Multilevel Graph Clustering with Density-Based Quality Measures

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$Eine Einführung in LaTeX-Beamer - studiy - Brandenburgische ...$

1 IntroductionFigure 1.1: Graph of the Mexican political elite [30] from 1910 to 1994. Each boxrepresents a Mexican politician. The family name and the main year of politicalactivity is printed for each. The names of actual presidents are printed in red color.The lines connect related politicians based on friendships, business partnerships,marriages, and similar. Finally the color of the boxes shows the best clusteringcomputed with the algorithms presented in this work.2
1.2 Objectives and Outline1.2 Objectives and OutlineThe main objective of this work is the implementation and evaluation of effective andefficient clustering algorithms for the modularity quality measure of Newman [60].Optimizing the modularity is a NP-complete problem [13]. Therefore it is unlikelythat a fast, polynomial time algorithm finds a clustering of optimal modularity inall graphs. Instead relatively good clusterings have to suffice.The major design goal for the clustering algorithm is the effectiveness. The developedalgorithm shall be able to find clusterings of very good modularity comparedto alternative methods. Secondly the efficiency is a concern. Complicated and expensivealgorithms should replaced by simpler alternatives in case they are not ableto produce significantly better clusterings.This work focuses on multi-level refinement heuristics. In the past these weresuccessfully applied on similar problems like minimum cut partitioning [41, 43, 78].However modularity clustering differs significantly from these. The optimizationaim is for example more complex because the number of clusters is not known inadvance but has to be discovered by the algorithm. To date no detailed study existson the adaption of multi-level refinement to modularity clustering. In this contextalso a comparison of the effectiveness and efficiency to other clustering algorithmsis necessary.Chapter 2 introduces the mathematical formulation of the modularity clusteringaim and discusses related concepts. This analysis provides the base for the laterdevelopment of strategies specific to the modularity. The chapter concludes in Section2.4 with the study of existing clustering algorithms. From these fundamentalstrategies and components are derived. The developed multi-level algorithm willcombine several particularly useful components to benefit from their advantages.Based on the insights in the mathematical structure and fundamental clusteringstrategies Chapter 3 presents the developed family of algorithms. The algorithms arebased on multi-level refinement methods adapted to the special needs of modularityclustering. The algorithm operates in two phases. First a hierarchy of coarsenedgraphs is produced. Then this hierarchy is used in the refinement of an initialclustering.The coarsening phase involves the selection of pairs of clusters to be merged. Asub-goal of this work is the exploration of alternative selection criteria in order toimprove the effectiveness and overall clustering results. This includes criteria derivedfrom random walks and spectral methods like presented in Section 3.3. In additionthe implementation should incorporate appropriate data structures and algorithmsto efficiently support these merge selector.The second major component of the algorithm are cluster refinement heuristics.These try to improve an initial clustering by exploring similar clusterings and selectingone of them. Here such neighbor clusterings are generated by moving singlevertices. The objective of Section 3.4 is the exploration of heuristics for the searchof local optima, i.e. clusterings without any neighbor clusterings of better modularity.Here the efficiency is an important concern because many traditional heuristics,like Kernighan-Lin refinement [45, 24], cannot be efficiently implemented for themodularity measure.3
Page 1: Brandenburgische Technische Univers
Page 5 and 6: ContentsList of FiguresList of Tabl
Page 7: List of Figures1.1 Graph of the Mex
Page 11: 1 IntroductionSince the rise of com
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
Page 57 and 58: 3.4 Cluster RefinementAlgorithm Sea
Page 59 and 60: 3.4 Cluster RefinementData: graph,c
Page 61 and 62: 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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