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Why Graph Mining and Searching?„ Graphs are ubiquitous „ Chemical compounds Cheminformatics „ Protein structures, biological pathways/networks Bioinformactics „ Program control flow, tra

Mining, Indexing and Searching

Graph Databases

Presenter: A/ Prof Do PhucSource: Jiawei Han , Vladimir Lipets

Graph, Graph, Everywhere

Aspirin Yeast protein interaction network

Why Graph Mining and Searching?

„ Graphs are ubiquitous

„ Chemical compounds (Cheminformatics)

„ Protein structures, biological pathways/networks (Bioinformactics)

„ Program control flow, traffic flow, and workflow analysis

„ XML databases, Web, and social network analysis

„ Graph is a general model

„ Trees, lattices, sequences, and items are degenerated graphs

„ Graph Isomorphism, Subgraph Isomorphism

„ Mining frequent graph patterns

„ Graph indexing methods

„ Similairty search in graph databases

„ Biological network analysis

„ Graph, Subgraph isomorphism is important and

very general form of pattern matching that finds practical application in areas such as:

„ pattern recognition and computer vision,

A hierarchy of pattern matching problems

„ Graph isomorphism

„ Approximate subgraph isomorphism

„ Graph edit distance

Isomorphic Graphs

Graph Isomorphism

Subgraph of a given graph

Subgraph Isomorphism

Subgraph Isomorphism and Related

Problems

„ Given a pattern graph G and a target graph H

„ Decision problem: Answer whether H contains a

subgraph isomorphic to G

„ Search problem: Return an occurrence of G as a

subgraph of H

„ Counting problem: Return a count of the number

of subgraphs of H that are isomorphic to G

„ Enumeration problem: Return all occurrences of

G as a subgraph of H

„ Graph Isomorphism, Subgraph Isomorphism

„ Mining frequent graph patterns

„ Graph indexing methods

„ Similairty search in graph databases

„ Biological network analysis

Graph Pattern Mining

„ Frequent subgraphs

„ A (sub)graph is frequent if its support (occurrence

frequency) in a given dataset is no less than a

minimum support threshold

„ Applications of graph pattern mining

„ Mining biochemical structures

„ Program control flow analysis

„ Mining XML structures or Web communities

„ Building blocks for graph classification, clustering,

comparison, and correlation analysis

Example: Frequent Subgraphs

S OH

O O

O N

O N

HO

O N

O N

O N

Frequent Subgraph Mining Approaches

„ Apriori-based approach

„ AGM/AcGM: Inokuchi, et al (PKDD’00)

„ FSG: Kuramochi and Karypis (ICDM’01)

„ PATH: Vanetik and Gudes (ICDM’02, ICDM’04)

„ FFSM: Huan, et al (ICDM’03)

„ Pattern growth-based approach

„ MoFa, Borgelt and Berthold (ICDM’02)

„ gSpan: Yan and Han (ICDM’02)

„ Gaston: Nijssen and Kok (KDD’04)

Properties of Graph Mining Algorithms

„ Search order

„ breadth vs depth

„ Generation of candidate subgraphs

„ apriori vs pattern growth

„ Elimination of duplicate subgraphs

„ passive vs active

„ Support calculation

„ embedding store or not

„ Discover order of patterns

„ path Æ tree Æ graph

„ Mining frequent graph patterns

„ Graph indexing methods

„ Similairty search in graph databases

„ Biological network analysis

Graph Search: Querying Graph Databases

„ Querying graph databases:

„ Given a graph database and a query graph, find all graphs containing this query graph

N N

OH O

N

O

N

OH O

S OH

S

HO O

O N

N O O

query graph graph database

S OH

S HO O O

N N O

O

OH O

Query graph

„ Index substructures of a query graph to prune graphs that do not contain these substructures

„ Two steps in processing graph queries

Step 1 Index Construction

database, build an inverted index between structures and graphs

Step 2 Query Processing

these structures

performing subgraph isomorphism test

„ Mining frequent graph patterns

„ Graph indexing methods

„ Similairty search in graph databases

„ Biological network analysis

„ Some recent progress on graph mining

Graph Clustering

„ Graph similarity measure

„ Feature-based similarity measure

„ Each graph is represented as a feature vector

„ The similarity is defined by the distance of their corresponding vectors

„ Frequent subgraphs can be used as features

„ Structure-based similarity measure

„ Maximal common subgraph

„ Graph edit distance: insertion, deletion, and relabel

Graph Classification

„ Local structure based approach

„ Local structures in a graph, e.g., neighbors

surrounding a vertex, paths with fixed length

„ Graph pattern-based approach

„ Subgraph patterns from domain knowledge

„ Subgraph patterns from data mining

„ Kernel-based approach

„ Random walk (Gärtner ’02, Kashima et al ’02,

ICML’03, Mahé et al ICML’04)

„ Optimal local assignment (Fröhlich et al

Structure Similarity Search

(a) caffeine (b) diurobromine (c) viagra

Some “Straightforward” Methods

„ Method1: Directly compute the similarity between the

graphs in the DB and the query graph

„ Sequential scan

„ Subgraph similarity computation

„ Method 2: Form a set of subgraph queries from the

original query graph and use the exact subgraph

search

„ Costly: If we allow 3 edges to be missed in a

20-edge query graph, it may generate 1,140 subgraphs

Index: Precise vs Approximate Search

„ Precise Search

„ Use frequent patterns as indexing features

„ Select features in the database space based on their selectivity

„ Build the index

„ Approximate Search

„ Hard to build indices covering similar subgraphs—

explosive number of subgraphs in databases

„ Idea: (1) keep the index structure

(2) select features in the query space

Substructure Similarity Measure

„ Query relaxation measure

„ The number of edges that can be relabeled or

missed; but the position of these edges are

not fixed

QUERY GRAPH

…

Substructure Similarity Measure

„ Feature-based similarity measure

„ Each graph is represented as a feature vector

X = {x1, x2, …, xn}

„ The similarity is defined by the distance of

their corresponding vectors

„ Advantages

„ Easy to index

„ Fast Rough measure

Query Processing Framework

„ Three steps in processing approximate graph

queries

Step 1 Index Construction

„ Select small structures as features in a graph database, and build the feature-graph matrix between the features

and the graphs in the database

Framework (cont.)

Step 2 Feature Miss Estimation

„ Determine the indexed features belonging

to the query graph

„ Calculate the upper bound of the number

of features that can be missed for an approximate matching, denoted by J

„ On the query graph, not the graph database

Framework (cont.)

Step 3 Query Processing

„ Use the feature-graph matrix to calculate the difference in the number

of features between graph G and query

Q, FG – FQ

„ If FG – FQ > J, discard G The remaining graphs constitute a candidate answer set

„ Mining frequent graph patterns

„ Graph indexing methods

„ Similairty search in graph databases

„ Biological network analysis

Data Mining Across Multiple Networks

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Data Mining Across Multiple Networks

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j a

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j

Identify Frequent Co-expression Clusters

across Multiple Microarray Data Sets

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CODENSE: Mine Coherent Dense Subgraphs

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summary graph Ĝ

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(2) Identify dense subgraphs of the summary graph

dense subgraph in the summary graph However, the

reverse is not true

CODENSE: Mine Coherent Dense Subgraphs

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Applying CoDense to 39 Yeast Microarray Data Sets

MRPL51

MRP49 YDR115W

PHB1

PET100

Discovery of New Genes Based on Similar Genes

Brown: YDR115W, FMC1, ATP12, MRPL37, MRPS18

MRPL32

ACN9

MRPL51 MRP49

YDR115W

PHB1

PET100 PET100

Network of Known Similar Genes

ACN9

MRPL51

MRP49 YDR115W

PHB1

PET100

Network Involved in the New Genes

„ Mining frequent graph patterns

„ Graph indexing methods

„ Similairty search in graph databases

„ Biological network analysis

„ Graph mining has wide applications

„ Frequent and closed subgraph mining methods

„ gSpan and CloseGraph: pattern-growth depth-first search

approach

„ Graph indexing techniques:

„ Frequent and discirminative subgraphs as indexing fatures

„ Similairty search in graph databases

„ Indexing and approximate matching help similar subgraph search

„ Biological network analysis

„ Mining coherent, dense, multiple biological networks

Many new developments along the line of graph pattern mining

Thanks and Questions