Multilevel Graph Clustering with Density-Based Quality Measures

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2 Graph Clusteringthe global quality until no such pair is left. The merges are directed by a selectorthat tries to predict the optimal cluster structure. As the method alone is unableto correct wrong merges later, good predictors are necessary. These should lead toa faithful cluster structure with a high quality. As side product the merge historyprovides a hierarchical clustering.One of the most used agglomeration methods is due to Ward [79]. He developeda hierarchical clustering method to assign US Air Force personnel to tasks whileoptimizing multiple objectives. By design agglomeration is very similar to the graphcoarsening used in multi-level graph partitioning algorithms [43, 78, 41]Mark Newman was the first using agglomeration to directly optimize the modularitymeasure [60]. He called it greedy joining and selected clusters by the maximalincrease of modularity. Later the algorithm was accelerated by using sorted treesfor faster lookups [15]. Recently Wakita and Tsurumi [76] proposed a different implementation.Like in [17] they observed an unbalanced growth behavior with theoriginal selector. They introduced various consolidation ratios as relative size ofcluster pairs. Then as selection criterion the quotient of modularity increase andconsolidation ratio was applied.Cluster Refinement Refinement methods improve an initial clustering by applyinglocal modifications. They are an ideal way to correct small errors in the clustering.Generally refinement methods construct several similar clusterings from the current.The most common type of such neighborhood is formed by moving single vertices intoother clusters. As the next clustering the best from this neighborhood is selected.Greedy refinement accepts only new clusterings with a better quality than thecurrent. Thus the refinement ends in a local optimum where all clusterings of theneighborhood have worse quality. Kernighan and Lin [45] proposed a very popularmethod that tries to escape such local optima by accepting small sequences of slightlyworse clusterings. Fiduccia and Mattheyses [24] simplified this strategy by justconsidering single vertex moves instead of constructing and moving whole groups.Most of these methods were developed for minimum cut graph partitioning. Inthe context of modularity clustering recently simple refinement methods based ongreedy and Kernighan-Lin refinement were proposed [57, 5, 69]. Unfortunately nodetailed study exists how the refinement perform in modularity clustering althoughthe objective is quite different from minimum cut partitioning.2.4.3 Meta-StrategiesFinally meta-strategies combine construction methods from the previous subsection.The idea is to exploit advantages of single components while compensating disadvantages.The strategies discussed in the next paragraphs are recursive subdivisions,multi-level approaches, basin hopping, and genetic algorithms.Recursive Subdivision and Hierarchical Clusterings Some algorithms can only finda fixed number of clusters, for example [80, 81, 80, 58]. In order to find more clustersand better clusterings the algorithms are applied recursively on each previouslycomputed cluster. Subdivisions into exactly two groups are called bisection and22
2.4 Fundamental Clustering Strategiesfirst bisectionsecond bisection(a) Recursive Bisection(b) Hierarchical Clustering and DendrogramFigure 2.3: Recursive Subdivision and Hierarchical Clusterings. The red dashed linemarks the cut through the dendrogram. The corresponding clustering is drawn byred circles.an example recursive bisection is shown in Figure 2.3a. The recursive applicationresults in a hierarchical clustering.Like agglomeration methods recursive subdivision is unable to correct errors.Moreover the subdivision methods work with a fixed number of clusters. This imposesa structure on the clusterings which may hide the correct structure [57]. Forexample a bisection of a graph with three clusters might falsely split one of theclusters into two parts. Further bisections will not correct this.Also other algorithms producing hierarchical clusterings exist. For example themerge steps in agglomeration methods and the connectivity components of dissectionmethods provide a cluster hierarchy. Such hierarchies are often drawn as a tree calleddendrogram. The root is a cluster containing all vertices and each leaf contains justa single vertex.To derive a normal, flat clustering a cut through this tree is searched as shown inFigure 2.3b. The simplest variants select the level with best modularity [59]. Moreelaborate implementations search for better clusterings by shifting the level up anddown in subtrees [63]. Of course this search is inherently limited by the underlyingtree.Multi-Level Approaches Multi-level methods were developed to improve the searchrange and speed of partition refinement methods by moving small vertex groups atonce. These groups and the connections between the groups are represented as acoarse graph. In this graph it is sufficient for the refinement to move just singlecoarse-vertices. Applying this idea recursively leads to multi-level refinement methodslike [41, 78]. Karypis and Kumar [43] were the first proposing to use agglomerationmethods to already optimize the clustering quality during the constructionof coarse graphs. However up to now no multi-level refinement algorithm exists todirectly optimize the modularity.23
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 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 29 and 30: 2.4 Fundamental Clustering Strategi
Page 31: 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
Page 63 and 64: 3.5 Further Implementation NotesInd
Page 65 and 66: 3.5 Further Implementation NotesBOO
Page 67: 3.5 Further Implementation Notesfor
Page 70 and 71: 4 Evaluationparameter component des
Page 72 and 73: 4 Evaluationsignificance scale also
Page 74 and 75: 4 EvaluationModularity by Match Fra
Page 76 and 77: 4 Evaluation5% 10% 30% 50% 100%G-no
Page 78 and 79: 4 Evaluationmean modularity0.50 0.5
Page 80 and 81: 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
Page 94 and 95:
4 Evaluationadministrators, and gra
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5 Results and Future WorkThe object
Page 99 and 100:
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.
Page 106 and 107:
BIBLIOGRAPHY[42] H. Jeong, B. Tombo
Page 108 and 109:
BIBLIOGRAPHY[71] A. J. Soper and C.
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A The Benchmark Graph Collectionsub
Page 112 and 113:
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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