# Reliable Design for a Network of Networks with Inspiration from Brain Functional Networks
## 4 Evaluation
### Network Construction
#### 4.1.2. Internetwork Assortativity  
As well as configuring the connectivity within each VN, we must also consider how to map the dependence of V-nodes because traffic fluctuation along the VNs causes interference. For example, we must investigate whether a V-node with high influence in one VN should be interdependent with a V-node of high influence in another VN. We therefore introduce the variable \(\theta\) to evaluate the assortativity among VNs, and we configure the NoN structure from the perspective of mapping V-nodes on the VNs. Although we use the degree assortativity for connectivity within a VN because of its conditional limitation, we use the CI as a centrality measurement for the assortativity between networks, which is described in Section 2.  
In a previous study [47], we developed a method for measuring the assortativity between networks to evaluate the interdependence of information networks. The assortativity of a set of links is represented as the sum of each link's contribution to the assortativity of the entire network. Consequently, we begin by rewriting the definition of network assortativity described by Equation (17) as  
\[\eta=\frac{1}{\sigma_{q}^{2}}\bigg{(}E[(I-U_{q})(K-U_{q})]\bigg{)}, \tag{18}\]  
where \(U_{q}\) is the expected value of the remaining degree, and \(J\) and \(K\) are variables of the remaining degree that have the same expected value \(U_{q}\). To expand the definition of assortativity for centrality metrics other than degree centrality, we introduce \(p^{\prime}(c)\) as distribution of any kind of centrality metrics \(c\) on a VN. As for internetwork assortativity, the centrality of endpoint nodes of an internetwork link when the link is removed is equal to the centrality of those nodes on each VN topology. Hence, generalized assortativity \(\eta^{\prime}\) can be defined based on degree assortativity \(\eta\) in Equation (18) as follows:  
\[\eta^{\prime}=\frac{1}{\sigma_{p^{\prime}_{j}}\sigma_{p^{\prime}_{k}}}\bigg{(} E[(C_{j}-U_{p^{\prime}_{j}})(C_{k}-U_{p^{\prime}_{k}})]\bigg{)}, \tag{19}\]  
where \(p^{\prime}_{j}\) denotes the centrality distribution on VN \(j\). \(U_{p^{\prime}_{j}}\) and \(\sigma_{p^{\prime}_{j}}\) denote the expected value and the standard deviation of the centrality distribution \(p^{\prime}_{j}\), respectively. \(C_{j}\) denotes variables of the node centrality on VN \(j\) that have the same expected value \(U_{p^{\prime}_{j}}\). Based on Equation (19), the contribution \(\theta_{I}\) of link \(l\) to the assortativity \(\eta^{\prime}\) of the entire network is defined as follows:  
\[\theta_{l}=\frac{(c_{j}-U_{p^{\prime}_{j}})(c_{k}-U_{p^{\prime}_{k}})}{\sigma _{p^{\prime}_{j}}\sigma_{p^{\prime}_{k}}}, \tag{20}\]  
where \(c_{j}\) and \(c_{k}\) are the node centrality of the two endpoints of link \(l\). Finally, the internetwork assortativity \(\theta\) (i.e., the assortativity of the set of links \(L_{set}\) between two networks) is given by  
\[\theta=\sum_{l\in L_{set}}\theta_{l}=\sum_{l\in L_{set}}\frac{(c_{j}-U_{p^{ \prime}_{j}})(c_{k}-U_{p^{\prime}_{k}})}{\sigma_{p^{\prime}_{j}}\sigma_{p^{ \prime}_{k}}}. \tag{21}\]  
To map interdependent V-nodes among VNs, we begin by deploying the VNs randomly upon a PN. Then, similarly to the configuration of connectivity within a VN, we set a target value \(\theta_{target}\) and repeatedly re-map until the \(\theta\) calculated from the current VN interconnectivity approximates \(\theta_{target}\) sufficiently.