# A Constrained Louvain Algorithm with a Novel Modularity
## 4 Results  
In this section, we evaluate the performance of the constrained Louvain algorithm with different objective functions and compare the constrained Louvain algorithm with \(F2\) and two classical algorithms by the normalized mutual information (\(NMI\)) [47]. The better the algorithm, the higher the value of the \(NMI\). First of all, we execute the constrained Louvain algorithm with different objective functions on the ground truth networks and the \(LFR\) benchmark networks. For the ground truth networks, the result is as shown in Table 2. The \(NMI\) values of \(F2\) are the highest for the networks, except for the karate network. For the karate network, the constrained Louvain algorithm with \(F2\) and \(Q\) divides the network into four communities. For the constrained Louvain algorithm with \(F2\), after merging two small \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Networks** & \(N\) & \(L\) & \(M\) & \(\langle k\rangle\) & **References** \\ \hline Karate & 34 & 78 & 2 & 4.5882 & [44] \\ Dolphins & 62 & 159 & 4 & 5.129 & [46] \\ Football & 115 & 613 & 12 & 10.6609 & [6] \\ DBLP & 317,180 & 1,049,866 & 4770 & 6.62 & [10] \\ Amazon & 33,4863 & 925,872 & 1015 & 5.52986 & [10] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The structure parameters and the community number of networks. For each network, parameters \(N\), \(L\) and \(m\) represent the numbers of nodes, edges and standard communities respectively, and \(\langle k\rangle\) represents the average degree of all nodes in the network. communities, the node 10 is different than the ground truth network. As a result, The \(NMI\) value of the constrained Louvain algorithm with \(F2\) is smaller than the one with \(Q\). For all other ground truth networks, the constrained Louvain algorithm with \(F2\) is the best. In particular, for DBLP and the Amazon network, the constrained Louvain algorithm with \(Q\) identifies 179 and 201 communities, respectively. However, the constrained Louvain algorithm with \(F2\) obtains 7916 and 1819 communities, respectively. Obviously, the \(Q\) has a resolution limit problem and the \(F2\) has not had this problem. For the constrained Louvain algorithm with \(F\) and \(M\), there are many communities that are not satisfying the definition of a weak community, or the sizes are less than 4. Finally, these communities are disbanded. Therefore, the result is inferior to the constrained Louvain algorithm with \(F2\). We also ran the constrained Louvain algorithm on the \(LFR\) benchmark networks. In Figure 2a, the \(NMI\) values of the constrained Louvain with \(F2\) are highest when \(\mu<0.4\). With the increase in \(\mu\), the community structure becomes more fuzzy and harder to identify, all \(NMI\) values decrease and the \(NMI\) value of the constrained Louvain algorithm with \(F2\) becomes less than the one with \(Q\). This demonstrates that the constrained Louvain algorithm with \(F2\) has the best performance for the networks with a clear community structure. Here, it is not necessary to calculate the cases of \(\mu>0.5\), because the communities are not satisfying the definition of a weak community. In Figure 2b, for the small value of \(\gamma\), the \(NMI\) values of the constrained Louvain algorithm with \(Q\) are the highest. However, with the increase in \(\gamma\), the heterogeneity of networks becomes stronger and the \(NMI\) values of the constrained Louvain algorithm with \(F2\) becomes higher than the constrained Louvain algorithm with other objective functions. Obviously, the stronger the heterogeneity of the networks are, the better is the performance of \(F2\). In Figure 2c, for the high values of the parameter \(\beta\), the constrained Louvain algorithm with \(F2\) has the highest \(NMI\) values. With the increase in \(\beta\) value, the distribution of the community sizes becomes heterogeneous so that the resolution limit problem appears for some objective functions.