Efficient Mining of GraphBased Data

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Efficient Mining of Graph-Based Data
Jesus Gonzalez, Istvan Jonyer, Larry Holder and
Diane Cook University of Texas at
Arlington Department of Computer Science and
Engineering http//cygnus.uta.edu/subdue
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Motivation

• Structural/relational data
• Ease of graph representation

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Graph-Based Discovery
Input Database
Substructure S1 (graph form)
Compressed Database
T1
triangle
shape
C1
C1
S1
B1
object
R1
R1
on
square
S1
S1
S1
shape
T2
T3
T4
object
B2
B3
B4
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Algorithm

• Create substructure for each unique vertex label

Substructures
triangle (4), square (4), circle (1), rectangle
(1)
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Algorithm

• Expand best substructure by an edge or
  edgeneighboring vertex

Substructures
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Algorithm

• Keep only best beam-width substructures on queue
• Terminate when queue is empty or discovered
  substructures gt limit
• Compress graph and repeat to generate
  hierarchical description
• Note polynomially constrained

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Evaluation Metric

• Substructures evaluated based on ability to
  compress input graph
• Compression measured using minimum description
  length (DL)
• Best substructure S in graph G minimizes DL(S)
  DL(GS)

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Examples
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Inexact Graph Match

• Some variations may occur between instances
• Want to abstract over minor differences
• Difference cost of transforming one graph to
  isomorphism of another
• Match if cost/size lt threshold

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Parallel/Distributed Discovery

• Divide graph into P partitions using Metis,
  distribute to P processors
• Each processor performs serial Subdue on local
  partition
• Broadcast best substructures, evaluate on other
  processors
• Master processor stores best global substructures
• Close to linear speedup

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Graph-Based Concept Learning

• One graph stores positive examples
• One graph stores negative examples
• Find substructure that compresses positive graph
  but not negative graph
• (PosEgsNotCovered) (NegEgsCovered)
• Multiple iterations implements set-covering
  approach

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Concept-Learning Example
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Concept-Learning Results

• Chess endgames (19,257 examples)
• Black King is () or is not (-) in check
• 99.8 FOIL, 99.21 Subdue

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More Concept-Learning Results

• Tic-Tac-Toe endgames
• is win for X (958 examples)
• 100 Subdue, 92.35 FOIL
• Bach chorales
• Musical sequences (20 sequences)
• 100 Subdue, 85.71 FOIL

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Graph-Based Clustering

• Iterate Subdue until single vertex
• Each cluster (substructure) inserted into a
  classification lattice

Root
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Clustering Example Animals
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Graph-Based Clustering Results
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Cobweb Results

• Comparison of Subdue and Cobweb results
• Subdue lattice produced better generalization,
  resulting in less clusters at higher levels
• Subdue lattice identifies overlap between
  (reptile) and (amphibian/fish)

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Clustering Example DNA
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Graph-Based Clustering Results

• Coverage
• 61
• 68
• 71

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Evaluation of Clusterings

• Traditional evaluation
• Not applicable to hierarchical domains
• Does not make sense to compare clusters in
  different subtrees
• Not applicable to relational clusterings

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Properties of Good Clusterings

• Small number of clusters
• Large coverage ? good generality
• Big cluster descriptions
• More features ? more inferential power
• Minimal or no overlap between clusters
• More distinct clusters ? better defined concepts

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New Evaluation Heuristic for Hierarchical
Clusterings

• Clustering rooted at C with c children Hi having
  Hi instances Hi,k
• distance() measured by inexact graph match
• Animals SubdueCQ2.6, CobwebCQ1.7

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Graph-Based Data Mining Application Domains

• Biochemical domains
• Protein data
• DNA data
• Toxicology (cancer) data
• Spatial-temporal domains
• Earthquake data
• Aircraft Safety and Reporting System
• Telecommunications data
• Program source code
• Web topology

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Theoretical Analysis

• Galois lattice Lequiere et al.
• Conceptual graphs Sowa et al.
• PAC analysis Jappy et al.

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Graph-based Data Mining

• Pattern (substructure) discovery
• Hierarchical discovery
• Distributed discovery
• Concept learning
• Clustering
• Compression heuristic based on minimum
  description length

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Future Work

• Concept learning
• Theoretical analysis
• Comparison to ILP systems
• Clustering
• Classification lattice
• Hierarchical relational conceptual clustering
  evaluation metric
• Probabilistic substructures
• Domains WWW, source code

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Subdue Source Code and Data

• http//cygnus.uta.edu/subdue