Close Graph(Improvements of gSpan)

Sayeed Mahmud

gSpan – where we left of

• Two graph pattern mining techniques– Apriori based – Pattern growth based

• gSpan is based on pattern growth philosophy• Reduced two main bottlenecks of apriori

based techniques– Join 2 K-edge graph to generate k+1 edge graph– Checking frequency of generated candidate

separately

Pattern Growth before gSpan • gSpan improved several other aspects of

previous pattern growth approach (like NaiveGraph).

• A n-edge graph can be extended in n ways

• There are also isomorphism• These extended graph were duplicate graphs• With naïve approaches it was necessary to

check these duplicate graphs

gSpan over Naive

• Reduced the generation of duplicate graph– only expands rightmost

• No need to search for duplicate checking– Can tell it by dfs codes

• Eliminates duplicate graph and still gurantees that no frequent candidate is left out– DFS Code Covering property

Backlogs of gSpan

• Competitive performance edge over previous pattern growth algorithms

• Performance throttle is seen when using large graph size

• A graph with 64 edges may have 264 frequent sub-graphs.

• gSpan may have to extend a lot of them.• Implementation throttles (DFS based

implementation)

Backlogs of gSpan

• Most of the generated frequent candidates contain no helpful information in analysis– Just shares the common frequency– Example: AIDS Antiviral screen dataset

• 1,000,000 frequent dataset 5% min support• Only 422 is meaningful

• gSpan has no terminating mechanism, will generate all these frequent candidates.

• Will be difficult to analyse with huge number of frequent candidates.

Closed Frequent Itemset

• Instead of mining all frequent sub-graphs, we can get away with mining frequent close-graphs.

• An graph is close frequent if its frequent and none of its proper super graph has same frequency.

Is not closed

Frequency 2

Supergraph

Frequency 2

In this case

Is closed

No supergraph having same frequency

Close Frequent Itemset

• The notion of non-close frequent item set is wherever the super-graph occurs, the sub-graph occurs (frequency same).

• So we don’t need to extend them again leading to meaningless candidates

• And as the sub graph occurs whenever super-graph occurs, we are not ruling them out either.

• This reduces the amount of item generated.– Among 1,000,000 frequent candidates only 2000 are

closed in AIDS antiviral screen dataset.

CloseGraph

• Mines all close frequent patterns given a graph dataset.

• Pattern growth approach• Built on top of gSpan• Features added to gSpan– Equivalent Occurrence and Extended frequency

counting– Early termination.

• Outperforms gSpan by a factor of 4 to 10.

Equivalent Occurrence

• The target is to return recursive call as early as possible – early termination

• Equivalent Occurrence is the condition for early termination.

• If we find certain super-graph (or isomorphic super-graph) that has the same support count of the sub-graph we return.

• The same support count super graph means wherever the large graph occurred the small graph also occurred (Equivalent Occurrence)

Equivalent Occurrence

• CloseGraph introduces one more paramter than gSpan – Extended frequency count– L(g, g’, G)

• Number of super-graph (or its isomorphism) g’ that contains g or one of its isomorphic as sub-graph.– The super graph (g’) must be one edge extended

from g.

Early termination using Equivalent Occurrence

Graph Dataset D:

Frequecy 2 + 1 + 0 = 3

Frequecy 2 + 1 + 0 = 3

Extended from g1 Extended from g2

Frequecy 2 + 1 + 0 = 3

Extended from g1

Frequecy Apparently 3 !!

1. We didn’t extend g1 and yet its generatedWith appropriate frequency count

2. This is because of equivalent occurrenceWe don’t need to extend g1 we can extend the super-graph g2

g1 g2 g3

Early Termination Failure

Support Count Blue-Red = 2

Support Count Blue-Red –Yellow = 2

False Assumption : Equivalent OccurrenceExtend Blue-Red-Yellow instead of Blue-Red !!

We shall miss this frequent pattern

Can only be extended from Blue-Red

Detecting Failure

• Passively – do not explicitly look for it• Assume there is no failure and continue• After generation look for certain characteristics

Frequent - Extend

Frequent and Equivalent, extend this instead of g1

Both non frequent and have a common edge !!!

Performance Comparison

• All comparison between gSpan and CloseGraph is done in the following configuration– Intel Pentium IV 1.7GHz– 1GB RAM– RedHat Linux– G++ compiler with STL Support implementation– AIDS Antiviral screen compound dataset

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Thank You