## 4 Discussion  
Simulations have been performed considering a weighted multiplex network \(M\) as explained in Section 3. To build the model, do computation and obtain our results, we used the programming language R and the IDE RStudio. The figures were generated thanks to the package Plotly and Linkcomm [40; 41; 42; 43]. We consider a population \(N=61\) in the two layers \(l_{1},l_{2}\) representing two distinct kinds of weighted interactions and connectivity between nodes. In the first layer \(l_{1}\), the interactions are based on the graph extracted from a real contact network [33], while in \(l_{2}\), we consider a theoretical scheme for the same population of nodes \(N\). In Figure 5, we can observe how the structural heterogeneity in the multiplex representation of the networks leads to highlight several key aspects for the structural roles of nodes. For a heterogeneous weighted multiplex network, we study how certain metrics are distributed across different layers and in the whole structure. We have computed the degree \(k_{i}^{l}\), the strength \(s_{i}^{l}\) and the overlapping degree \(o_{i}\) as indicated in the two heath-map plots. Both plots display metrics for the multiplex \(M\) composed by the first layer based on real contact network while through the second layer we can compare the case of a scale-free network (in the top-left plot) with a small-world network (in top-right plot). For the sake of clarity, we indicate the first resulting multiplex \(M\) as \(RC-SF\), and the second one as \(RC-SW\). As shown in Figure 5, our findings exhibit two different levels of heterogeneity. We can notice how in the \(RC-SW\) case, there is a small heterogeneity due to a high clustering and modularity, meaning that there are groups of nodes more highly connected than the rest of the network with an abundance of high degree nodes \begin{table}
\begin{tabular}{c c c} \hline \hline
**Community Profiling** & **Formalism** & **Description** \\ \hline Parameters [9; 13] & \(\overrightarrow{m}_{ij}\) & Multilink between \(i,j\) \\  & \(\bar{G}^{w}\) & Weighted Aggregated Network \\  & \(A_{ij}\) & Elements of \(\bar{G}^{w}\) \\  & \(k_{i}^{\overline{n}l}\) & Multidegree of node \(i\) \\  & \(n_{ci}\) & Number of cooperations \\  & \(c_{b}\) & Between-ness Centrality \\ \hline Defined Metrics & \(hc_{ij|\bar{G}}\) & Score Function for dissimilarity structure \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Community Profiling Parameters and Defined Metrics. Here, we summarize the general formalism linked to the mesoscale analysis applied to the multiplex network \(M\). We list the key parameters and some defined metrics used to compose the community profiling of nodes in a 6G scenario represented and analyzed through the proposed framework and the mentioned above mathematical tools, based on complex systems and networks. that act as hubs of the network. Instead, in the \(RC-SF\) case, there are few hubs in the network since it exhibits a highly heterogeneous and high degree correlations. In the other two panels below in Figure 5, the curves correspond to the distribution of the participation coefficient \(P_{i}\), in the range \([0,1]\), considering this variation in function of the Z-score of the overlapping degree, \(z(o_{i})\). We extract profiling features structurally classifying the nodes of the multiplex in three classes, focused, mixed and multiplex nodes, putting in evidence the fitness based on the two cases \(RC-SF\) and \(RC-SW\), the awareness distribution of \(aw_{i}\) and the homophily distribution of the values \(h_{ij}\), in accordance with the modeling approach. Our findings highlight an increasing of the \(P_{i}\) value in both cases (\(RC-SF\)-\(RC-SW\)), and a higher density of regular multiplex nodes. Decreasing the homophily (from the left plot to the right plot below), the findings exhibit a decreasing of \(P_{i}\) for the \(RC-SW\) case that has a more homogeneous distribution with a higher density of regular mixed nodes.