Multilevel Graph Clustering with Density-Based Quality Measures

4.7 Comparison to Published Results(a) polBooks(b) afootballFigure 4.12: Reference Graphs (polBooks and afootball)dolphins The dolphin social network is an undirected graph of frequent associationsbetween 62 dolphins in a community residing in Doubtful Sound in New Zealand [51,50]. For a picture see Figure 4.11b.This graph just recently appeared in the context of modularity clustering. Thusonly very few published results exist. With Fractional Linear Programming a clusteringof modularity 0.529 and the upper bound 0.531 was found [2]. Also the exactInteger Linear Programming gave an optimum clustering with modularity 0.529 [13].Using multi-level refinement 5 clusters with modularity 0.52586 (ML-KL-density)and 0.52772 (ML-KL-rw) were found. The best clustering found with experimentalconfigurations had modularity 0.52851 and the same clustering was found by thespinglass algorithm in the previous section.polBooks A network of books about recent US politics sold by the online booksellerAmazon.com. Edges between books represent frequent co-purchasing of books bythe same buyers (Fig. 4.12a). The network was compiled by Valdis Krebs [47].With Fractional Linear Programming a clustering of modularity 0.5272 and anupper bound 0.528 was found [2]. This result was confirmed by the exact IntegerLinear Programming method with modularity 0.527 [13]. The multi-level algorithmML-KL-rw and very many other configurations found 5 clusters with modularity0.52723. Surprisingly the ML-KL-density variant just found 0.52560 with 6 clustersalthough greedy refinement in the same configuration found the better clustering.afootball A real-world graph with known group structure is the games schedule ofDivision I of the United States college football for the 2000 season [31]. The 115teams are grouped into conferences of 8-12 teams and some independent teams. In81