# Reliable Design for a Network of Networks with Inspiration from Brain Functional Networks
## 3 Network of Networks in Virtualized Networks
### Interdependence of Virtualized Networks
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Variable** & **Value** & **Description** & **Equation** \\ \hline \(n_{i}\) & \{0,1\} & Existing state of node \(i\) & – \\ \hline \(\sigma_{i}\) & \{0,1\} & Local effectiveness of node \(i\) & Equation (1) \\ \hline \(\rho_{i}\) & \{0,1\} & Global effectiveness of node \(i\) & Equations (3) and (5) \\ \hline \(\rho_{i\to j}\), \(\varphi_{i\to j}\) & \{0,1\} & Probability information for message passing sent from node \(i\) to \(j\) & Equations (2) and (4) \\ \hline Cl\({}_{i}\) & [0,\(\infty\)] & Collective influence of node \(i\) & Equations (6) and (7) \\ \hline \(r_{x}^{P}\), \(r_{x}^{V_{x}}\) & – & Index for P-node \(x\) and V-node \(x\) on VN \(k\) & – \\ \hline \(i_{x,y^{P}}^{V_{x}}\) & – & Index for P-interface \(y\) of \(r_{x}^{P}\) and V-interface \(y\) of \(r_{x}^{V_{x}}\) & – \\ \hline \(B_{x,y}^{P}\) & _given_ & Buffer capacity for P-interface \(y\) of \(r_{x}^{P}\) & – \\ \hline \(n_{x,y}^{P}\) & \{0,1\} & Availability state for P-interface \(y\) of \(r_{x}^{P}\) & – \\ \hline \(n_{x,y}^{V_{x}}(t)\) & [0,\(\infty\)] & Traffic state for V-interface \(y\) of \(r_{x}^{V_{x}}\) at time \(t\) & – \\ \hline \(\mathrm{R}_{x,y}^{V_{x}}(t)\) & \{0,1\} & Vacancy state for V-interface \(y\) of \(r_{x}^{V_{x}}\) at time \(t\) & Equations (8) and (10) \\ \hline \(\rho_{x,y}^{V_{x}}(t)\) & \{0,1\} & Traffic state for V-interface \(y\) of \(r_{x}^{V_{x}}\) at time \(t\) & Equations (9), (11) and (12) \\ \hline CI\((r_{x}^{P})\), CI\((r_{x}^{V_{x}})\) & [0,\(\infty\)] & Collective influence of \(r_{x}^{P}\) and \(r_{x}^{V_{x}}\) & Equations (13)–(15) \\ \hline \(\eta\) & [\(-\)1,1] & Assortativity for intra-network connectivity & Equation (17) \\ \hline \(\theta\) & [\(-\)1,1] & Assortativity for inter-network connectivity & Equation (21) \\ \hline \(N^{P}\), \(N^{V_{x}}\) & _given_ & Number of P-nodes and V-nodes on VN \(k\) & – \\ \hline \(L^{P}\), \(L^{V_{x}}\) & _given_ & Number of P-links and V-links on VN \(k\) & – \\ \hline \(W\) & _given_ & Bandwidth for P-interfaces & – \\ \hline \(\lambda\) & _given_ & Arrival rate of packets & – \\ \hline \(R\) & _given_ & Number of packet re-transmission & – \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of variables used for the PV-NoN model and evaluation. Various approaches have been studied for partitioning physical resources, the aim being to improve the flexibility and efficiency of the virtualized networks [32; 33; 34]. Here, we assume three basic types of interface partitioning (Table 3). The most fundamental is _type-SD_ (Statically Divided), in which resources (i.e., the buffer and network I/O) for the P-interface are divided statically for each VN; the upper limit of the number of packets stored and sent out from a V-interface is divided statically beforehand. With this type, constant performance is guaranteed for each VN and no traffic interference occurs, but the efficiency of resource utilization is not optimal. In contrast to type-SD, we also assume _type-UD_ (UnDivided), in which the physical resource for each VN is not partitioned; the resource capacity for each VN is not guaranteed, and traffic in a VN could even occupy all the buffers and network I/O on the PN, but the physical resources can be utilized completely. As the third case, we assume _type-DD_ (Dynamically Divided), in which the physical resources are allocated dynamically depending on the changing traffic conditions; the physical resources are not partitioned under normal conditions (as with type-UD), but they are partitioned when traffic congestion occurs. Therefore, type-DD allows physical resources to be used more efficiently while guaranteeing the resources for each VN. With all three types, resource virtualization makes the PN and VNs interdependent. Furthermore, with type-UD and type-DD, the sharing of physical resources among multiple VNs makes the latter interdependent.