## 3 Results
### Structural Profiling  
A complex network involves basic units connected to each other with multiple kinds of relationships among them. Embedding the interactions in different layers in accordance to their types, we can consider the multiplex social network representation and analysis [5; 9; 10; 15]. Following the scheme presented in Figure 2, we consider a multiplex social network \(M\), assuming that it can represent a 6G sub-network and it is referred to different kinds of interactions among nodes. A node represents an aggregation of different basic units of the network (as wearable or hand-held devices, IoT systems, etc.). The multiplex network consists of a population of \(N\) nodes and \(l=l_{1},l_{2},\ldots,l_{M}\) layers. Figure 2: Modeling Approach: a technical scheme of nodes’ profiling in a 6G network scenario. The figure graphically describes the various steps and the key aspects to profile the elements that constitute the 6G scenario into a complex network heterogeneous environment of cognitive elements. The approach starts from the application of the complexity and the multiplex representation and analysis to extract structural features as detailed, respectively, in (**a,b**). The scheme shows the collective profiling based on the analysis of diffusion and competition, representing two co-evolving and interdependent dynamics, as, respectively, detailed in (**c,d**). The community profiling, as in (**e,f**), is characterized by the detection of communities based on node and link hierarchical clustering obtained through a multilink graph. Here, we consider \(M=2\) layers, where the layer \(l_{1}\) is referred to a real contact network, extracted from a real case [33], while the second one is referred to a virtual layer following a theoretical scheme as, for example, a Scale-Free network [34] or a Small-World network [35]. In the literature, the multiplex network is defined as a set of graphs \(G\), with \(G=(V,E)\), with vertices \(V\) and edges \(E\). The set of nodes \(N\) is the same for each layer, whereas the set of links \(E\) changes according to the layer [9]. Each network \(G_{l}\) is described by the adjacency matrix, denoted by \(A^{l}\) with elements \(a^{l}_{ij}\), where \(a^{l}_{ij}=1\), if the node \(i\) and the node \(j\) are connected through a link on a layer \(l\), otherwise \(a^{l}_{ij}=0\). Alternatively, in the weighted case, \(a^{l}_{ij}=w^{l}_{ij}>0\), if there is a link between \(i\) and \(j\) with a weight \(w^{l}_{ij}\), otherwise \(a^{l}_{ij}=0\), with \(w_{ij}\) a real positive number, namely the weight of the link between \(i\) and \(j\). In this case, a weighted multiplex network is completely specified by the weighted adjacency matrix \(W^{l}\), with elements \(w^{l}_{ij}\)[13]. With \(k^{l}_{i}\), we indicate the degree of the node \(i\) in a given layer \(l\), that considers the total number of connections. Moreover, other basic features that we include in the profiling are represented by the edge overlap \(o^{[l_{1},l_{2}]}_{ij}\), that take into consideration the fact that a pair of nodes can be connected by ties in different layers, and the overlapping degree \(o_{i}\)[9]. In the case of weighted multiplex networks, the latter are, respectively, expressed as weighted overlap \(o^{w}_{ij}\) and weighted overlapping degree \(o^{w}_{i}\)[9; 13]. The weights, which can be distributed heterogeneously, are strongly correlated with the structure [13], shedding light on the relevance of the links in a layer of the multiplex network representation. For that reason and with the aim at embodying social aspects into structural profiling, we apply the definition of the weights of the links as function of some key metrics, as follows:  
\[w^{l}_{ij}=h^{l}_{ij}|aw_{i}-aw_{j}| \tag{1}\]  
where \(h^{l}_{ij}\) indicates the tendency to interact with similar nodes (as equal devices) or with that ones that are in similar conditions (capacity, resources, etc.) [5; 9; 13]. Furthermore, we include the gap value, between \(i\) and \(j\), of the node awareness \(aw\).