## 6 Simulation Evaluation of the Dynamic VWSN
### Discussion of Memory Utilization  
In this section, we estimate the amount of memory needed to store the tables used for the routing algorithm shown in this paper. For the \(N\)-th-tier routing table, the number of entries that each node needs to hold is the sum of the number of \(N\)-th-tier modules that belong to the \((N+1)\)-th-tier module of that node (excluding the \(N\)-th-tier module of the node) and the number of neighboring \(N\)-th-tier modules that belong to a different \((N+1)\)-th-tier module. Therefore, the number of entries \(Z_{RT}^{N}(n)\) of the \(N\)-th-tier routing table when node \(n\) belongs to an \(N\)-th-tier module \(M_{a}^{N}\) and an \((N+1)\)-th-tier module \(M_{x}^{N+1}\) is defined as follows:  
\[Z_{RT}^{N}(n)=|Sub^{N}(M_{x}^{N+1})|-1+|\mathcal{N}^{N}(M_{a}^{N})\setminus Sub ^{N}(M_{x}^{N+1})| \tag{4}\]  
Here, \(n\in M_{a}^{N}\) and \(n\in M_{x}^{N+1}\). \(Sub^{N}(M_{x}^{N+1})\) is the set of \(N\)-th-tier modules that belong to the \((N+1)\)-th-tier module \(M_{x}^{N+1}\), and \(|Sub^{N}(M_{x}^{N+1})|\) is the cardinality of \(Sub^{N}(M_{x}^{N+1})\). The second term means that \(M_{a}^{N}\) is not included as the destination module in the \(N\)-th-tier routing table. \(\mathcal{N}^{N}(M_{a}^{N})\) is the set of \(N\)-th-tier modules that neighbor \(M_{a}^{N}\), and \(\mathcal{N}^{N}(M_{a}^{N})\setminus Sub^{N}(M_{x}^{N+1})\) is the relative complement of \(Sub^{N}(M_{x}^{N+1})\) in \(\mathcal{N}^{N}(M_{a}^{N})\). Then, \(|\mathcal{N}^{N}(M_{a}^{N})\setminus Sub^{N}(M_{x}^{N+1})|\) is the number of \(N\)-th-tier modules that are connected to \(M_{a}^{N}\) and belong to an \((N+1)\)-th-tier module other than \(M_{x}^{N+1}\). From the above, the total number of entries \(Z_{RT}(n)\) of the routing tables held by node \(n\) is  
\[Z_{RT}(n)=\sum_{i}Z_{RT}^{i}(n), \tag{5}\]  
where \(i\) is the identity of each tier composing the VWSN topology. Because the number of zeroth-tier modules, which are nodes belonging to the same first-tier module, is the largest of all of the tiers, the size of the zeroth-tier routing table is dominant in many cases. For the \(N\)-th-tier connection table, the number of entries that each node needs to hold is the number of \((N-1)\)-th-tier virtual links that connect an \((N-1)\)-th-tier module belonging to the \(N\)-th-tier module of the node with another \((N-1)\)-th-tier module belonging to a neighboring \(N\)-th-tier module. In our proposal, the number of \((N-1)\)-th-tier virtual links added by mapping the \(N\)-th-tier virtual link depends on the number of \((N-1)\)-th-tier virtual links embedded in the \(N\)-th-tier VWSN topology. Therefore, the number of entries \(Z_{CT}^{N}(n)\) of an \(N\)-th-tier connection table for node \(n\) that belongs to \(M_{a}^{N}\) is as follows:  
\[Z_{CT}^{N}(n)=\sum_{M_{x}^{N}\in\mathcal{N}^{N}(M_{x}^{N})}\lceil C_{\text{ inter}}^{N}(E^{M_{a}^{N}}+E^{M_{y}^{N}})\rceil \tag{6}\]  
Here, \(n\in M_{a}^{N}\). Therefore, the total number of entries \(Z_{CT}(n)\) of the connection tables held by \(n\) is  
\[Z_{CT}(n)=\sum_{i}Z_{CT}^{i}(n), \tag{7}\]  
where \(i\) is the identity of each tier composing the VWSN topology. Although the number of zeroth-tier virtual links is the largest among all of the tiers, we can tune the parameter \(C_{\text{inter}}^{N}\) separately for each tier. In this paper, because we set \(C_{\text{inter}}^{N}\) to a small value, the size of the connection tables is smaller than that of the routing tables. For the long-link table, the number of entries that each node needs to hold, denoted by \(Z_{LT}(n)\), is the number of virtual links to which it is assigned as a relay node or an end node. Because the number of entries of a long-link table depends entirely on the specific node, we cannot easily estimate the size of the long-link table. Some nodes will have an empty long-link table; others will have a large number of entries in the long-link table.