# A Set-Theoretic Approach to Modeling Network Structure
## 2 The Interior
### The Network Interior
There exists \(y_{3}\
eq y_{1}\in\mathcal{I}^{\prime}\) such that \(y_{2}.\eta\subseteq y_{3}.\eta\), and so forth. Since \(\mathcal{I}\) is finite, this construction must halt with some \(y_{n}\). The points \(\{y_{0},y_{1},y_{2},\ldots y_{n}\}\) constitute a complete graph \(Y_{n}\) with \(\{y_{i}\}.\eta=Y_{n}.\eta\), for \(i\in[0,n]\). In any reduction, all \(y_{i}\in Y_{n}\) reduce to a single point. All possibilities lead to mutually isomorphic maps. Proposition 2 assures us that, even though which nodes are preserved in \(\mathcal{I}\) is completely dependent on the order in \(\omega\) that they are visited, the output must be effectively identical. For example, in Figure 2, assume the nodes \(x\) and \(z\) are irreducible elements of \(\mathcal{I}\). In each case, if \(y_{0}\in\mathcal{I}\), then \(y_{1}\) or \(y_{2}\) could be as well. They are the equivalent nodes defining the isometry. Each set of equivalent nodes must be a \"complete\" subgraph of \(\mathcal{N}\). (A graph, or network, \(K\) is said to be complete if for all \(x,y\in K\), there is a link \(\{x,y\}\). A complete graph on \(n\) nodes is denoted by \(K_{n}\).) A sequence, \(\dot{\rho}=\langle y_{0},\ldots,y_{n}\rangle\) of \(n+1\) nodes, where \(\{y_{i-1},y_{i}\}\in L\), or a set of \(n\) links \(\ddot{p}=\langle\{y_{0},y_{1}\},\ldots,\{y_{n-1},y_{n}\}\rangle\) is called a **path**\(\rho(y_{0},y_{n})\) of length \(n\). It is often easier to describe a path in terms of its nodes, \(\dot{p}\) rather than \(\ddot{p}\), which is more precise. By \(|\rho(x,z)|\), we mean the length of the path independent of whether we are counting nodes or links. A cycle \(\dot{C}=\langle y_{0},y_{1},\ldots,y_{n}\rangle\), where \(y_{n}=y_{0}\), of length \(n\geq 4\) is said to have a **bridge** if there exists a path \(\ddot{p}(y_{i},y_{k})\in L\) where \((k-i)\ mod(n)\
eq 1\)[2]. If the path consists of a single link, it is called a **chord**. If \(C\) has no such chords, it is said to be a **chordless** cycle. Graphs, in which every cycle of length \(\geq 4\) must have a chord, are called \"chordal graphs\" [7].) Proposition 3: _The nodes of a chordless cycle are irreducible._  
Proof: Let \(\{y_{i-1},y_{i}\},\{y_{i},y_{i+1}\}\in L\). Suppose \(y_{i}\in y_{i-1}.\varphi_{\eta}\) implying \(y_{i+1}\in\{y_{i}\}.\eta\subseteq\{y_{i-1}\}.\eta\) or \(\{y_{i+1},y_{i-1}\}\in L\) contradicting chordless assumption. Proposition 4: _Let \(\mathcal{N}\) be a finite network with \(\mathcal{I}=\mathcal{N}.\omega\) being an irreducible subset. If \(y\in\mathcal{I}\) is not an isolated point, then either:_  
_(1) there exists a chordless \(k\)-cycle \(\dot{C}\), \(k\geq 4\) such that \(y\in\dot{C}\); or_  
_(2) there exist chordless \(k\)-cycles \(\dot{C}_{1},\dot{C}_{2}\) each of length \(\geq 4\) with \(x\in\dot{C}_{1}\,z\in\dot{C}_{2}\) and \(y\) lies on a path from \(x\) to \(z\)._  
Proof: Let \(y_{1}\in\mathcal{I}\). Since \(y_{1}\) is not isolated, let \(y_{0}\in y_{1}.\eta\), so \(\{y_{0},y_{1}\}\in L\). \(\geq 4\). Since \(y_{1}\) is not subsumed by \(y_{0}\), \(\exists y_{2}\in y_{1}.\eta,y_{2}\
ot\in y_{0}.\eta\), and since \(y_{2}\) is not subsumed by \(y_{1}\), \(\exists y_{3}\in y_{2}.\eta\), \(y_{3}\
ot\in y_{0}.\eta\). Since \(y_{2}\
ot\in y_{0}.\eta\), \(y_{3}\
ot\in y_{0}\). Suppose \(y_{3}\in y_{0}.\eta\), then \(\langle y_{0},y_{1},y_{2},y_{3},y_{0}\rangle\) constitutes a \(k\)-cycle \(k\geq 4\), and we are done. Suppose \(y_{3}\
ot\in y_{0}.\eta\). We repeat the same path extension. \(y_{3}.\eta\
ot\subseteq y_{2}.\eta\) implies \(\exists y_{4}\in y_{3}.\eta\), \(y_{4}\
ot\in y_{2}.\eta\). If \(y_{4}\in y_{0}.\eta\) or \(y_{4}\in y_{1}.\eta\), we have the desired cycle. If not \(\exists\ y_{5},\ldots\) and so forth. Because \(\mathcal{N}\) is finite, this path extension must terminate with \(y_{k}\in y_{i}.\eta\), where \(0\leq i\leq n-3\), \(n=|N|\). The preceding establishes that any link sequence in \(\mathcal{I}\) terminates in a cycle of length \(\geq\)4.