Multilevel Graph Clustering with Density-Based Quality Measures

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4 Evaluation1 2 3 4RWreach-none 1 0.56852 0.57150 0.57218 0.57380RWreach-sgrd 1 0.59678 0.59616 0.59663 0.59676RWreach-none 3 0.57530 0.57672 0.57468RWreach-sgrd 3 0.59658 0.59609 0.59760RWreach-none 5 0.57720 0.57102 0.55942RWreach-sgrd 5 0.59732 0.59464 0.59016Table 4.6: Mean modularity with Random Walk Reachability. Columns contain thelength of random walks and rows contain the algorithm combined with the numberof iterations. Empty cells are impossible combinations.number of iterations. Figure 4.4a shows a plot of the mean modularities vs. randomwalk length. Again the dashed lines mark the mean modularity produced by theweight density selector. Appendices B.2 and B.3 contain two detailed tables of themodularities gained on each graph with walks of length 2 and 3.In addition Figure 4.4a shows the runtime measured on the graph DIC28 main.Again the runtime difference between the sorted greedy refinement and no refinementprovides a visual relation to other components of the algorithm.In the mean modularity plot some small improvements are visible without refinement.However with refinement these completely disappear. Three iterations seemto be slightly better then just one iteration. Using five iterations has a negativeimpact when combined with longer walks.In summary no clear tendency or improvements were measurable. Maybe the usedbenchmark graphs are structurally unsuited for these merge selectors. In the furtherevaluation simply two steps long walks with three iterations (RW reachability (2,3))are chosen as default. This combination still can be computed quickly and does notreally degrade the mean modularity.4.3.3 Comparison of Merge SelectorsTo find the best merge selector in terms of effectiveness and efficiency this sectioncompares mean modularities produced with each merge selector and their runtime.The larger graph collection is used to gain more reliable mean values. As a side effectthis allows to verify whether the results from the reduced graph set are representative.For the same reason greedy matching is considered again checking whether itworks better with other merge selectors than the weight density.The studied configurations are composed of three components: Merge selector,coarsening method, and refinement method using the naming scheme Selector-Coarsening-Refinement. Cluster pairs are selected either by modularity increaselike used in the greedy joining algorithm of Newman (MI ), or by the weight density(WD), random walk reachability (RWR), spectral length (SPL), spectral length difference(SPLD), or the spectral angle (SPA). As coarsening method greedy grouping(G) and greedy matching (M ) are used. The third component is the refinement algorithm.Like before no refinement (none) and sorted greedy refinement (sgrd) isused.70
4.3 Effectiveness of the Merge SelectorsModularity by Merge Selector (large set)mean modularity0.44 0.46 0.48 0.50 0.52 0.54●●●●●●●G−noneM−noneG−sgrdM−sgrdSPL SPLD SPA MI WD RWRmerge selectorFigure 4.5: Mean Modularity of the Merge Selectors (large set)Runtime by Merge Selector (DIC28_main)Runtime by Merge Selector (Lederberg_main)runtime [s]20 50 100 200 500 1000 2000● ● ●●●●G−noneM−noneG−sgrdM−sgrd●runtime [s]20 50 100 200 500●●●●●●G−noneM−noneG−sgrdM−sgrd●SPL SPLD SPA MI WD RWRmerge selector(a) Runtime by Merge Selector on DIC28 mainSPL SPLD SPA MI WD RWRmerge selector(b) Runtime by Merge Selector onLederberg mainFigure 4.6: Runtime of the Merge Selectors71
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Brandenburgische Technische Univers
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ContentsList of FiguresList of Tabl
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List of Figures1.1 Graph of the Mex
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1 IntroductionSince the rise of com
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1.2 Objectives and Outline1.2 Objec
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2 Graph ClusteringThis chapter intr
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2.2 The Modularity Measure of Newma
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2.3 Density-Based Clustering Qualit
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2.4 Fundamental Clustering Strategi
Page 29 and 30: 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 82 and 83: 4 EvaluationG-none M-none G-sgrd M-
Page 84 and 85: 4 Evaluationmean modularity time DI
Page 86 and 87: 4 EvaluationRuntime vs. Graph SizeR
Page 88 and 89: 4 EvaluationComparison of Modularit
Page 90 and 91: 4 Evaluation(a) karate(b) dolphinsF
Page 92 and 93: 4 Evaluation(a) jazz(b) celegans me
Page 94 and 95: 4 Evaluationadministrators, and gra
Page 97 and 98: 5 Results and Future WorkThe object
Page 99 and 100: 5.3 Directions for Future Workstrat
Page 101: 5.3 Directions for Future Workties
Page 104 and 105: 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.
Page 110 and 111: A The Benchmark Graph Collectionsub
Page 112 and 113: B Clustering ResultsRWreach-sgrd 1
Page 114: B Clustering Resultswalktrap leadin
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Multilevel Graph Clustering with Density-Based Quality Measures

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