# A Set-Theoretic Approach to Modeling Network Structure
## 4 Network Generation by Expansion  
The interior, \(\mathcal{I}\), of a network \(\mathcal{N}\) represents its global structure. If the \(\beta\)-count is appended to each node of \(\mathcal{I}\), how well does \(\mathcal{I}\) represent \(\mathcal{N}\) as a whole? In effect, what is the information content of \(\mathcal{I}\), so augmented? One measure of the information content of any collection of network properties is the ability to construct, or generate, similar networks based on those properties. For example, given a network \(\mathcal{N}=(N,L)\) one can construct many different networks \(\mathcal{N}^{\prime}=(N^{\prime},L^{\prime})\) such that \(|N^{\prime}|=|N|\) and \(|L^{\prime}|=|L|\). However, they need not be at all similar to \(\mathcal{N}\). Here, we are using \"similar\" in its colloquial sense. A formal notion of \"similarity\" would require it to be an equivalence relation. One way of determining the nature of networks with a given interior, \(\mathcal{I}\), and known \(\beta\)-counts is to randomly generate some. Let \(\mathcal{I}\) be given. Suppose the \(\beta\)-count of a node \(y\) is greater than one. New nodes can be attached to replace those of the original \(\beta\)-set. Let \(y\):\(n\) be the node to be expanded (\(n>1\)) and let \(z\) denote the new node. Our code generates artificial node names of the form 'A0, B0, \(\ldots\), Z0, A1, \(\ldots\)'. The last generated node in the expansion of Figure 5 is \(M11\). Besides the link \(\{y,z\}\), we require \(\{z\}.\eta\subseteq\{y\}.\eta\). A random number determines how many of the other nodes in \(\{y\}.\eta\) will be linked to \(z\), and which, if any, of those are also randomly chosen. In the reduction process, \(\omega\), nodes with considerable \(\beta\)-sets may be subsequently reduced themselves. In the re-expansion, a portion of the \(\beta\)-count of \(y\) may be transferred to the \(\beta\)-count of \(z\). Pseudocode for a procedure expand to implement an operator \(\varepsilon\) that generates new nodes relative to the interior is given in Pseudocode III. (\(\varepsilon\), as shown here, is a round-robin procedure expanding one node in a \(\beta\)-set at a time. An alternative, and slightly faster, process can be found in [22]). As a test, the interior \(\mathcal{I}\) of \(\mathcal{N}_{2}\), Figure 3, was expanded three times (using different random number seeds) to yield \(exp.1\), \(exp.2\) and \(exp.3\) of Figure 8.  
**Proposition 8**. : _Let \(\mathcal{I}\) be the interior of a network \(\mathcal{N}\), that is \(\mathcal{I}=\mathcal{N}.\omega\), then \(\mathcal{I}.\varepsilon.\omega=\mathcal{I}\)._Proof. : The expansion procedure of Pseudocode III was written to make this true. Consider \(z_{n}\), the last node appended by \(\varepsilon\). By construction, \(x_{n}.\eta\subseteq y_{n}.\eta\) for some \(y_{n}\); thus, \(z_{n}\) can be subsumed into \(y_{n}.\beta\). A finite induction \(z_{n},\ldots,z_{1}\) completes the proof. Considered as operators, \(\varepsilon\) is a left-inverse of \(\omega\) since \(\varepsilon\cdot\omega=1\), where \(1\) denotes the identity operator. However, \(\omega\cdot\varepsilon\
eq 1\), as shown by Figure 8.  
To what extent are the network features of \(\mathcal{N}\) enumerated in the preceding section preserved in the randomly generated networks, \(\mathcal{N}.\omega.\varepsilon\)? Readily, the generation process \(\varepsilon\) was constrained so that \(|\mathcal{N}.\omega.\varepsilon|=|\mathcal{N}|\) and \(\mathcal{N}.\omega.\varepsilon\omega=\mathcal{I}=\mathcal{N}.\omega\), thus the path-based centers of Section 3.4 are preserved. Some other network properties are illustrated in Table 1.  
Table 2 presents the principal eigenvector associated with the nodes of \(\mathcal{N}_{2}\) in Figure 3 and for the three expansions shown in Figure 8. Note that, except for the ten nodes of \(\mathcal{I}\), node values for generated expansions are not comparable with node values of the original \(\mathcal{N}\).