Multilevel Graph Clustering with Density-Based Quality Measures

4.6 Comparison to Reference Algorithmswalktrap leadingev wakita HE wakita HN fgj ML-none spinglass ML-sgrd ML-KLChain8 0.31633 0.31633 0.37755 0.31633 0.35714 0.35714 0.37755 0.35714 0.37755Star9 -0.21875 0.00000 -0.078125 -0.078125 -0.00781 -0.00781 -0.00781 0.00000 0.00000K55 0.00000 0.30000 0.30000 0.30000 0.00000 0.30000 0.30000 0.30000 0.30000Tree15 0.47449 0.50510 0.51276 0.50510 0.50510 0.50510 0.50510 0.50510 0.51276ModMath main 0.34641 0.42287 0.31712 0.35488 0.41051 0.42018 0.44880 0.42905 0.44880SouthernWomen 0.00000 0.21487 0.22775 0.24157 0.31467 0.31530 0.33121 0.31972 0.33601karate 0.39908 0.37763 0.35947 0.41880 0.38067 0.40869 0.41979 0.41979 0.41979mexican power 0.22869 0.29984 0.33772 0.31949 0.33089 0.32811 0.35119 0.34776 0.35952Grid66 0.47833 0.55000 0.42222 0.49236 0.49597 0.51319 0.53250 0.55000 0.54125Sawmill 0.40153 0.44745 0.49102 0.48660 0.55008 0.55008 0.53863 0.55008 0.55008dolphins 0.44043 0.48940 0.48687 0.44838 0.49549 0.51786 0.52852 0.52587 0.52587polBooks 0.49927 0.39922 0.46585 0.49138 0.50197 0.50362 0.52640 0.52724 0.52561adjnoun 0.17640 0.22153 0.24365 0.25766 0.29349 0.28428 0.31336 0.31078 0.31078sandi main 0.78157 0.78667 0.80995 0.82054 0.82729 0.82083 0.81619 0.82773 0.82773USAir97 0.28539 0.26937 0.34439 0.31526 0.32039 0.34353 0.36597 0.36824 0.36824circuit s838 0.70256 0.70215 0.76499 0.79488 0.80472 0.79041 0.81481 0.81551 0.81551CSphd main 0.88971 0.74219 0.91382 0.92001 0.92470 0.92484 0.90584 0.92558 0.92558Erdos02 0.62534 0.59221 0.66991 0.67756 0.67027 0.68484 0.70985 0.71592 0.71611DIC28 main 0.70285 0.70923 0.73570 0.74179 0.78874 0.80154 0.81747 0.84747 0.84781average 0.39630 0.43927 0.45803 0.46445 0.47180 0.49272 0.50502 0.50753 0.51100Table 4.10: Clustering Results of Reference Algorithmscluster size |C i | and the wakita HE variant the of number of edges leaving thecluster. A few other algorithms are available through the igraph library of Csárdiand Nepusz [16]. 4 The following were used. The spinglass algorithm of Reichardtand Bornholdt [67] optimizes modularity by simulated annealing on a physical energymodel. The implementation requires an upper bound on the number of clusters. Inthis evaluation the upper bound 120 was used for all graphs. A spectral method isNewman’s [58] recursive bisection based on the first eigenvector of the modularitymatrix (leadingev). Finally an agglomeration method based on random walks is thewalktrap algorithm of Pons and Latapy [64].For comparison the standard configuration with three refinement variants waschosen. All use greedy grouping by weight density with 10% reduction factor. Thevariant without any refinement (ML-none) is similar to other pure agglomerationmethods, namely fgj, wakita-HN, and wakita-HE. The two other refinement variantsare sorted greedy refinement by density-fitness (ML-sgrd) and the improvedKernighan-Lin refinement (ML-KL).Table 4.10 lists the modularity of the clusterings found by the implementations.The graphs are sorted by number of vertices and the algorithms by mean modularity.The last row shows the arithmetic mean modularity for each algorithm. Foreach graph the algorithms with the best result are marked with a bold font. A similartable of comparing the runtime of the algorithms is included in Appendix B.6.Figure 4.10 summarizes the mean modularity gained by each algorithm and themeasured runtime.It is visible that in average greedy grouping by density (ML-none) is better than allother agglomeration methods (fgj, wakita HE, wakita HN ) even without refinement.Yet on a few graphs other agglomeration methods perform slightly better. Looking atthe multi-level refinement methods (ML-sgrd, ML-KL) only the spinglass algorithm4 available at http://igraph.sourceforge.net/77