Multilevel Graph Clustering with Density-Based Quality Measures

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Contents3.5.2 Data Management . . . . . . . . . . . . . . . . . . . . . . . . 564 Evaluation 594.1 Methods and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1.1 Configuration Space . . . . . . . . . . . . . . . . . . . . . . . 594.1.2 Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.1.3 Efficiency and Scalability . . . . . . . . . . . . . . . . . . . . 634.2 Effectiveness of the Graph Coarsening . . . . . . . . . . . . . . . . . 644.2.1 Match Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.2 Coarsening Methods . . . . . . . . . . . . . . . . . . . . . . . 654.2.3 Reduction Factor . . . . . . . . . . . . . . . . . . . . . . . . . 674.3 Effectiveness of the Merge Selectors . . . . . . . . . . . . . . . . . . . 674.3.1 Random Walk Distance . . . . . . . . . . . . . . . . . . . . . 674.3.2 Random Walk Reachability . . . . . . . . . . . . . . . . . . . 694.3.3 Comparison of Merge Selectors . . . . . . . . . . . . . . . . . 704.4 Effectiveness of the Cluster Refinement . . . . . . . . . . . . . . . . . 724.4.1 Greedy Refinement . . . . . . . . . . . . . . . . . . . . . . . . 734.4.2 Kernighan-Lin Refinement . . . . . . . . . . . . . . . . . . . . 744.5 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.6 Comparison to Reference Algorithms . . . . . . . . . . . . . . . . . . 764.7 Comparison to Published Results . . . . . . . . . . . . . . . . . . . . 794.7.1 The Graphs and Clusterings . . . . . . . . . . . . . . . . . . . 794.7.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 Results and Future Work 875.1 Results of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 By-Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.3 Directions for Future Work . . . . . . . . . . . . . . . . . . . . . . . 895.3.1 Pre-Coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3.2 Study of Merge Selectors . . . . . . . . . . . . . . . . . . . . 905.3.3 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . 905.3.4 Multi-Pass Clustering and Randomization . . . . . . . . . . . 905.3.5 High-Level Refinement Search . . . . . . . . . . . . . . . . . . 90Bibliography 93A The Benchmark Graph Collection 99B Clustering Results 101VI
List of Figures1.1 Graph of the Mexican political elite . . . . . . . . . . . . . . . . . . 22.1 Example Volume Models . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Graph of US Airports . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Recursive Subdivision and Hierarchical Clusterings . . . . . . . . . . 233.1 Operations for local cluster modification . . . . . . . . . . . . . . . . 263.2 The Multi-Level Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 The Multi-Level Algorithm . . . . . . . . . . . . . . . . . . . . . . . 273.4 Coarsening Method: Greedy Grouping . . . . . . . . . . . . . . . . . 293.5 Merging two Vertices and their Edges . . . . . . . . . . . . . . . . . 303.6 Coarsening Method: Greedy Matching . . . . . . . . . . . . . . . . . 313.7 Contribution of Neighbor Vertices to the Visit Probability . . . . . . 363.8 Spectral vertex vectors and two cluster vectors . . . . . . . . . . . . 403.9 Dependencies when Moving a Vertex . . . . . . . . . . . . . . . . . . 453.10 Refinement Method: Complete Greedy . . . . . . . . . . . . . . . . . 483.11 Refinement Method: Sorted Greedy . . . . . . . . . . . . . . . . . . 493.12 Refinement Method: basic Kernighan-Lin . . . . . . . . . . . . . . . 503.13 Kernighan-Lin Refinement Creating Clusters on Graph Epa main . . 513.14 Effective Search Depth of Kernighan-Lin Refinement . . . . . . . . . 523.15 Index Spaces: Class and Concept Diagram . . . . . . . . . . . . . . . 533.16 Index Maps: Class and Concept Diagram . . . . . . . . . . . . . . . 543.17 C++ Example: Calculation of Vertex Degrees . . . . . . . . . . . . . 554.1 Mean Modularity by Match Fraction (reduced set) . . . . . . . . . . 644.2 Modularity and Runtime by Reduction Factor . . . . . . . . . . . . . 654.3 The Random Walk Distance . . . . . . . . . . . . . . . . . . . . . . . 684.4 The Random Walk Reachability . . . . . . . . . . . . . . . . . . . . . 694.5 Mean Modularity of the Merge Selectors (large set) . . . . . . . . . . 714.6 Runtime of the Merge Selectors . . . . . . . . . . . . . . . . . . . . . 714.7 Mean Modularity by Refinement Method (reduced set) . . . . . . . . 734.8 Mean Modularity by Refinement Method (large set) . . . . . . . . . 754.9 Runtime by Graph Size . . . . . . . . . . . . . . . . . . . . . . . . . 764.10 Clustering Results and Runtime of the Reference Algorithms . . . . 784.11 Reference Graphs (karate and dolphins) . . . . . . . . . . . . . . . . 804.12 Reference Graphs (polBooks and afootball) . . . . . . . . . . . . . . 814.13 Reference Graphs (polBooks and celegans metabolic) . . . . . . . . . 824.14 Reference Graphs (circuit s838 and email) . . . . . . . . . . . . . . . 83VII
Page 1: Brandenburgische Technische Univers
Page 5: ContentsList of FiguresList of Tabl
Page 11 and 12: 1 IntroductionSince the rise of com
Page 13 and 14: 1.2 Objectives and Outline1.2 Objec
Page 15 and 16: 2 Graph ClusteringThis chapter intr
Page 17 and 18: 2.2 The Modularity Measure of Newma
Page 19 and 20: 2.3 Density-Based Clustering Qualit
Page 27 and 28: 2.4 Fundamental Clustering Strategi
Page 35 and 36: 3 The Multi-Level Refinement Algori
Page 37 and 38: 3.1 The Multi-Level Schemeas starti
Page 39 and 40: 3.2 Graph CoarseningData: graph,sel
Page 41 and 42: 3.2 Graph Coarseningnearly no edges
Page 43 and 44: 3.3 Merge SelectorsExtent Name Desc
Page 45 and 46: 3.3 Merge Selectorsdifferent size.
Page 47 and 48: 3.3 Merge SelectorsThe probability
Page 49 and 50: 3.3 Merge SelectorsAs selection qua
Page 51 and 52: 3.3 Merge Selectorsvectors the eige
Page 53 and 54: 3.4 Cluster Refinementleave the loc
Page 55 and 56: 3.4 Cluster Refinementmoving v from
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3.4 Cluster RefinementAlgorithm Sea
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3.4 Cluster RefinementData: graph,c
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3.4 Cluster RefinementModularity0.2
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3.5 Further Implementation NotesInd
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3.5 Further Implementation NotesBOO
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3.5 Further Implementation Notesfor
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4 Evaluationparameter component des
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4 Evaluationsignificance scale also
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4 EvaluationModularity by Match Fra
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4 Evaluation5% 10% 30% 50% 100%G-no
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4 Evaluationmean modularity0.50 0.5
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4 Evaluation1 2 3 4RWreach-none 1 0
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4 EvaluationG-none M-none G-sgrd M-
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4 Evaluationmean modularity time DI
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4 EvaluationRuntime vs. Graph SizeR
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4 EvaluationComparison of Modularit
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4 Evaluation(a) karate(b) dolphinsF
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4 Evaluation(a) jazz(b) celegans me
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4 Evaluationadministrators, and gra
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5 Results and Future WorkThe object
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5.3 Directions for Future Workstrat
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5.3 Directions for Future Workties
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BIBLIOGRAPHY[14] B.L. Chamberlain.
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BIBLIOGRAPHY[42] H. Jeong, B. Tombo
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BIBLIOGRAPHY[71] A. J. Soper and C.
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A The Benchmark Graph Collectionsub
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B Clustering ResultsRWreach-sgrd 1
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B Clustering Resultswalktrap leadin
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Multilevel Graph Clustering with Density-Based Quality Measures

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