## 5 Experiments and Discussions
### Evaluation Index  
In this paper, we use two widely adopted evaluation metrics for community detection algorithms to assess the quality of the algorithm's community detection results. For labeled real network datasets, we utilize Normalized Mutual Information (NMI) [52] and modularity [36] to evaluate the community detection results of each algorithm. For unlabeled real-world network datasets, since the ground-truth community structures of these network datasets are still unknown, we assess the quality of the detected results in terms of the modularity only. For LFR networks, we use NMI to evaluate the community detection results of each algorithm.  
The NMI score is defined as shown in Equation (17):  
\[NMI(X,Y)=\frac{-2\sum\limits_{i=1}^{C_{X}}\sum\limits_{j=1}^{C_{Y}}C_{ij}\text {lb}\left(\frac{C_{ii}N}{C_{i}C_{j}}\right)}{\sum\limits_{i=1}^{C_{X}}C_{i} \text{lb}\left(\frac{C_{i}}{N}\right)+\sum\limits_{j=1}^{C_{Y}}C_{j}\text{lb} \left(\frac{C_{i}}{N}\right)} \tag{17}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset** & **Nodes** & **Edges** & **Community** \\ \hline Karate & 34 & 78 & 2 \\ Dolphins & 62 & 159 & 2 \\ Polbooks & 105 & 441 & 3 \\ Football & 115 & 613 & 12 \\ Lesmis & 77 & 254 & - \\ Jazz & 198 & 5484 & - \\ Email & 1133 & 5451 & - \\ Nestscience & 1589 & 2742 & - \\ Power Grid & 4941 & 6594 & - \\ Facebook & 4039 & 88,234 & - \\ PGP & 10,680 & 24,316 & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Basic information of real network datasets.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Network** & \(n\) & \(\overline{d}\) & \(d_{\text{max}}\) & min\(c\) & max\(c\) & \(tau1\) & \(tau2\) & \(\mu\) \\ \hline LFR1 & 1000 & 15 & 50 & 20 & 100 & 2 & 1.1 & 0.1–0.8 \\ LFR2 & 4000 & 15 & 50 & 20 & 100 & 2 & 1.1 & 0.1–0.8 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The parameters for LFR network construction.  
where \(X\) represents the ground truth community structure and \(Y\) represents the community detection results of the algorithm. \(C_{X}\) denotes the number of true communities in the ground truth, and \(C_{Y}\) represents the number of communities detected by the algorithm. \(\mathbf{C}\) represents the confusion matrix, where rows represent the ground truth community structure and columns represent the algorithm's community detection results. \(C_{ij}\) represents the number of common nodes between the true community \(i\) in \(X\) and the community \(j\) detected in \(Y\). \(C_{i}\) represents the sum of row \(i\) in matrix \(\mathbf{C}\), and \(C_{j}\) represents the sum of row \(j\) in matrix \(\mathbf{C}\). \(N\) stands for the total number of nodes in the network. \(NMI\in[0,1]\). A higher NMI value indicates a better agreement between the algorithm's community detection results and the true community structure, thus implying a better performance of community detection.  
The modularity(Q) is defined as shown in Equation (18):  
\[Q=\frac{1}{2m}{\sum_{i,j}}\bigg{(}a_{ij}-\frac{d(v_{i})\times d(v_{j})}{2m} \bigg{)}\times\delta\big{(}c_{i},c_{j}\big{)} \tag{18}\]  
where \(m\) denotes the number of edges in the network, \(a_{ij}\) denotes the connection status between node \(v_{i}\) and node \(v_{j}\); \(c_{i}\) denotes the community label of node \(v_{i}\), \(c_{j}\) denotes the community label of node \(v_{j}\); \(\delta(c_{i},c_{j})\) denotes the Kronecker function, which takes the value 1 if \(c_{j}\) and \(c_{j}\) are the same; otherwise, it takes the value 0. Generally, a larger modularity value implies a more distinct community structure.