## 2 Information Granulation and Classification Systems
### Proposed Approach  
#### 2.2.1 Embedding  
Let \(\mathcal{D}=\{\mathcal{G}_{1},\ldots,\mathcal{G}_{N_{P}}\}\) be a dataset of \(N_{P}\) graphs, where each graph has the form \(\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{L}_{v})\), where \(\mathcal{L}_{v}\) is the set of vertices labels. For the sake of argument, let us consider a supervised problem, thus let \(\mathcal{L}\) be the corresponding ground-truth class labels for each of the \(N_{P}\) graphs in \(\mathcal{D}\). Further, consider \(\mathcal{D}\) to be split into three non-overlapping training, validation and test sets (\(\mathcal{D}_{\text{TR}}\), \(\mathcal{D}_{\text{VAL}}\), \(\mathcal{D}_{\text{TS}}\), respectively) and, by extension, the labels \(\mathcal{L}\) are split accordingly (\(\mathcal{L}_{\text{TR}}\), \(\mathcal{L}_{\text{VAL}}\), \(\mathcal{L}_{\text{TS}}\)). Let \(q\) be the number of classes for the classification problem at hand. The first step is to evaluate the simplicial complex separately for all graphs in the three datasets splits, hence  
\[\mathcal{D}_{\text{TR}}^{\text{SC}} =\{sc(\mathcal{G}),\ \forall\mathcal{G}\in\mathcal{D}_{\text{TR}}\} \tag{1}\] \[\mathcal{D}_{\text{VAL}}^{\text{SC}} =\{sc(\mathcal{G}),\ \forall\mathcal{G}\in\mathcal{D}_{\text{VAL}}\}\] (2) \[\mathcal{D}_{\text{TS}}^{\text{SC}} =\{sc(\mathcal{G}),\ \forall\mathcal{G}\in\mathcal{D}_{\text{TS}}\} \tag{3}\]  
where \(sc(\mathcal{G}):\mathcal{G}\rightarrow\mathcal{S}\) is a function that evaluates the simplicial complex starting from the 1-skeleton \(\mathcal{G}\). However, the embedding is performed on the concatenation of \(\mathcal{D}_{\text{TR}}\) and \(\mathcal{D}_{\text{VAL}}\) or, specifically, \(\mathcal{D}_{\text{TR}}^{\text{SC}}\) and \(\mathcal{D}_{\text{VAL}}^{\text{SC}}\). In other words, the alphabet sees the concatenation of the simplices belonging to the simplicial complexes evaluated starting from all graphs in \(\mathcal{D}_{\text{TR}}\) and \(\mathcal{D}_{\text{VAL}}\). In cases of large networks and/or large datasets, this might lead to a huge number of simplices which are hard to match. For example, let us consider any given node belonging to a given graph to be identified by a progressive unique number. In this case, it is impossible to match two simplices belonging to possibly two different simplicial complexes (i.e., determine whether they are equal or not). In order to overcome this problem, node labels \(\mathcal{L}_{v}\) play an important role. Indeed, a simplex can dually be described by the set of node labels belonging to its vertices. This conversion from'simplices-of-nodes' to'simplices-of-node-labels' has a three-fold meaning, especially if node labels belong to a categorical and finite set:  
1. the match between two simplices (possibly belonging to different simplicial complexes) can be done in an exact manner: two simplices are equal if they have the same order and they share the same set of node labels
2. simplicial complexes become multi-sets: two simplices (also within the same simplicial complex) can have the same order and can share the same set of node labels
3. the enumeration of different (unique) simplices is straightforward.