# Link Pruning for Community Detection in Social Networks
## 4 Experiments
### Link Attribute Distribution
#### 4.2.2 Real-World Networks  
As in Section 4.2.1, we calculated the link attributes of each network and analyzed the distribution of these attributes in real-world networks. In Table 2, links connecting nodes in the same community are classified as intra-community, while the links connecting nodes in different communities are classified as inter-community. Intra-community and inter-community are marked in blue and red, respectively, in a histogram. Accordingly, the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset** & **\# of Nodes** & **\# of Links** & **Clustering Coefficient** \\ \hline Karate & 34 & 78 & 0.571 \\ \hline Football & 115 & 613 & 0.403 \\ \hline Twitter & 348 & 4831 & 0.475 \\ \hline DBLP & 317,080 & 1,049,866 & 0.632 \\ \hline Amazon & 334,863 & 925,872 & 0.397 \\ \hline YouTube & 1,134,890 & 2,987,624 & 0.081 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Real-world network datasets.  
Figure 1: Distribution of link attributes in synthetic networks: (**a**) Jaccard; (**b**) CommonTriangles; and (**c**) Forman–Ricci.  
values of the mean and standard deviation are indicated in Table 2. As shown in Table 2, the mean value of the intra-community was higher than that of the inter-community in 14 out of the entire 18 cases. Regarding the Football and Twitter networks, the values of intra-community were higher than those of inter-community in the entire case. For the Amazon network, the mean number of nodes that belonged to the corresponding community was only 4.46, despite low mixing. Consequently, the values of the intra-community were unlikely to be distinguished from those of the inter-community.  
We conducted case studies on the Football network with three link attributes. We compared the patterns of intra-community links with inter-community links. Figure 2 presents the distribution of link attributes in the Football network. As shown in the histograms of the three link attributes, the distribution of intra-community links was separated from that of inter-community links. Figure 3 presents the results of graph sparsification of the Football network according to the pruning rate from 0 to 0.4 using Jaccard's index. Here, the graph with pruning rate 0 represents the original graph. As shown in Figure 3, we confirm enhanced community structure with link pruning.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Dataset** & **Link Type** & **Jaccard** & **CommonTriangles** & **Forman-Ricci** \\ \hline \multirow{2}{*}{Karate} & intra & \(0.34\pm 0.13\) & \(\mathbf{3.90\pm 1.63}\) & \(\mathbf{-5.63\pm 5.25}\) \\ \cline{2-5}  & inter & \(\mathbf{0.46\pm 0.15}\) & \(2.73\pm 1.05\) & \(-10.7\pm 5.64\) \\ \hline \multirow{2}{*}{Football} & intra & \(\mathbf{0.47\pm 0.11}\) & \(\mathbf{7.52\pm 1.28}\) & \(\mathbf{-1.06\pm 3.56}\) \\ \cline{2-5}  & inter & \(0.17\pm 0.11\) & \(3.16\pm 1.60\) & \(-13.7\pm 5.11\) \\ \hline \multirow{2}{*}{Twitter} & intra & \(\mathbf{0.26\pm 0.11}\) & \(\mathbf{21.1\pm 13.0}\) & \(\mathbf{-39.4\pm 28.8}\) \\ \cline{2-5}  & inter & \(0.16\pm 0.07\) & \(16.5\pm 10.3\) & \(-67.3\pm 25.2\) \\ \hline \multirow{2}{*}{DBLP} & intra & \(0.36\pm 0.31\) & \(\mathbf{6.95\pm 13.53}\) & \(\mathbf{-21.96\pm 40.0}\) \\ \cline{2-5}  & inter & \(\mathbf{0.43\pm 0.30}\) & \(0.36\pm 0.31\) & \(0.36\pm 0.31\) \\ \hline \multirow{2}{*}{Amazon} & intra & \(\mathbf{0.31\pm 0.20}\) & \(\mathbf{2.22\pm 2.30}\) & \(-12.49\pm 25.70\) \\ \cline{2-5}  & inter & \(0.29\pm 0.20\) & \(1.66\pm 1.81\) & \(\mathbf{-12.32\pm 25.22}\) \\ \hline \multirow{2}{*}{YouTube} & intra & \(0.06\pm 0.09\) & \(\mathbf{9.49\pm 21.07}\) & \(\mathbf{-472.96\pm 691.84}\) \\ \cline{2-5}  & inter & \(\mathbf{0.09\pm 0.15}\) & \(2.78\pm 12.14\) & \(-998.67\pm 3324.71\) \\ \hline \end{tabular}
\end{table}
Table 2: Distribution of link attributes in real-world networks.  
Figure 2: Distribution of link attributes in Football: (**a**) Jaccard; (**b**) CommonTriangles; and (**c**) Forman–Ricci.