Multilevel Graph Clustering with Density-Based Quality Measures

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4 EvaluationG-none M-none G-sgrd M-sgrdSPL 0.43653 0.44014 0.48521 0.51962SPLD 0.45479 0.48348 0.49926 0.53231SPA 0.52405 0.52252 0.54041 0.54341MI 0.52963 0.51985 0.53968 0.54369WD 0.52305 0.51866 0.54608 0.54529RWR 0.52971 0.52768 0.54718 0.54744Table 4.7: Mean Modularity of Merge Selectors. Columns contain the algorithmsand rows the merge selectors.Table 4.7 displays the mean modularities of all 24 configurations on the large graphset. The rows contain the six merge selectors and the columns the four algorithmvariants. The rows are sorted by their overall mean modularity. Figure 4.5 shows aplot of the same data.In order to compare the runtime of the configurations the times measured on thegraphs DIC28 main and Lederberg main are shown in Figures 4.6a and 4.6b. Alogarithmic scale is used to easier differentiate the runtime.The first two spectral merge selectors SPL and SPLD did not perform well interms of runtime and clustering results. This could also be expected from theirmathematical derivation as approximation of the modularity increase. Howeverthe modularity values on these two show that the matching (M ) method is lesssensitive against bad selectors than grouping (G). All other merge selectors SPA,WD, MI, and RWR yielded very similar clustering results when compared to themodularity improvement possible by refinement. This is an interesting observationbecause the modularity increase selector (MI ) was expected to produce worse resultsthan the weight density (WD). Very small modularity improvements over the otherselectors were gained by the random walk reachability (RWR). On the better mergeselectors greedy matching (M ) turned out to be as good as greedy grouping (G)when refinement was used. Without refinement grouping produced slightly betterresults. On DIC28 main matching with refinement was slightly faster than grouping.But on the graph Lederberg main with its extremely skewed distribution of vertexdegrees greedy grouping always was considerably faster.Altogether the weight density selector (WD) often had the lowest runtime whilestill producing good mean modularities. Therefore it will be used as default merge selector.On the large graph set the grouping and matching methods were comparablygood in terms of mean modularity. This result slightly differs to the measurementsfrom the reduced graph set.4.4 Effectiveness of the Cluster RefinementThis section compares the multi-level refinement algorithms against raw graph coarseningwithout refinement. The first subsection concentrates on the greedy refinementvariants to select the best in terms of modularity improvement and search speed.And finally the second subsection compares the best greedy refinement against the72
4.4 Effectiveness of the Cluster RefinementModularity by Refinement MethodRuntime by Refinement Method (DIC28_main)mean modularity0.570 0.575 0.580 0.585 0.590 0.595runtime [s]0 1000 2000 3000 4000none SGR−mod SGR−eo CGR SGR−density KL(a) Modularity by Refinement Methodnone SGR−mod SGR−eo CGR SGR−density KL(b) Runtime by Refinement Method onDIC28 mainFigure 4.7: Mean Modularity by Refinement Method (reduced set)presented Kernighan-Lin method.4.4.1 Greedy RefinementThe complete greedy refinement method moves the vertex with the best modularityincrease in each move step until no further improvement is possible. However itis computational expensive to find the best vertex because it requires to evaluateall possible moves of all vertices in each step. For this reason the sorted greedyrefinement was introduced. It saves search time by trying to move the vertices ina specific order according to a vertex selector. In the following evaluation the bestof the available vertex selectors is selected. And finally it is studied whether thecomplete greedy refinement can be replaced by the sorted variant without degradingthe results too much.The three sorted greedy refinement methods are SGR-density using the densityfitnessvertex selector, SGR-mod using the mod-fitness, and SGR-eo using the eofitness.The complete greedy refinement is named CGR. To provide a significancescale Kernighan-Lin refinement (KL) and no refinement (none) is included. All otherparameters are set to their default values, i.e. greedy grouping by weight density isused for the coarsening.Table 4.8 summarizes the mean modularity produced by each refinement algorithmon the reduced graph set. The second column displays the runtime of the algorithmson the graph DIC28 main. The rows are sorted by their mean modularity. Inaddition to the table the bar plot 4.7a visually compares the mean modularities.The runtime is also plotted in Figure 4.7b.Between the three sorted greedy variants (SGR-density, SGR-mod, and SGR-73
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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
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2.4 Fundamental Clustering Strategi
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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
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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