Mining for Tree-Query Associations in a Graph

Jan Van den Bussche Hasselt University, Belgium

joint work with Bart Goethals (U Antwerp, Belgium)and Eveline Hoekx (U Hasselt, Belgium)

2

Graph Data

A (directed) graph over a set of nodes N is a set G of edges: ordered pairs ij with ij N.

Snapshot of a graph representing the metabolic pathway of a human.

Applications: life sciences, biology, social sciences, WWW, ...

3

Graph Mining

Transactional category– dataset: set of many small graphs (transactions)

– frequency: transactions in which the pattern occurs (at least once)

– ILP: Warmr

[AGM, FSG, TreeMiner, gSpan, FFSM, Horvath-Ramon-Wrobel]

Single graph category– dataset: single large graph

– frequency: copies of the pattern in the large graph

[Subdue, Vanetik-Gudes-Shimony, SEuS, SiGraM, Jeh-Widom]

Focus on pattern mining, few work on association rule mining!

4

Tree-Query Pattern

• powerful tree-shaped pattern• inspired by conjunctive database queries

• special features:– existential nodes– parameterized nodes

• occurrence of the pattern in G is any homomorphism from the pattern in G

0

8 x frequency: x z: 0z G z8 G zx G

5

Association rules

0

8 x

x1

x3 x2

⇒

(x1,x2,x3) (0, ,8)x

• fully fledged associations over tree-query patterns• example:

6

Experimental results: Real-life datasets

• Food web nodes1 edges30

x1

20

x2

20

frequency = 176

x1

x2

⇒

x1

20

x2

confidence = 89%

7

Experimental results: Real-life datasets

• Food web nodes1 edges30

x1

20

x2

frequency = 176

x1

x2

⇒

x1

20

x2

confidence = 89%

8

Experimental results: Food web

nodes1 edges30

⇒ x4

x3

x2

(x1,x2,x3,x4,x5)

x1

x5

x4

x3

x2

(0,x2,x3,x4,x5)

0

x5

⇒ x4

x3

x2

(x1,x2,x3,x4,x5)

x1

x5

x3

x2

x1

(x1,x2,x3,x4,x5)

0

x4

x5

45% 55%

9

Experimental results: Real-life datasets

• Protein interactions graph nodes211 edges80

x1

x2

⇒

x1

22

x2 confidence = 10%

10

Experimental results: Protein interaction graph

nodes211 edges80

⇒

746

x2

(x1,x2)

x1

(x1,x2)

746

x2

x1

376

90%

11

Outline rest of the talk

• Formal problem definition• Algorithm

– overall approach– levelwise generation of tree patterns– generation of containment mappings– generation of parameter assignments

• Equivalent association rules• Certhia• Performance and Experimental results• Future work

12

Tree pattern

0

8 x

13

Tree pattern

0

8 x

14

Tree pattern

0

8 x

15

Tree pattern

0

8 x

16

Tree pattern

0

8 x

select distinct G3.to as xfrom G G1, G G2, G G3where G1.from=5 and G1.to=G2.from

and G1.to=G3.from and G2.to=8

17

Matching

0

8 x

P:

0

1

5 4

2 3

7

6 8

G: z y z x

18

Matching

0

8 x

P:

0

1

5 4

2 3

7

6 8

G: z y z x

19

Matching

0

1

5 4

2 3

7

6 8

G: z y z x

h1 0 1 8 4

0

8 x

P:

20

Matching

0

1

5 4

2 3

7

6 8

G: z y z x

h 0 1 8 4

h 0 1 8 8

0

8 x

P:

21

Matching

0

1

5 4

2 3

7

6 8

G: z y z x

h 0 1 8 4

h 0 1 8 8

h 0 2 8 4

0

8 x

P:

22

Matching

0

1

5 4

2 3

7

6 8

G: z y z x

h 0 1 8 4

h 0 1 8 8

h 0 2 8 4

h 0 2 8 5

0

8 x

P:

23

Matching

0

1

5 4

2 3

7

6 8

G: z y z x

h 0 1 8 4

h 0 1 8 8

h 0 2 8 4

h 0 2 8 5

h 0 2 8 8

0

8 x

P:

24

Frequency

0

1

5 4

2 3

7

6 8

G: z y z x

h 0 1 8 4

h 0 1 8 8

h 0 2 8 4

h 0 2 8 5

h 0 2 8 8

frequency = 3

0

8 x

P:

25

Tree Query

0

8 x

( , , 8)x x

P, body

H, head

• Q = (H,P)

26

Association Rule

• AR: Q1 Q2

Confidence (AR) = freq(Q2)/freq(Q1) Q2 Q1

x

8 6

0

( , , 6)x x

x1

1

8 x3

0

(x1, x2, x3)

x2

2

⇒

Q1 Q2

{ (x1,x2,x3) | Q1(x1,x2,x3) G} { (x,x,6) | Q2(x,x,6) G }

27

Examples of Association Rules

x1

x2

(x1, x2)

,) ⇒

x1

8

(x1, 8)

,)

x1

(x1)

,) ⇒

x1

8

(x1)

,)

(1) (2)

28

Association Rule

• AR: Q1 Q2

Confidence (AR) = freq(Q2)/freq(Q1) Q2 Q1

x

8 6

0

( , , 6)x x

x1

1

8 x3

0

(x1, x2, x3)

x2

2

⇒

Q1 Q2

{ (x1,x2,x3) | Q1(x1,x2,x3) G} { (x,x,6) | Q2(x,x,6) G }

29

Containment Mapping

x

8 6

0

( , , 6)x x

x1

1

8 x3

0

(x1, x2, x3)

x2

2

Q1 Q2

containment mapping

30

Containment Mapping

x

8 6

0

( , , 6)x x

x1

1

8 x3

0

(x1, x2, x3)

x2

2

Q1 Q2

containment mapping

31

Containment Mapping

x

8 6

0

( , , 6)x x

x1

1

8 x3

0

(x1, x2, x3)

x2

2

Q1 Q2

containment mapping

32

Containment Mapping

x

8 6

0

( , , 6)x x

x1

1

8 x3

0

(x1, x2, x3)

x2

2

Q1 Q2

containment mapping

33

Containment Mapping

x

8 6

0

( , , 6)x x

x1

1

8 x3

0

(x1, x2, x3)

x2

2

Q1 Q2

containment mapping

Q Q containment mapping from Q1 to Q2

34

Problem statement: Mining tree queries

Given a graph G and a threshold k, find all tree queries that

have frequency at least k in G, those queries are calledfrequent.

35

Problem statement: Association rules

• Input:– a graph G– minsup

– Qleft frequent in G

– minconf

• Output: All association rules Qleft Q– frequent in G– confident in G.

36

Algorithm: mining tree queries

Outer loop: Generate, incrementally, all possible trees of increasing sizes. Avoid generation of isomorphic trees.

Inner loop: For each newly generated tree, generate all queries based on that tree, and test their frequency.

...

x1

x4x3

x2

x2x1

x2x1

x1

2

37

Outer loop

• It is well known how to efficiently generate all trees uniquely up to isomorphism

• Based on canonical form of trees.

• [Scions, Li-Ruskey, Zaki, Chi-Young-Muntz]

38

Inner loop: Levelwise approach

• A query Q is characterized by Q set of existential nodes Q set of selected nodes– Labeling Q of the selected nodes by constants.

• Q11 1 1 specializes Q22 2 2 if 12, 1 2 and 1 agrees with 2 on 2.

• If Q1 specializes Q2 then freqQ1 freqQ2

• Most general query: T = (, , )

39

Inner loop: Candidate generation

• CanTab is a candidate queryFreqTab is a frequent query

• Q’=’ ’ is a parent of Q= if either:– ’ and has precisely one more node than

’, or– ’ and has precisely one more node than ’

• Join Lemma: Each candidacy table can be computed by taking the natural join of its parent frequency tables.

40

Inner loop: Frequency counting

• Each candidacy table can be computed by a single SQL query. (ref. Join lemma).

• Suppose: Gfrom to table in the database, then each frequency table can be computed with a single SQL query.

» formulate in SQL and count

» formulate in SQL E» natural join of E with CanTab

» group by » count each group

41

Inner loop: Example

0

8 x

:Q

x1

x2

x3 x4

T: x2

x1 x3

x10 x38

42

Inner loop: Example

0

8 x

:Q

x1

x2

x3 x4

T: x2

x1 x3

x10 x38

• Join expression:

CanTab{x2}{x1,x3} = FreqTabx2x1 ⋈ FreqTabx2x3 ⋈ FreqTabx1x3

43

Inner loop: Example

0

8 x

:Q

x1

x2

x3 x4

T: x2

x1 x3

x10 x38

• Join expression:

CanTab{x2}{x1,x3} = FreqTabx2x1 ⋈ FreqTabx2x3 ⋈ FreqTabx1x3

44

Inner loop: Example

0

8 x

:Q

x1

x2

x3 x4

T: x2

x1 x3

x10 x38

• Join expression:

CanTab{x2}{x1,x3} = FreqTabx2x1 ⋈ FreqTabx2x3 ⋈ FreqTabx1x3

45

Inner loop: Example

0

8 x

:Q

x1

x2

x3 x4

T: x2

x1 x3

x10 x38

• Join expression:

CanTab{x2}{x1,x3} = FreqTabx2x1 ⋈ FreqTabx2x3 ⋈ FreqTabx1x3

46

Inner loop: Example

0

8 x

:Q

x1

x2

x3 x4

T: x2

x1 x3

x10 x38

• Join expression:

CanTab{x2}{x1,x3} = FreqTabx2x1 ⋈ FreqTabx2x3 ⋈ FreqTabx1x3

47

Inner loop: Example

0

8 x

:Q

x1

x2

x3 x4

T: x2

x1 x3

x10 x38

• SQL expression E for x2

select distinct G1.from as x1, G2.to as x3, G3.to as x4

from G G1, G G2, G G3where G1.to = G2.from and G3.from = G2.from

48

Inner loop: Example

0

8 x

:Q

x1

x2

x3 x4

T: x2

x1 x3

x10 x38

• SQL expression for filling the frequency table:

select distinct E.x1, E.x3, count(E.x4)from E, CanTab{x2}{x1,x3} as CT

where E.x1 = CT.x1 and E.x3 = CT.x3group by E.x1, E.x3having count(E.x4) >= k

49

Algorithm: Mining association rules

Loop 1: Generate incrementally all possible trees T of increasing sizes.

Loop 2: For each T, generate all frequent tree patterns P based T.

Loop 3: For each P, generate all containment mappings f from Pleft to P.

Loop 4: For each f, generate Q=(f(Hleft),P) and all parameter instantiations for Qleft Q.

50

Pattern database

• For each P a table FreqTabP, that contains all frequent parameter instantiations.

Pattern Database

51

Loop 3: Generation of containment mappings

Efficiently solvable, due to tree shape.

52

Loop 4: Generation of parameter

instantiations single relational algebra expression (SQL)

plistσ FreqTabP .freq

FreqTabPleft .freq≥minconf

(σ ϑ left (FreqTab Pleft ) ><θ FreqTab P )

ϑ :σ ∈∑left

∧FreqTab Pleft.σ = FreqTab P . f (σ )

ϑ left :σ ∈∑left

∧FreqTab Pleft.σ = σ left (σ )

plist: all P.σ withσ ∈∑,FreqTabP .freq,

FreqTabP .freqFreqTabPleft

.freq

•

•

•

53

Example: Loop 4

x2

x

1

(x2, x, x)

x

Qleft

1

x2

4 5

P

54

Example: Loop 4

x2

x

1

(x2, x, x)

x

Qleft

1

x2

4 5

Q

(x2, x2, 5)

55

Example: Loop 4

x2

x

1

(x2, x, x)

x

Qleft

1

x2

4 5

Q

(x2, x2, 5)

select freqQleft.x1, freqQleft.x4, freqP.x1, freqP.x4, freqP.x5, freqP.freq, freqP.freq/freqQleft.freqfrom freqP, freqQleftwhere freqQleft.x1=freqP.x1 and freqQleft.x4=freqP.x4and freqP.freq/freqQleft.freq >= minconf

56

Equivalent queries

Queries Q1 and Q2 are equivalent if same result sets on all

graphs G (up to renaming of the distinguished variables)

• 2 cases of equivalent queries:1. Q1 has fewer nodes than Q2

2. Q1 and Q2 have the same number of nodes

57

Equivalence theorem

A containment mapping from Q1 to Q2 is a h: Q1 Q2 that

maps distinguished variables of Q1 one-to-one to distinguished

variables of Q2, and maps selected nodes of Q1 to selected

nodes of Q2, preserving labels

Two queries are equivalent if and only if there are containment mappings between them in both directions.

58

Q2 x1

x3

x2

Case 1: Q1 fewer nodes than Q2

Redundancy lemma: Let Q be a tree query without selected nodes. Then Q has aredundancy if and only if it contains a subtree C in the form of

alinear chain of nodes (possibly just a single node), such that

the parent of C has another subtree that is at least as deep as C.

Q1 x1

x3

x2

Redundantsubtree

59

Case 2: Q1 and Q2 same number of nodes

• Q1 and Q2 must be isomorphic.

• Canonical form of queries: refine the canonical ordering of the underlying unlabeled tree, taking into account node labels.

60

Equivalent Association Rules

(x1, x2, x3, x)

,) ⇒

x1

x3

x2

(x1, x2, x2, x3)

,)

x1

x3

x2

x

(x1, x2, x3, x)

,) ⇒

x1

x3

x2

(x1, x2, x3, x3)

,)

x1

x3

x3

x

(1)

(2)

61

Equivalence detection for rules

• Many cases efficiently checked.• But worst case still as hard as general graph isomorphism checking.

• Fast heuristics for graph isomorphism checking i.e. Nauty

62

Certhia

• Loop 1 + Loop 2: preprocessing step Pattern database

• Loop 3 + Loop 4: interactive browsing tool Certhia Demo session

63

Experimental results: Performance

• Fully implemented on top of IBM DB2

• Preliminary performance results:– adequate performance– huge number of patterns– constant overhead per discovered pattern

64

Performance: Association rules

• Loop 3 and Loop 4: –very fast–constant overhead per rule

65

Future work

• Serious scientific data mining• Loosen restriction to trees

66

Publications

• B. Goethals, E. Hoekx, J. Van den Bussche, “Mining tree queries in a graph”, KDD’05, p 61–69.

• E. Hoekx, J. Van den Bussche, “Mining tree-query associations in a graph”, ICDM’06 regular paper.

• “Certhia: Tree-query mining in large graphs”, ICDM’06 software demo.

• http://alpha.uhasselt.be/~vdbuss