Multilevel Graph Clustering with Density-Based Quality Measures

More documents

Recommendations

Info

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

5 Results and Future Work5.3.2 Study of Merge SelectorsDuring the course of this work a measure for the prediction quality of merge selectorswas developed. It counts the percentage of the vertex pairs having a high selectionquality that would actually select a pair of vertices in the same final cluster. Forthis purpose a reference clustering is necessary.However it turned out that this prediction quality alone does not much about thegeneral performance of a merge selector. In practice it is also important to studywhen and where coarsening errors, i.e. putting together vertices of different clusters,occur. For example it might be that the spectral angle merge selector performs wellon big, fine grained graphs but just fails on the coarser coarsening levels. On the finelevels all information is non-locally stretched while the coarsening joins informationabout the structure and may produce good local information later on. In that casethe weight density selector should be used on the coarser levels instead.5.3.3 Linear ProgrammingIt is possible to formulate the clustering problem as linear or quadratic program [13,2]. Instead of classic rounding techniques the computed distances could be used asmerge selection quality. This would enable multi-level refinement on the roundingresults.The fractional linear program is solvable in polynomial time but requires |V | 2space for the distance matrix and cubic space for the transitivity constraints. Thusit becomes impracticable already for medium-sized graphs unless the constraintsare replaced by a more compact implicit representation. However for the presentedmulti-level refinement approximate distances between adjacent vertices would suffice.Maybe such approximations could be faster computed using other representationsof the optimization aim. For example in [6] an embedding into higher-dimensionalunit spheres under square Euclidean norm was used for similar quality measures.5.3.4 Multi-Pass <strong>Clustering</strong> and RandomizationA meta-strategy similar to evolutionary search [72] is multi-pass clustering. Becausethe refinement corrects coarsening errors the computed clustering contains valuableinformation. In a second pass this can be fed back into the coarsening by ignoringall vertex pairs crossing previous cluster boundaries. This effectively producesa corrected coarsening hierarchy and allows further improvements by refinementheuristics. Coarsening and refinement are repeated until the clustering does notfurther improve. The multi-pass search may be widened by applying some kind ofrandomization during the coarsening.5.3.5 High-Level Refinement SearchIn the domain of cluster refinement several improvements might be possible. Forexample restarting the Kernighan-Lin search on intermediate clusterings allows toimprove the search depth. In this context a slight randomization would help breaking90
5.3 Directions for Future Workties and avoiding to cycle repeatedly through the same movement sequence. But atoo strong randomization prevents the detection of local optima again.Still the Kernighan-Lin approach is very slow because in modularity clusteringthe quality improvements of vertex moves are difficult to compute. On the otherhand in this work a very fast greedy refinement method was developed. Similarly afast, randomized method to leave local optima could be developed. Combining bothwould allow to walk between local optima like in the basin hopping method [52].However some open questions remain how this can be effectively combined <strong>with</strong> themulti-level strategy. For example currently a lot of information about the graph islost between each coarsening level because just the last best clustering is projectedto the finer level.91
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 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
2.4 Fundamental Clustering Strategi
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
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 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: 5.3 Directions for Future Workstrat
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
show all

Multilevel Graph Clustering with Density-Based Quality Measures

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?