## 2 Information Granulation and Classification Systems  
### An Introduction to Simplicial Complexes  
Let \(\mathcal{P}\) be a set of points in a multi-dimensional space equipped with a notion of distance \(d(\cdot,\cdot)\) and let \(\mathcal{X}\) be the topological space enclosing \(\mathcal{P}\). The topological space \(\mathcal{X}\) can be described by means of its _simplices_, that are multidimensional objects of different order (dimension) drawn from \(\mathcal{P}\). Formally, a \(k\)-simplex (simplex of order \(k\)) is a set of \(k+1\) points drawn from \(\mathcal{P}\), for example, \(0\)-dimensional simplices correspond to points, \(1\)-dimensional simplices correspond to lines, \(2\)-dimensional simplices correspond to triangles, \(3\)-dimensional simplices correspond to tetrahedrons and so on for higher-dimensional simplices. Every non-empty subset of the \((k+1)\) vertices of a \(k\)-simplex is a _face_ of the simplex: a face is itself a simplex. _Simplicial complexes_[50; 51] are properly constructed finite collections of simplices that are closed with respect to inclusions of the faces: if a given simplex \(s\) belongs to a given simplicial complex \(\mathcal{S}\), then all faces of \(s\) also belong to \(\mathcal{S}\). The order (dimension) of the simplicial complex is the maximum order of any of its simplices. A graph \(\mathcal{G}=(\mathcal{V},\mathcal{E})\), where \(\mathcal{E}\) is the set of edges and \(\mathcal{V}\) is the set of vertices, is also commonly-known as \"\(1\)-skeleton\" or \"simplicial complex of order \(1\)\" since the only entities involved are \(0\)-simplices (nodes) and \(1\)-simplices (edges). However, the modelling capabilities offered by graphs are often not sufficient as they only regard pairwise relations. Indeed, for some problems (ranging from bioinformatics [52; 53; 54] to signal processing [55; 56; 57]), multi-way relations are more suited, where two or more nodes are more conveniently connected by an _hyperedge_ (in this scenario, we are de facto talking about _hypergraphs_[58]). Simplicial complexes are an example of hypergraphs and therefore able to capture the multi-scale organisation in real-world complex networks [48; 49]. A straightforward example in order to focus hypergraphs and complexes may regard a scientific collaboration network in which nodes are authors and edges exist whether two authors co-authored a paper. This representation does not consider the case in which three or more authors wrote a paper together or, better, it would be ambiguous: three authors (for example) can be connected by \(3\cdot(3-1)/2\) edges in a graph but this scenario is ambiguous about whether the three authors co-authored a paper or each pair of authors co-authored a paper. By using hypergraphs, the same problem can be modelled where nodes are authors and hyperedges connect groups of authors that co-authored a paper together. A more biologically-oriented example include protein interaction networks, where nodes correspond to proteins and edges exist whether they interact. Yet, this representation does not consider protein complexes [52]. In literature, several simplicial complexes have been proposed, with the _Cech complex_, the _Alpha complex_ and the _Vietoris-Rips complex_ being the most studied [50; 51; 59; 60; 61]. In order to introduce the three simplicial complexes, let \(\mathcal{P}\) be a point cloud and let \(\epsilon>0\) be a real-valued number:  
**Cech complex:**: for each subset \(S\subset\mathcal{P}\) of points, form an \(\epsilon\)-ball (A ball with radius \(\epsilon\)) centred at each point in \(S\), and include \(S\) as a simplex if there is a common point contained in all of the balls created so far. **Alpha complex:**: for each point \(x\in\mathcal{P}\), evaluate its Voronoi region \(V(x)\) (i.e., the set of points closest to it). The set of Voronoi regions forms the widely-known Voronoi diagram and the nerve of the latter is usually referred to as Delaunay complex.