:

Graph ClassificationUsing Frequent Subgraph Patterns

The positive graphs should have

Some common subgraph patterns

that negative graphs don’t have

Generate classifiers

Frequent subgraph mining in the positive set

(frequency >= threshold)

Feature selection

High dimensional data points classification

Graph ClassificationUsing Frequent Subgraph Patterns

The positive graphs should have

Some common subgraph patterns

that negative graphs don’t have

Generate classifiers

Frequent subgraph mining in the positive set

Feature selection

High dimensional data points classification

Graph ClassificationUsing Discriminative Subgraph Patterns

Frequent subgraph mining in the positive set

Feature selection

Mining discriminative/significant

subgraph patternsmerge

Scoring function: Pattern redundancy:Pattern 1: found in positive graphs P1, P2 and in negative graphs N1, N2Pattern 2: found in positive graphs P1, P2, P3 and in negative graphs N1 Pattern 1 is redundant given pattern 2

Previous Discriminative Pattern Mining Methods

• Each tree node represents a subgraph pattern• Each node is a supergraph of its parent node, with one more edge• One subgraph pattern corresponds to only one node

Pattern redundancy:Pattern 1: found in positive graphs G1, G2 and in negative graphs G4, G5Pattern 2: found in positive graphs G1, G2, G3 and in negative graphs G4Pattern 1 is redundant given pattern 2

Scoring function:

1. Heuristic Exploration Order

Pattern 1

Pattern 2

Pattern redundancy:Pattern 1: found in positive graphs G1, G2 and in negative graphs G4, G5Pattern 2: found in positive graphs G1, G2, G3 and in negative graphs G4Pattern 1 is redundant given pattern 2

Heuristic Exploration Order: Delta Score

Pattern p

Pattern p’

Delta score of p = score of p – score of p’

Workflow of Pattern ExplorationCollect frequent edges in the positive set and insert

into a heap H

If H not empty

terminate

Pop from H the pattern p with the highest delta score

Extend pattern p and insert new non-redundant patterns into H

A frequency threshold tp is needed

2. Use Co-occurrences of PatternsD

DC

B

A

D

DC

B

A ACan be approximated by Co-occurrence

D

DC

B

A

D

DC

B

A AGraph G

Graph G’

When Co-occurrence Is Superior

Separately:A-B: N1, N2, P1, P2, P3, P4B-C: N3, N4, P1, P2, P3, P4

Co-occurrence of A-B and B-C:P1, P2, P3, P4No negative graphs

Co-occurrence Generation

Candidate co-occurrence 1

Candidate co-occurrence 2

Candidate co-occurrence 3

Candidate co-occurrence 4

Candidate co-occurrence n

For each new pattern p:

Pattern p

Union of pattern p and candidate co-occurrence k

insert

insert

merging candidate k and pattern p can improve the score of p

most significantly A co-occurrence is a set of subgraph patterns: {p1, p2, …, pm}

3. Use Association Rules to ClassifyAssociation Rule:{p1, p2, p3, …, pn} “positive”

Input of COM (Co-Occurrence rule Miner):Positive graph set, negative graph setFrequency threshold tp of classification rule in the positive set; frequency threshold tn in the negative set

Output of COM:A set of association rules

Association Rule Generation

Each candidate co-occurrence corresponds to a candidate association rule

If a rule satisfies >=tp and <=tn, it is a resulting rule

Terminate when each positive

graph is covered

Remove redundant rules

Experiments: Datasets

Protein datasets:Six SCOP families

Chemical datasets:Six PubChem bioassays

Experiments: Parameters & Evaluation

Protein datasets: tp = 30%, tn = 0%

Chemical datasets:tp = 1%, tn = 0.4%

Experimental Results: Protein Datasets

Experimental Results: Chemical Datasets

Conclusions

• Using heuristic pattern exploration order and co-occurrences can improve runtime efficiency of mining discriminative patterns

• Using association rules can achieve competitive classification accuracy

Questions & Suggestions