# A Set-Theoretic Approach to Modeling Network Structure
## 5 Observations  
This paper might have been titled \"An Operator Approach to...\" since the operators \(\eta\), \(\varphi_{\eta}\), \(\omega\) and \(\varepsilon\) play such an important role. This aspect is briefly suggested by Proposition 8, but not enlarged. However, surely, interesting networks are dynamic; they change over time which demands an operator approach. Thus, one might ask: \"Is a transformation \(\tau:\mathcal{N}\to\mathcal{N}^{\prime}\) continuous?\" [23] The operators \(\omega\) and \(\varepsilon\) are, in fact, \"continuous\" with respect to \(\varphi_{\eta}\). Moreover, it appears that \(\mathcal{N}.\upsilon=\mathcal{N}-\mathcal{N}.\omega=\mathcal{N}-\mathcal{I}\) is a violator space in the sense of [24]. This could be expanded in the future. However, computability is such a dominant theme in current network analysis and understanding that we thought focusing on the use of set-theoretic computer procedures \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(|\mathcal{N}|\) & \(|\mathcal{L}|\) & \(Density\) & \(Triangles\) & \(2\_Tresses\) & \(3\_Tresses\) \\ \hline \(\mathcal{N}_{2}\) & 21 & 44 & 2.095 & 21 & 2 & 4 \\ \hline exp.1 & 21 & 49 & 2.333 & 31 & 1 & 3 \\ exp.2 & 21 & 46 & 2.190 & 25 & 2 & 3 \\ exp.3 & 21 & 37 & 1.762 & 13 & 2 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Network properties of networks in Figure 8 generated from \(\mathcal{I}=\mathcal{N}_{2}.\omega\), Figure 3.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & **a** & **b** & **c** & **d** & **e** & **f** & **g** & **h** & **i** & **j** & **k** \\ \hline \(\mathcal{N}\) & 0.179 & 0.182 & 0.123 & 0.350 & 0.293 & 0.155 & 0.226 & 0.234 & 0.194 & 0.231 & 0.120 \\ \hline  & **A0** & **B0** & **C0** & **D0** & **e** & **f** & **E0** & **h** & **i** & **j** & **F0** \\ \hline exp.1 & 0.170 & 0.295 & 0.225 & 0.033 & 0.355 & 0.129 & 0.053 & 0.306 & 0.202 & 0.265 & 0.162 \\ exp.2 & 0.048 & 0.095 & 0.203 & 0.021 & 0.262 & 0.183 & 0.026 & 0.192 & 0.254 & 0.285 & 0.303 \\ exp.3 & 0.125 & 0.212 & 0.056 & 0.187 & 0.265 & 0.120 & 0.093 & 0.353 & 0.270 & 0.243 & 0.195 \\ \hline  & **1** & **m** & **n** & **o** & **P** & **q** & **r** & **s** & **t** & **u** & \\ \hline \(\mathcal{N}\) & 0.291 & 0.293 & 0.159 & 0.174 & 0.271 & 0.220 & 0.280 & 0.187 & 0.104 & 0.022 & \\ \hline  & **G0** & **m** & **n** & **o** & **p** & **H0** & **r** & **10** & **J0** & **K0** & \\ \hline exp.1 & 0.054 & 0.190 & 0.387 & 0.133 & 0.112 & 0.265 & 0.224 &  
such as reduce, count_triangles and expand was more important. Programming with set operators is not widespread. However, these set-theoretic procedures appear to be fast and quite scalable. The reduction, \(\omega\), of the [PERSON] co-authorship network to Figure 5 took 0.008 s; reduction of the smaller networks (Figures 1 and 3) were each less than 0.001 s. Calculation of the eigenvectors of Figure 5 exceeded 5 s. Such anecdotal evidence is suggestive, but far from definitive. Only standard set-theoretic reasoning was used to develop the reduction process, \(\omega\), which leads to the concept of the \"interior\", \(\mathcal{I}\), of a network, \(\mathcal{N}\), and its \(\beta\)-set. It is a powerful concept that effectively captures the essence of many networks, as shown by Section 4, in which very similar networks can be generated from \(\mathcal{I}\) alone. Moreover, by reducing a network to its interior, one effectively partitions the network into it constituent \(\beta\)-set communities. However, the reduction process has its limitations. Some networks are nearly irreducible to start with. The sparse network of Norwegian corporate directors [25] is an example. Hierarchical networks reduce to a single node, that is a single node interior with a very large \(\beta\)-set. Other networks can be too dense. The complete network \(K_{n}\) also reduces to a single node. However, we believe that the easily computed interior is a most effective network descriptor and possibly should be an automatic first step in network description and understanding.