# A Numerical Approach for the Filtered Generalized Cech Complex
## 3 Filtered Generalized Simplicial Structures for Disk Systems
Thus, for any disk system \(M\), the generalized Cech complex \(\mathcal{C}(M)\) has a filtered simplicial complex structure,  
\[\mathcal{C}_{M}(0)\subset\mathcal{C}(\lambda_{1})\subset\cdots\subset \mathcal{C}_{M}(\lambda_{s}).\]  
Of course, when we vary the scale \(\lambda\) on a interval the above filtration contains only a finite number of different sets. Moreover, those sets only change when the Cech scale of some disk system is reached, and therefore it is enough to compute all sets corresponding to Cech scales to characterize entirely the filtration. The goal of the next sections is the construction of algorithms to numerically estimate the Cech scale of every Cech subsystem of \(M\). The filtered generalized Cech complex can be \"approximated\" by the Vietoris-Rips structure, in the following sense. The inclusion \(\mathcal{C}(M)\subset\mathcal{VR}(M)\) holds clearly; in consequence, \(\mathcal{C}_{M}(\lambda)\subset\mathcal{VR}_{M}(\lambda)\) for any scale \(\lambda\geq 0\), then by Lemma 2 any Vietoris-Rips \(d\)-system \(\sigma\in\mathcal{VR}(M)\) rescaled by a factor of \(\sqrt{2d/(d+1)}\) is also a Cech \(d\)-system: \(\sigma_{\sqrt{2d/(d+1)}}\in\mathcal{C}(M)\). Therefore, for any \(d\)-disk system \(M\) the following relation is fulfilled:  
\[\mathcal{VR}_{M}(\lambda^{\prime})\subset\mathcal{C}_{M}(\lambda)\subset \mathcal{VR}_{M}(\lambda),\]  
where \(\sqrt{2d/(d+1)}\cdot\lambda^{\prime}\leq\lambda\). To any disk system \(M\), the simplicial substructure \(\mathcal{C}(M)^{(1)}\) given by the 1-skeleton of the generalized Cech complex of \(M\) is a basic combinatorial structure (actually, a graph) that can be easily defined, it just takes the relationship into account if every two vertices are neighbors: the set of vertices is \(M\), and there exists an edge \(\{D_{i},D_{j}\}\) whenever \(D_{i}\cap D_{j}\
eq\emptyset\). The Cech-weight function restricted to \(\mathcal{C}(M)^{(1)}\) is, in fact: \(\omega(\{D_{i}\})=0\) to every vertice, and \(\omega(\{D_{i},D_{j}\})=\|c_{i}-c_{j}\|/(r_{i}+r_{j})\) to any edge. In Algorithm 1, we calculate the Cech-weight function \(\omega:\mathcal{C}_{M}(\lambda)^{(dim)}\to\mathbb{R}\), for the _dim_-skeleton of a \(\lambda\)-rescaled disk system \(M\). To do this, we assume an arbitrary linear order in the disk system \(M\), and for every disk \(D\in M\) we consider the following set. \[\lambda\text{-}\text{LowerNbrs}(D)=\{\tilde{D}\in M\mid\tilde{D}<D,\;\omega( \{D,\tilde{D}\})\leq\lambda\}.\]  
The following algorithm (based on work in [7]), is a standard expansion algorithm for simplicial complexes, and we are including the Cech-weight function value of each simplex when it is calculated. ```
Input : A \(d\)-disk system \(M\), a non-negative parameter \(\lambda\) and an integer \(dim\geq 2\). Output : The Cech-weight function \(\omega:\mathcal{C}_{M}(\lambda)^{(dim)}\rightarrow\mathbb{R}\). 1 for\(n\gets 0\)to\(dim-1\)do
2 for\(\sigma\in\mathcal{C}_{M}(\lambda)^{(n)}\langle\mathcal{C}_{M}(\lambda)^{(n-1)}\)do
3\(\mathrm{LN}_{\sigma}\leftarrow\bigcap_{D\in\sigma}\lambda\text{-}\textsc{LowerNbrs}(D)\);
4 for\(D\in\mathrm{LN}_{\sigma}\)do
5\(N\leftarrow\{D\}\cup\sigma\);
6 if\(n\leq d-1\)then
7 Calculate \(\omega(N)\leftarrow\mu_{N}\);
8 else
9\(\omega(N)\leftarrow\max\{\omega(\tau)\mid\tau\subsetneq N\}\);
10
11 end if
12 if\(\omega(N)\leq\lambda\)then
13 Update \(\mathcal{C}_{M}(\lambda)^{(dim)}\leftarrow\mathcal{C}_{M}(\lambda)^{(dim)}\cup\{N\}\);
14
15 end if
16
17 end for
18
19 end for return\((\mathcal{C}_{M}(\lambda)^{(dim)},\omega:\mathcal{C}_{M}(\lambda)^{(dim)} \rightarrow\mathbb{R})\)
```  
**Algorithm 1**Cech-weight function of a \(d\)-disk system. We conclude the section with an application of Algorithm 1 to a 2-disk system. **Example 2**.