## 4 Results and Discussion
### Simulated Multiplex Networks  
In this experiment, the adopted simulated multiplex networks are developed in [32]. The structure of these networks is controlled by two parameters: \(\mu_{0}\) and \(p_{0}\). \(\mu_{0}\) is the community mixing parameter that controls the network's modularity, while \(p_{0}\) is the value of the probability of the inter-layer dependency tensor which is used to control the dependency across the layers of the network. The values of \(\mu_{0}\) and \(p_{0}\) can vary in the range of \([0,1]\). As the value of \(\mu_{0}\) increases, the noise level in the network increases. The simulated multiplex networks are generated by varying the parameter \(\mu_{0}\) to control the noise level in the multiplex network so that the robustness of the proposed **Mx-CRTSA** can be evaluated. In our experiment, we kept the value of \(p_{0}=1\) and varied the value of \(\mu_{0}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Objective function** & **Constraints** \\ \hline CentroidCoreg & \(\max\limits_{\mathbf{U}^{(m)},\mathbf{U}}\sum\limits_{m=1}^{M}\left[tr(\mathbf{ U}^{(m)}\top^{\mathbf{L}}\mathbf{L}^{(m)}\mathbf{U}^{(m)}+\right.\) & \(\mathbf{U}^{*}\top^{\mathbf{U}}\mathbf{u}^{*}=\mathbf{I}\), \(\mathbf{U}^{(m)}\top^{\mathbf{U}}\mathbf{U}^{(m)}=\mathbf{I}\). \\  & \(\lambda_{m}tr(\mathbf{U}^{(m)}\mathbf{U}^{(m)})^{\top}\mathbf{U}^{*}\mathbf{U}^ {*}\top))\right]\) & \(\mathbf{U}^{*}\top^{\mathbf{U}}\mathbf{u}^{*}=\mathbf{I}\). \\ \hline SCML & \(\min\limits_{\mathbf{U}}\sum\limits_{m=1}^{M}tr(\mathbf{U}^{\top}\mathbf{L}^{( m)}\mathbf{U})-\lambda\sum\limits_{m=1}^{M}tr(\mathbf{UU}^{*}\top\mathbf{U}^{(m)} \mathbf{U}^{(m)}\top^{\mathbf{\top}})\) & \(\mathbf{U}^{*}\top^{\mathbf{U}}\mathbf{u}^{*}=\mathbf{I}\). \\ \hline CSNMF & \(\min\limits_{\mathbf{H}}\|\mathbf{A}^{(m)}-\mathbf{U}^{*}\mathbf{U}^{*}\top^{ \mathbf{I}}\|_{F}^{2}+\|\mathbf{UU}^{*}\top^{-\mathbf{U}}-\mathbf{U}^{(m)} \mathbf{U}^{(m)}\top^{\mathbf{\top}}\|_{F}^{2}\) & \(\mathbf{U}\geq 0\), \(\mathbf{U}^{*}\top^{\mathbf{U}}\mathbf{u}^{*}=\mathbf{I}\). \\ \hline CPNMF & \(\min\limits_{\mathbf{H}}\|\mathbf{A}^{(m)}-\mathbf{U}^{*}\mathbf{U}^{*}\top^{ \mathbf{A}}(m)\|_{F}^{2}+\|\mathbf{UU}^{*}\top^{-\mathbf{U}}-\mathbf{U}^{(m)} \mathbf{U}^{(m)}\top^{\mathbf{\top}}\|_{F}^{2}\) & \(\mathbf{U}\geq 0\), \(\mathbf{U}^{*}\top^{\mathbf{U}}\mathbf{u}^{*}=\mathbf{I}\). \\ \hline Mx-CRTSA & \(\min\limits_{\mathbf{U}^{*}\top^{\mathbf{U}}=\mathcal{Z},\mathbf{E},\mathbf{w}} \|\mathcal{Z}\|_{\oplus}+\gamma\|\mathbf{E}\|_{2,1}-\) & \\ Mx-CRTSA & \(\lambda\sum\limits_{m=1}^{M}\left[tr\Big{(}\mathbf{U}^{(m)}\top^{\mathbf{Z}} \mathbf{Z}^{(m)}_{N}\mathbf{U}^{(m)}\Big{)}+\right.\) & \(\mathbf{A}^{(m)}=\mathbf{A}^{(m)}\mathbf{Z}^{(m)}+\mathbf{E}^{(m)},\mathbf{U}^ {*}\top^{\mathbf{U}}\mathbf{u}^{*}=\mathbf{I},\mathbf{U}^{(m)}\top^{\mathbf{ U}}\mathbf{u}^{(m)}=\mathbf{I}\), & \(\|\mathbf{w}\|_{2}=1\), \(\mathbf{w}\geq 0\) \\  & \(\left.w_{m}tr\Big{(}\mathbf{U}^{*}\top^{\mathbf{U}}\mathbf{U}^{(m)}\mathbf{U}^ {(m)}\top^{\mathbf{\mathbf{\top}}}\mathbf{u}^{*}\Big{)}\right]\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between the objective functions of the different methods considered in the experiments. The effectiveness of the different algorithms, including **Mx-CRTSA**, in discovering the community structure in multiplex networks as \(\mu_{0}\) changes in the range \([0.1,0.8]\) was validated by NMI and AR under multiple scenarios. In the first scenario, 2-layer multiplex networks with \(n=128\) nodes and \(c=4\) communities were generated. Examples of some of the networks tested in the first scenario for different values of \(\mu_{0}\) are shown Figure 2a-d. In the second scenario, the number of layers was increased to three, while the number of nodes and clusters were fixed, i.e., \(n=128\) and \(c=4\), to explore the performance of the proposed **Mx-CRTSA** method with networks with a larger number of layers. Furthermore, some examples of the generated networks from scenario 2 are shown in Figure 3a-f.