## 4 Community Detection on Induced Metric Space
### Computational Results  
In this subsection, we compare two groups of algorithms for network community detection with our proposed algorithm based on metric space. The experiment is performed on a large list of network datasets. Two versions of the experiment are developed for comparison purposes based on two different quality measures: conductance and modularity. The results based on conductance are shown in the Table 3, and the results based on modularity are shown in the Table 4, respectively. Regarding the two groups of algorithms, the first group contains algorithms based on semi-definite programming, and the second group contains algorithms based on graph traversal approaches. For each group, we have taken the best value of conductance in Table 3 and the best value of modularity in Table 4 among all of the algorithms in the groups. The results obtained with our approach are very competitive with most of the well-known algorithms in the literature, and this is justified over the large collection of datasets. On the other hand, it can be observed that time taken (Table 5) by our algorithm is quite less compared to other methods and justifies the theoretical findings described in Sections 3 and 4. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Name** & **Type** & **No. of Nodes** & **No. of Edges** & **Diameter** & **DCC** & **k** \\ \hline Facebook & U & 4039 & 88,234 & 4.7 & 0.72498 & 164 \\ Gplus & D & 107,614 & 13,673,453 & 3 & 0.50073 & 457 \\ Twitter & D & 81,306 & 1,768,149 & 4.5 & 0.57072 & 213 \\ Epinions1 & D & 75,879 & 508,837 & 5 & 0.14001 & 128 \\ LiveJournal1 & D & 4,847,571 & 68,993,773 & 6.5 & 0.27432 & 117 \\ Pokec & D & 1,632,803 & 30,622,564 & 5.2 & 0.10971 & 246 \\ Slashdot0811 & D & 77,360 & 905,468 & 4.7 & 0.05884 & 81 \\ Slashdot0922 & D & 82,168 & 948,464 & 4.7 & 0.06340 & 87 \\ Friendster & U & 65,608,366 & 1,806,067,135 & 5.8 & 0.16231 & 833 \\ Orkut & U & 3,072,441 & 117,185,083 & 4.8 & 0.16689 & 756 \\ Youtube & U & 1,134,890 & 2,987,624 & 6.5 & 0.08090 & 811 \\ DBLP & U & 317,080 & 1,049,866 & 8 & 0.63307 & 268 \\ Arxiv-AstroPh & U & 18,772 & 396,160 & 5 & 0.65841 & 23 \\ web-Stanford & D & 281,903 & 2,312,497 & 9.7 & 0.60034 & 69 \\ Amazon0601 & D & 403,394 & 3,387,388 & 7.6 & 0.41890 & 92 \\ P2P-Gnutella31 & D & 62,586 & 147,892 & 6.5 & 0.00710 & 35 \\ RoadNet-CA & U & 1,965,206 & 5,533,214 & 500 & 0.40458 & 322 \\ Wiki-Vote & D & 7115 & 103,689 & 3.8 & 0.17048 & 21 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Complex network datasets and the values of their parameters. DCC, data complexity for community detection. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Name** & **Spectral** & **SDP** & **GT** & **Metric** \\ \hline Facebook & 0.4487(1) & 0.5464(4) & 0.5434(5) & 0.5472 \\ Gplus & 0.2573(1) & 0.4047(3) & 0.3998(5) & 0.4056 \\ Twitter & 0.3261(3) & 0.3706(1) & 0.3691(7) & 0.3709 \\ Epinions1 & 0.0280(1) & 0.1440(3) & 0.1401(5) & 0.1447 \\ LiveJournal1 & 0.0791(1) & 0.1455(5) & 0.1432(5) & 0.1458 \\ Pokec & 0.0129(3) & 0.0294(1) & 0.0288(5) & 0.0295 \\ Slashdot0811 & 0.0038(1) & 0.0130(4) & 0.0127(7) & 0.0131 \\ Slashdot0922 & 0.0045(1) & 0.0176(5) & 0.0171(5) & 0.0176 \\ Friendster & 0.0275(4) & 0.0536(5) & 0.0526(7) & 0.0536 \\ Orkut & 0.0294(3) & 0.0689(4) & 0.0675(5) & 0.0690 \\ Youtube & 0.0096(1) & 0.0934(2) & 0.0903(5) & 0.0936 \\ DBLP & 0.4011(5) & 0.4214(1) & 0.4207(5) & 0.4215 \\ Arxiv-AstroPh & 0.4174(3) & 0.5079(3) & 0.5045(5) & 0.5081 \\ web-Stanford & 0.3595(5) & 0.3908(4) & 0.3896(7) & 0.3908 \\ Amazon0601 & 0.1768(1) & 0.2649(4) & 0.2615(7) & 0.2650 \\ P2P-Gnutella31 & 0.0009(1) & 0.0522(2) & 0.0503(5) & 0.0523 \\ RoadNet-CA & 0.0212(3) & 0.1690(4) & 0.1633(5) & 0.1692 \\ Wiki-Vote & 0.0266(1) & 0.2093(1) & 0.2023(5) & 0.2095 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of our approaches with other best methods in terms of modularity; the numbers inside the brackets denote the algorithm of the group.