Multilevel Graph Clustering with Density-Based Quality Measures

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4 Evaluationadministrators, and graduate students. Two users are connected by an edge if bothexchanged e-mails with each other (Fig. 4.14b). Thus the graph is undirected andunweighted.In the last years also this graph became popular for the evaluation of graph clusteringmethods. The first published result is for Extremal Optimization. It found15 clusters and modularity 0.5738 [23]. Danon’s greedy joining with modified mergeselector just yielded modularity 0.5569 [17]. The recursive spectral bisection of Newmanfound a clustering with modularity 0.572 [58]. The best published clusteringwith modularity 0.579 was found by recursive bisection with Quadratic Programming[2]. The multi-level algorithm ML-KL-density produced 11 clusters with modularity0.57767. A better clustering with 9 clusters and modularity 0.58137 wasobtained from the random walk variant ML-KL-rw.hep-th main A scientist collaboration network in high-energy physics. The graphcontains the collaborations of scientists posting preprints on the high-energy theoryarchive at http://www.arxiv.org between 1995 and 1999. The 5835 vertices representauthors and edges connect co-authors. The edges are weighted as describedin [61].The only published clustering is from a recursive spectral bisection method. Itproduced 114 clusters with modularity 0.707 [21]. The multi-level algorithm ML-KL-density found 57 clusters of 0.85523. A better clustering was found with therandom-walk variant ML-KL-rw. It contains 59 clusters and has modularity 0.85629.Erdos02 This is the year 2002 version of the collaboration network around Erdös [34].The vertices represent authors and the unweighted edges connect co-authors. Onlyauthors with a co-author distance from Erdös up to 2 are included.The only published clustering was found with greedy joining combined with randomwalk based pre-coarsening. It contains 20 clusters and has modularity 0.6817 [65].The multi-level algorithm ML-KL-density produced 39 clusters with modularity0.71611. A slightly better clustering with the same number of clusters and modularity0.716396 was found by the random-walk variant ML-KL-rwPGPgiantcompo A giant component of the network of users of the Pretty-Good-Privacy algorithm for secure information interchange compiled by Arenas et al. [12,35]. The component contains 10680 vertices. Edges connect users trusting eachother.The modified greedy joining of Danon found the worst clustering with modularity0.7462 [17]. Extremal Optimization found 365 clusters with modularity 0.8459 [23].The best published modularity got Newman’s spectral bisection with 0.855 [58].The best of both multi-level algorithms was ML-KL-rw. It found 96 clusters withmodularity 0.88462. The greedy grouping by density ML-KL-density yielded 102clusters with 0.88392. Finally the best clustering found in own experiments so farhas 94 clusters and modularity 0.885450.cond-mat-2003 main A collaboration network of scientists posting preprints onthe condensed matter archive at http://www.arxiv.org. This version contains data84
4.7 Comparison to Published Resultsgraph |V | |E| article modularity ML-KL-density ML-KL-rwkarate 34 78 [2] 0.4197 (4) 0.41978 (4) 0.41978 (4)dolphins 62 159 [2] 0.529 0.52586 (5) 0.52772 (5)polBooks 105 441 [2] 0.5272 0.52560 (6) 0.52723 (5)afootball 115 613 [2] 0.6046 0.59419 (7) 0.60582 (10)jazz 198 2742 [23] 0.4452 (5) 0.44487 (4) 0.44514 (4)celegans metabolic 453 2040 [2] 0.450 0.45090 (9) 0.44992 (8)circuit s838 512 819 [69] 0.815 (13) 0.81551 (15) 0.81550 (15)email 1133 5451 [2] 0.579 0.57767 (11) 0.58137 (9)hep-th main 5835 13815 [21] 0.707 (114) 0.85523 (57) 0.85629 (59)Erdos02 6927 11850 [65] 0.6817 (20) 0.71611 (39) 0.71639 (39)PGPgiantcompo 10680 24316 [58] 0.855 0.88392 (102) 0.88462 (96)cond-mat-2003 main 27519 116181 [65] 0.7251 (44) 0.81377 (77) 0.81586 (78)Table 4.11: Comparison to Published Resultsfrom 1995 to 2003. Vertices represent authors and edges connect co-authors. Theedges are weighted as described in [61]. Here only the largest component containing27519 vertices is used.Extremal Optimization discovered 647 clusters with modularity 0.6790 [23]. Latermuch better clusterings were found. Newman’s recursive spectral bisection yielded0.723 [58] and greedy joining combined with pre-coarsening found 44 clusters withmodularity 0.7251 [65]. Of the own algorithms the multi-level refinement ML-KLdensityproduced 77 clusters with 0.81377. A better clustering with 78 clustersand modularity 0.81586 was found by ML-KL-rw. However even better clusteringswere discovered in experiments. For example one with 82 clusters and modularity0.81718.4.7.2 SummaryThe Table 4.11 summarizes the clustering results. For each graph the best modularityfound in the literature and the modularity computed with the two multi-levelalgorithms ML-KL-density and ML-KL-rw is given. The graphs are sorted by thenumber of vertices and the best modularity on each graph is marked in bold.On smaller graphs optimal or at least very good clusterings were obtained withthe Linear Programming methods from [2] and [13]. In these cases the multi-levelalgorithms found comparable good clusterings. Only few published results for biggraphs exist. There the multi-level algorithm ML-KL-rw performed much betterthan other methods. Still the clusterings found by experimental configurations showthat on some graphs better results are possible. But these were found more byfortunate circumstances than by a reliable, systematic search.85
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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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3 The Multi-Level Refinement Algori
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3.1 The Multi-Level Schemeas starti
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3.2 Graph CoarseningData: graph,sel
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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
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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 97 and 98: 5 Results and Future WorkThe object
Page 99 and 100: 5.3 Directions for Future Workstrat
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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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