Multilevel Graph Clustering with Density-Based Quality Measures

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3 The Multi-Level Refinement Algorithmbest vertexbest unmovedSearch Mode The search mode controls how vertices are selected based on thevertex ranking. It plays a similar rule like the grouping and matching algorithmsin graph coarsening. The most accurate search method is to consider always allpossible operations: In each step all possible moves of all vertices are evaluated andthe global best modularity increase is selected. This is a costly operation becauseevaluating all moves requires to visit all edges in each move step. For k vertex movesthe k|V | maxmod evaluations lead to O(k|E|+k|V | max |C|) average run-time wheremax |C| is the highest number of clusters the occurred during the moves.Fortunately this method can be divided into two stages by first selecting the bestvertex according to a vertex ranking and then evaluating further operations only forthat vertex (mode: best vertex). In combination with the maxmod vertex rankingthis is equal to the previous method. Instead the vertex ranking should be replacedby a constant-time predictor. Then performing k vertex moves requires at most k|V |evaluations of the vertex ranking and one maxmod target ranking per moved vertex.Thus the average run-time will be in O(k|V | + k|E|/|V | + k max |C|).Most predictors are likely to also select vertices which cannot be moved formodularity increase. In greedy refinement these vertices must be skipped. In orderto not select these vertices again all processed vertices are marked and ignored later(mode: best unmoved). This search mode thus selects one of the remaining verticesuntil no vertex is left. The whole process is restarted in case some early selectedvertices became movable after later moves.Ranking Updates An algorithm could simply iterate over a fixed ordering of thevertices or update the ranking after each move. The updated ranking is implementedby simply visiting all vertices in each move step. Alternatively a heap can be used toefficiently retrieve the best ranked vertex. However empirical comparisons suggestedthat all variants are similarly fast. Therefore visiting all vertices is the preferredmethod.On Modularity Decrease In case the selected operation does not improve themodularity following three choices are possible:skip, abort,acceptskip Mark the vertex as moved but leave it in the current cluster.abort Abort the refinement.accept Perform the operation anyway.Greedy heuristics are characterized by not accepting non-improving operations.Whether to directly abort the refinement depends on the search method. In caseother vertices still allow modularity increasing moves the currently selected vertexshould be skipped. Otherwise the refinement can be aborted.Classification of Basic Refinement Algorithms Table 3.2 provides an overviewand classification of refinement methods building on the components described in46
3.4 Cluster RefinementAlgorithm SearchMode Vertex Ranking Mod. Decrease Impl.greedy refinementcomplete greedy best maxmod abort Xsorted-maxmod best unmoved maxmod abortsorted-mod best unmoved mod-fitness skip Xsorted-eo best unmoved eo-fitness skip Xsorted-dens best unmoved dens-fitness skip Xdeterministic searchKL-maxmod best unmoved maxmod accept XKL-mod best unmoved mod-fitness acceptKL-eo best unmoved eo-fitness acceptKL-dens best unmoved dens-fitness acceptsimulatedannealingspinglasssystemextremaloptimizationrandomized searchall random random target ranking,accept withprobability dependingon modularityincreaseall random random target rankingwith an energymodel, accepts likesimulated annealingbest+random eo-fitness acceptTable 3.2: Classification of Refinement Algorithms47
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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 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: 3.4 Cluster Refinementmoving v from
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 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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