# Exploring Graph and Digraph Persistence
## 3 Graph-Theoretical Persistence
### Indexing-Aware Persistence Functions  
The difference between any of the categorical persistence functions introduced in [7] and the functions of the present subsection (presented originally in [8]) is that the former come from a functor defined on \(\mathbf{Gr}\), while the latter strictly depend on the filtration, thus descending from a functor defined on \((\mathbb{R},\leq)\). Definition 4: [8] (Def. 5) Let \(p\) be a map assigning to each filtered (di)graph \((G,f)\) a categorical persistence function \(p_{(G,f)}\) on \((\mathbb{R},\leq)\), such that \(p_{(G,f)}=p_{(G^{\prime},f^{\prime})}\) whenever an isomorphism between \((G,f)\) and \((G^{\prime},f^{\prime})\) compatible with the functions \(f,f^{\prime}\) exists. All the resulting categorical persistence functions \(p_{(G,f)}\) are called _indexing-aware persistence functions (_ip-functions for brevity_). The map \(p\) itself is called an _ip-function generator_. Definition 5: [8] (Def. 6, Prop. 1) Let \(p\) be an ip-function generator on \(\mathbf{Gr}\). The map \(p\) itself and the resulting ip-functions are said to be _balanced_ if the following condition is satisfied. Let \(G=(V,E)\) be any (di)graph, \(f\) and \(g\) two filtering functions on \(G\), and \(p_{G,f}\) and \(p_{(G,g)}\) their ip-functions. If a positive real number \(h\) exists, such that \(\sup_{e\in E}|f(e)-g(e)|\leq h\), then for all \((u,v)\in\Delta^{+}\) the inequality \(p_{(G,f)}(u-h,v+h)\leq p_{(G,g)}(u,v)\) holds. Proposition 1: [8] (Thm. 1) Balanced ip-functions are stable. We now present our main ip-function generators: steady and ranging sets. Definition 6: [8] (Def. 8) Given a (di)graph \(G=(V,E)\), any function \(\mathcal{F}\): \(2^{V\cup E}\to\{true,false\}\) is called a _feature_. Conventionally, for \(X\subset V\cup E\), we set \(\mathcal{F}(X)=false\) in a sub(di)graph \(G^{\prime}=(V^{\prime},E^{\prime})\) if \(X\
subseteq V^{\prime}\cup E^{\prime}\). Remark 1: _The second part of Definition 6 was added for use in some of the following proofs._  
Let \(\mathcal{F}\) be a _feature_. **Definition 7**. : [8] (Def. 8) _We call an \(\mathcal{F}\)-set any \(X\subset V\cup E\) such that \(\mathcal{F}(X)=\text{true}\). In a weighted (di)graph \(G\), we shall say that \(X\subset V\cup E\) is an \(\mathcal{F}\)-set at level \(w\in\mathbb{R}\) if it is an \(\mathcal{F}\)-set of \(G_{w}\)._  
**Definition 8**. : [8] (Def. 9) _We define the maximal feature \(m\mathcal{F}\) associated with \(\mathcal{F}\) as follows: for any \(X\subseteq(V\cup E)\), \(m\mathcal{F}(X)=\text{true}\) if and only if \(\mathcal{F}(X)=\text{true}\) and there is no \(Y\subseteq(V\cup E)\) such that \(X\subset Y\) and \(\mathcal{F}(Y)=\text{true}\)._  
**Definition 9**. : [8] (Def. 10) _A set \(X\subseteq V\cup E\) is a steady \(\mathcal{F}\)-set (s\(\mathcal{F}\)-set for brevity) at \((u,v)\) if it is an \(\mathcal{F}\)-set at all levels \(w\) with \(u\leq w\leq v\)._  
_We call \(X\) a ranging \(\mathcal{F}\)-set (r\(\mathcal{F}\)-set) at \((u,v)\) if there exist levels \(w\leq u\) and \(w^{\prime}\geq v\) at which it is an \(\mathcal{F}\)-set._  
Let \(\mathcal{S}^{\mathcal{F}}_{(G,f)}(u,v)\) be the set of s\(\mathcal{F}\)-sets at \((u,v)\) and let \(R^{\mathcal{F}}_{(G,f)}(u,v)\) be the set of r\(\mathcal{F}\)-sets at \((u,v)\). **Definition 10**. : [8] (Def. 11) _For any filtered (di)graph \((G,f)\), we define \(\sigma^{\mathcal{F}}_{(G,f)}\) (resp., \(\sigma^{\mathcal{F}}_{(G,f)}\)) as the function that assigns to \((u,v)\in\mathbb{R}^{2}\), \(u<v\), the number \(|\mathcal{S}^{\mathcal{F}}_{(G,f)}(u,v)|\) (resp., \(|R^{\mathcal{F}}_{(G,f)}(u,v)|\))._  
_We denote by \(\sigma^{\mathcal{F}}\) and \(\varrho^{\mathcal{F}}\) the maps assigning \(\sigma^{\mathcal{F}}_{(G,f)}\) and \(\varrho^{\mathcal{F}}_{(G,f)}\), respectively, to \((G,f)\)._  
**Proposition 2**. : [8] (Prop. 2) _The maps \(\sigma^{\mathcal{F}}\) and \(\varrho^{\mathcal{F}}\) are ip-function generators._  
The next definition will be generalized in Section 4. **Definition 11**. : [8] (Def.