Multilevel Graph Clustering with Density-Based Quality Measures

4 Evaluation(a) karate(b) dolphinsFigure 4.11: Reference Graphs (karate and dolphins)computed by any configuration of the own multi-level algorithm is shown as vertexcolor.karate This is Zachary’s karate club study [82], maybe the most studied socialnetwork. A picture of the graph colored with the optimum clustering is shown inFigure 4.11a. During the 2 years long study, 34 members of a karate club wereobserved and the number of contacts outside of the club were counted. Like inmost other studies of this network the simple unweighted version of the network isused here. The instructor of the club (vertex #1) resigned because of conflicts withthe club administrator (vertex #34) about lesson fees. His supporters followed himforming a new organization with the members 1 to 9, 11 to 14, 17, 18, 20, 22. Person9 joined this faction although he was a supporter of the club president, because hewas only three weeks away from the test for black belt when the split occurred, andhad to follow the instructor to retain this rank.The greedy joining algorithm of Newman found 2 clusters with modularity 0.381 [60].By modifying the merge selector Danon found a clustering with modularity 0.4087 [17].The spectral clustering algorithms of Donetti et al. yielded modularity 0.412 [21]and 0.419 [20] in the improved version. Also Newman’s recursive spectral bisectionfound modularity 0.419 [58]. Surprisingly the algorithms based on random walksjust found clusterings with 0.3937 [65] and 0.38 [64]. Extremal Optimization with4 clusters of modularity 0.4188 [23] fell short of the optimum. As verified by [2]the global optimum clustering has 4 clusters with modularity 0.4197. The sameclustering was found by both multi-level algorithms.80