# A Set-Theoretic Approach to Modeling Network Structure
## 2 The Interior
### The Network Interior  
Consider any node \(y\in N\), and suppose there exists \(z\in\{y\}.\varphi_{\eta}\) implying \(\{z\}.\eta\subseteq\{y\}.\eta\). Such a node, \(z\) whose \"horizon\" is contained in that of \(y\), contributes very little to the information content of the network so that its removal from \(\{y\}.\eta\) will result in little information loss. This node \(z\in\{y\}.\varphi_{\eta}\) can be reduced. The node \(y\) is **irreducible** if \(\{y\}.\varphi_{\eta}=\{y\}\). A sub-network, \(\mathcal{I}\subseteq\mathcal{N}\), of irreducible nodes is called the network's **interior**. In the remainder of this section we define an operator, \(\omega\), which reduces any network to its irreducible core, and prove that it is almost an interior operator. If \(\{y\}\) is not closed, only elements \(z\) in \(\{y\}.\eta\) could possibly be in \(\{y\}.\varphi_{y}\) so only those need be considered. If \(\{z\}.\eta\subseteq\{y\}.\eta\) so that \(z\in\{y\}.\varphi_{y}\), we say \(z\) is subsumed by \(y\), or \(z\)**belongs** to \(y\). We can remove \(z\) from \(N\), together with all its connections, and add \(z\) to \(\{y\}.\beta\), the set of all nodes belonging to \(\{y\}\). This set \(\{y\}.\beta\) is called its \(\beta\)**-set**. Of course, \(y\in\{y\}.\beta\). The cardinality \(|\{y\}.\beta|\) is called its \(\beta\)**-count**. The pseudocode reduce Pseudocode I was used to implement a process \(\omega\) that reduces any network \(\mathcal{N}\) to its irreducible core, \(\mathcal{I}=\mathcal{N}.\omega\). ```
whilethereexistreduceablenodes {  reducible=0  for_each{y}inN{  for_each{z}in{y}.nbhd-{y}{  if({z}.nbhd contained_in{y}.nbhd { //zissubsumedbyyremovezfromnetwork; {y}.beta={y}.betaunion{z}.beta}  reducible=1}}}} ) ) ) ) )
```  
Pseudocode I, \(\omega\),reduce_network  
Applied to \(\mathcal{N}_{1}\), the well-known \"Karate\" network [6], this reduction code yields the interior depicted by bolder links in Figure 1.  
In this figure, two nodes of the interior have been suffixed by \":\(n\)\" to denote their \(\beta\)-count. Only nodes 1 and 33 have non-trivial \(\beta\)-sets of 12 and 8 elements, respectively, which have been delimited by dotted lines. (The \(\beta\)-set of node 33 might equally well have been the \(\beta\)-set of node 34; however, 33 precedes 34 in the reduction process). Proposition 1: _The process \(\omega\) described above is (11) contractive and (13) idempotent._  
Proof: Readily, \(\omega\) is contractive and it is idempotent because, when \(\mathcal{I}=\mathcal{N}.\omega\) is irreducible, the loop is not executed, so \(\mathcal{N}.\omega.\omega=\mathcal{I}=\mathcal{N}.\omega\). One can show that \(\mathcal{N}\subset\mathcal{N}^{\prime}\) need not imply that \(\mathcal{N}.\omega\subset\mathcal{N}^{\prime}.\omega\), so \(\omega\) is not an interior operator, even though we call \(\mathcal{I}=\mathcal{N}.\omega\) the \"interior\". Proposition 2: _Let \(\mathcal{I}=\mathcal{N}.\omega\) and \(\mathcal{I}^{\prime}=\mathcal{N}.\omega^{\prime}\) be irreducible subsets of a finite network \(\mathcal{N}\), then \(\mathcal{I}\cong\mathcal{I}^{\prime}\)._  
Proof: Let \(y_{0}\in\mathcal{I}\), \(y_{0}\
ot\in\mathcal{I}^{\prime}\). Then, \(y_{0}\) belongs to some point \(y_{1}\) in \(\mathcal{I}^{\prime}\) and \(y_{1}\
ot\in\mathcal{I}\) else because \(y_{0}.\eta\subseteq y_{1}.\eta\) implies \(y_{0}\in\{y_{1}\}.\varphi\) so \(\mathcal{I}\) would not be irreducible. Figure 1: The interior \(\mathcal{I}\) of \(\mathcal{N}_{1}\), the Karate network, is shown with bolder links. Similarly, since \(y_{1}\in\mathcal{I}^{\prime}\) and \(y_{1}\
ot\in\mathcal{I}\), there exists \(y_{2}\in\mathcal{I}\) such that \(y_{1}\) belongs to \(y_{2}\). Now, we have two possible cases; either \(y_{2}=y_{0}\), or not. Suppose \(y_{2}=y_{0}\) (which is often the case), then \(y_{0}.\eta\subseteq y_{1}.\eta\) and \(y_{1}.\eta\subseteq y_{0}.\eta\) or \(y_{0}.\eta=y_{1}.\eta\). Hence, \(i(y_{0})=y_{1}\) is part of the desired isometry, \(i\). Now, suppose \(y_{2}\
eq y_{0}\).