# Deep Learning-Based Community Detection Approach on Multimedia Social Networks
## 3 Methodology
### Multimedia Social Network Model  
Our MSN model is characterized by two entities: **users**, persons or organizations, characterized by some attributes (i.e., profiles, interests, preferences), belonging to one or more communities, and **multimedia objects**, a set of multimedia entities (i.e., images, text, or videos), described by metadata or low-level features, that can be shared within a social network. Different relationships can be established between these entities; for instance, a user can establish a relationship with another one (_friendship_ or _following_); a user can publish a photo, or video or comment on other multimedia objects or two images can be connected according to their similarity. **Definition 1**. : _Let \(U\) and \(O\) be, respectively, the set of users and multimedia objects. A multimedia social network can be defined as a triple \(G=(V,E=\{e_{i}:i\in I\},\omega)\), where \(V=U\cup O\) is the set of vertices, \(\mathbf{H}\) is the set of hyperarcs, and \(\omega:E\rightarrow[0,1]\) is a function that assigns a weight to each hyperarc._  
Each hyperarc is defined as an ordered pair \(e_{i}=(e_{i}^{+}=(V_{e_{i}^{+}}^{+},i));e_{i}^{-}=(i,V_{e_{i}^{-}}^{-}))\), where \(e_{i}^{+}\) is called the tail of the hyperarc and \(e_{i}^{-}\) is the head of the hyperarc. However, it turns out that the set \(V=V_{e_{i}^{+}}^{+}\cup V_{e_{i}^{-}}^{-}\) is the set of vertices representing the entire hypergraph. Furthermore, we define **the degree of the hyperarc**\(d_{e_{i}}\) as the cardinality of contained vertices that have incident arcs and **the degree of a vertex**\(d_{v}\) as the cardinality of hyperarcs that are incident to vertex \(v\). It is also possible to define **oriented and undirected** arcs: in the first case, each arc can be seen as a function that maps two disjoint and nonempty sets of \(V_{e_{i}^{+}}^{+}\) and \(V_{e_{i}^{-}}^{-}\). Furthermore, we can define the main relationships of an MSN, which are classified into the following three main categories:  
1. **User-to-user**: representing a user's actions with another user, which are, formally, defined as \(e_{i}=((V_{e_{i}^{+}}^{+},i);(i,V_{e_{i}^{-}}^{-}))\), where \(V_{e_{i}}^{+}=\{u_{k}\}\) and \(\{u_{k}\}\subseteq U\), where \(V_{e_{i}^{-}}^{-}=\{U-\{u_{k}\}\}\);
2. **User-to-multimedia object**: representing the relationships a user has with other media objects, possibly associated with a weight (depending to the analyzed MSN). The weight \(\omega(e_{i})\) assigned to such hyperarcs indicates the importance of the relationship of an MSN. Mathematically, it can be described as \(e_{i}=((V_{e_{i}^{+}}^{+},i);(i,V_{e_{i}^{-}}^{-}))\) with \(V_{e_{i}^{+}}^{-}=\{u_{k}\}\), where \(u_{k}\subseteq U\) and \(V_{e_{i}^{-}}^{-}=\{o_{i}\}\), where \(o_{i}\subseteq O\);
3. **Similarity**: represented by the similarity relations between two users or between two objects. Mathematically, it is described as \(e_{i}=((V_{e_{i}^{+}}^{+},i);(i,V_{e_{i}^{-}}^{-}))\) with \(V_{e_{i}^{+}}^{+}=\{v_{k}\}\) and \(V_{e_{i}^{-}}^{-}=\{v_{j}\}\) with \(i\
eq k\). The weight associated with this class of hyperarcs is a function of different factors, such as the metric used and the type of vertex considered. We chose these types of relationships in our model for representing the complex and heterogeneous interactions between users and/or multimedia objects considering the characteristics of the most diffused OSNs (i.e., Twitter, Facebook, Flickr, Youtube, Instagram, etc. ), but also more specific social networks (LinkedIn, ResearchGate, etc.). In our model, each kind of n-ary relationship can be easily implemented; thus, a user can tag another one within an image, different users can belong to the same group, and son on. We can observe that the weight of a hyperarc is calculated on the basis of the type of relationships that exist between two vertices; in the case of similarity between multimedia data, the weight will correspond to their similarity according to low- and high-level features.