## 3 Results
### Collective Profiling
Taking into consideration the nature of the phenomena or content, we can diagrammatically express the first process, in terms of reaction-diffusion equations, following two different hypothesis as indicated below as the \(S^{h}IS^{h}\) or \(S^{h}IR\) spreading processes:  
\[S^{h}IS\Rightarrow S^{h}\xrightarrow{\beta_{i}^{h}}I\xrightarrow{\mu}S^{h} \tag{3}\]  
\[S^{h}IR\Rightarrow S^{h}\xrightarrow{\beta_{i}^{h}}I\xrightarrow{\mu}R \tag{4}\]  
The \(S^{h}IS^{h}\) can occurs when a node, based on its interactions, becomes prone to participate to the diffusion, and after that it returns in the condition to the starting pools of nodes. Differently, in the \(S^{h}IR\) the last transition is replaced by the step which occurs when a node have acquired a permanent condition and it is not available to participate again in the diffusion. Both models are characterize by the states \(S^{h}\), defined as \"heterogeneous susceptible state\", where a node is predisposed to be involved in the spreading process [15]. The state \(I\) indicates the condition in which a node is involved or infected, while \(R\) represents the recovered state. The probability of call off the diffusion is equal to \(\mu\), while the \(\beta_{i}^{h}\) is the diffusion rate for each node \(i\) at each layer \(\alpha\) in the multiplex network \(M\). The latter represents the probability that a node \(i\) in the layer \(\alpha\) is predisposed or susceptible to be involved in the diffusion process. We assume that, the involvement for a node in the network means that a node \(i\) is in the state of informed or infected. The heterogeneous diffusion rate depends on the weighted structural connectivity through the measures of inverse participation ratio \(Y_{i}^{\alpha}\)[13] and the rate of awareness \(\lambda_{i}^{\alpha}\), defined in Equation (8). \[\beta_{i}^{\alpha}=\tfrac{1}{1+\lambda_{i}^{\alpha}}\cdot\tfrac{1}{Y_{i}^{ \alpha}} \tag{5}\]  
Similarly, the second spreading process, focused on awareness diffusion, is diagrammatically expressed in terms of reaction-diffusion equation also in two possible different cases, as follows:  
\[U^{h}AU^{h}\Rightarrow U\xrightarrow{\lambda_{i}}A\xrightarrow{\delta}U^{h} \tag{6}\]  
\[U^{h}AF\Rightarrow U\xrightarrow{\lambda_{i}}A\xrightarrow{\delta}F \tag{7}\]  
with \(\lambda_{i}^{\alpha}\) and \(\delta\), defined as transition rates. We distinguish between the rate of awareness \(\lambda_{i}^{\alpha}\) for each node \(i\) at each layer \(\alpha\) of the multiplex \(M\) and the rate of awareness after being aware. The \(U^{h}\) state expresses a condition in which a node is heterogeneously unaware, while in \(A\), it is aware about the phenomena that spreads inter-personally in the network. In the second case, the state \(F\) is introduced as the \"faded state\", in which a node decreases the attention and the interest to improve its knowledge about the phenomena of interest. The rate of awareness is expressed as follows:  
\[\lambda_{i}^{\alpha}=\tfrac{s_{i}^{\alpha}}{1+s_{i}^{\alpha}}\cdot\lambda \tag{8}\]  
where \(s_{i}^{\alpha}\) is the strength of node \(i\) in the layer \(\alpha\) of the multiplex network \(M\)[13]. Starting from the assignment of a state probability for each node \(i\) in \(M\), to be in one of the initial states as (SU)-(SA)-(IA), and choosing the suitable spreading models for the co-evolution process, as indicated in Figure 4 and expressed in Equations (3) and (4), we add, in the profiling process, some features extracted from the analysis of the co-evolution in the multiplex network considered. To obtain the contagion threshold, we need to apply the MMCA method, investigating the steady state of the system, indicated as \(\beta_{c}\). The Dynamic Microscopic Markov Chain Approach (MMCA) enables us to explore the dynamics of the co-evolution on a weighted multiplex network.