1 Mining Tree Queries in a Graph Bart Goethals, Eveline Hoekx and Jan Van den Bussche KDD ’ 05 presentor: Ming Jing Tsai.

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Mining Tree Queries in a Graph

Bart Goethals , Eveline Hoekx and Jan Van den Bussche

KDD’05presentor: Ming Jing Tsai

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Introduction

mining tree pattern T in a single graph Incremental in the number of nodes Unordered, rooted

For each tree T, all conjunctive queries are generated

SQL

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Tree query pattern example

Selected node(constant):0,8 Existential node:∃ Distinguished node: x

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matching A query Q matchs in a graph G Homomorphism h

(i,j) ∈ Q , (h(i), h(j)) ∈ G Verify value on x to distinguish them

Don’t care existential nodes on different values

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∃0 8

Q

G

Frequency = 3(4,5,8)

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Generate all trees

Increasing number of nodes Canonically ordered

Level sequence ith number is the depth of the ith node in preord

er Lexicagraph:Maximal one

Level sequence 012212 > 012122

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queries

Levelwise Fix a tree T, and find all queries based o

n T whose frequency in G is at lease k Q{∏, ∑, λ}

∏: existential nodes ∑: selected nodes λ: label of selected nodes

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To generate candidate in an efficient manner,using of candidacy tables and frequency tables

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CanTab ∏, ∑

parents

Each candidacy table can be computed by taking the natural join of its parent’s(∏’, ∑’) frequency tables

CanTabφ,{x} as the table with a single column x,holding all nodes of the graph G being mined

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∏=x2,formulate expression->SQL

∑={x1,x3} Candidacy table

Frequency table

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Equivalent queries

To avoid query Q2 equivalent to an earlier query Q1

Containment mapping Q1 to Q2 is a homomorphism the distinguished variables of Q1 is mapping

one-to-one to those of Q2 So as selected nodes

Case1:Q1 has fewer nodes than Q2 Case2:Q1 and Q2 have the same number

of nodes

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Case1 redundancy checking

Q2 contains redundant subtrees such that removing them yields an equivalent query

Redundancy a subtree C in the form of a linear chain of exist

ential nodes such that parent of C has another subtree that is at least as deep as C

Q1Q2Q2

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Case 2 canonical forms

Q1 and Q2 are tree isomorphism Canonical forms

Existential nodes-> ∃ Selceted nodes ->c Distinguished nodes->X

C, ∃

∃,C

∃,X

C,X

X,C

X,X

C, ∃

∃,C

∃,X

C,X

X,C

X,X

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experiment

Pentium4 2.8GHz 1GB main memory Linux 2.6 C++ embedded SQL Relational database:DB2 UDB v8.2

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Real dataset

A food web, a protein intersactions graph, and a citation graph

k: frequency threshold Size: maximal size of trees in the run It all takes several hours

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Food web

154 species dependent on Scotch Broom Label 20 occurs in many frequent patterns->

Orthotylus adenocarpi( 什麼都吃的植物害蟲 )

Frequency 176

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Protein interaction graph

1870 種 Saccharomyces cerevisiae 發酵酵母菌 ( 幫助麵包發酵 )

A small number of highly connected nodes occur

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Citation graph

Kdd cup 2003 2500 papers high-energy physics 350,000 cross-references

Frequency 1655

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Synthetic data,web graphs Tree size 5 Minsup 4,10,25

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Uniform random graphs

Dense, uniform minsup: 10,25 edges:47,264,997