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GRAPH SAMPLING

Ori Raz

November 2009

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Abstract

Isomorphism between graphs is a knownproblem

One of the latest ideas in the area is graph

sampling, or features selectionWe will try to find correlation between the

graphlet distribution and the

regression/classification tasks

SAT Problem DATASET 3000 formulas

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Graph representations

3 ways to represent a graph- Clause Graph

Each vertex represents a clause and an edge between two

clauses when they share a negated literal- Variable Clause Graph

Each vertex represents clause or variable (vertices =

clauses + variables). An edge when a variable in a clause.

Graph is bipartite graph.- Variable Graph

Each vertex represents a variable and an edge when two variables

occur in the same clause.

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Graph Sampling

Graph Random walk

2 methods: Random and MCMC

- Random: For each sample we will chooseall vertices randomly

- MCMC: Replacing a random vertex with a

new vertex

Total of 630K samples for each graph

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Graph representations - example

F= (x1 or x2 or x3) and (x1 or x2 or x5) and (x3 or x4 or x5) and (x2 or x3 or x5)

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Algorithm

- For each formulacreate representations

- For each created fileget samples

- For each sampling fileexecute

- Summarize all results in svm format- For each svm file

create learn and target files- For each learn file

generate learn-strong-features svm files- For each learn-strong-features file

use to learn the fileuse to classify

- For each result filefind correlation with original result

Nauty

svmLightsvmLight

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Results - Regression

MCMCVs. Random

0.46

0.465

0.47

0.475

0.48

0.485

0.49

0.495

0.5

0.505

20 40 100 200 400 1000

Number Of Features

Correlation

MCMC

RANDOM

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

20 40 100 200 400 1000

Number Of Features

Correlation

MCMC

RANDOM

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

20 40 100 200 400 1000

Number Of Features

Correlation

MCMC

RANDOM

UNSAT VARIABLE CLAUSE GRAPH

UNSAT VARIABLE GRAPH

SAT VARIABLE CLAUSE GRAPH

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Results - Regression

Representation Comparison

MCMC

RANDOM

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

20 40 100 200 400 1000

Number Of Features

Co

rrelation

UNSAT VARIABLE CLAUSE

UNSAT VARIAVLE GRAPH

SAT VARIABLE GRAPH

0

0.1

0.2

0.3

0.4

0.5

0.6

20 40 100 200 400 1000

Number Of Features

Correlation

UNSAT VARIABLE CLAUSE

UNSAT VARIAVLE GRAPH

SAT VARIABLE GRAPH

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Results - Regression

Graphlet size affection

UNSAT VARIABLE CLAUSE GRAPH

SAT VARIABLE GRAPH

0

0.1

0.2

0.3

0.4

0.5

0.6

20 40 100 200 400 1000

Number Of Features

Correlation

4 VERTICES

4+5 VERTICES

4+5+6 VERTICES

4+5+6+7 VERTICES

4+5+6+7+8 VERTICES

4+5+6+7+8+9 VERTICES

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

20 40 100 200 400 1000

Number Of Features

Correlation

4 VERTICES

4+5 VERTICES

4+5+6 VERTICES

4+5+6+7 VERTICES

4+5+6+7+8 VERTICES

4+5+6+7+8+9 VERTICES

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Results - Classification

MCMC Vs. Random

CLAUSE GRAPH

VARIABLE GRAPH

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

20 40 100 200 400 1000

Number Of Features

Correlation

MCMC

RANDOM

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

20 40 100 200 400 1000

Number Of Features

Correlation

MCMC

RANDOM

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Results - Classification

Representation Comparison

MCMC

RANDOM

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

20 40 100 200 400 1000

Number Of Features

Correlation

CLAUSE GRAPH

VARIABLE CLAUSE

VARIABLE GRAPH

-0.06

-0.04

-0.020

0.02

0.04

0.06

0.08

0.1

0.12

20 40 100 200 400 1000

Number Of Features

Correlation

CLAUSE GRAPH

VARIAVLE CLAUSE

VARIABLE GRAPH

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Results - Classification

Graphlet size affection

VARIABLE CLAUSE GRAPH

VARIABLE GRAPH

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

20 40 100 200 400 1000

Number Of Features

Correlation

4 VERTICES

4+5 VERTICES

4+5+6 VERTICES

4+5+6+7 VERTICES

4+5+6+7+8 VERTICES

4+5+6+7+8+9 VERTICES

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

20 40 100 200 400 1000

Number Of Features

Correlation

4 VERTICES

4+5 VERTICES

4+5+6 VERTICES

4+5+6+7 VERTICES

4+5+6+7+8 VERTICES

4+5+6+7+8+9 VERTICES

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Summary

Regression problem - We found high and solidcorrelation (reaching to 0.5)

Classification problem - We found good correlation(reaching to 0.1)

Representation methods - Both Variable-Clauseand Variable-Graph gave good results. Variable-Clause used least space

Sampling methods - Random method gave better

results for the SAT formulasGraphlet size - In most cases, increasing the

graphlet size caused an higher correlation

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Thank you

Questions?

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Nauty

Isomorphism testing program

We use it to create a canonicalrepresentation for each sample

Very fast and efficient

Written in C

http://cs.anu.edu.au/people/bdm/nauty/

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SVMLight

Implementation for Pattern Recognition

Very easy to use

Written in C

http://svmlight.joachims.org/