## 3 Heterogeneous Network Model of the Proposed Models
### Network Topology for the Proposed Protocols  
The network topology for the proposed model consists of the central controller residing at the BS, placed at the boundary of the network region, and a limited number of sensor nodes being scattered in the network area. Initially, the whole network region is assumed to be a two-dimensional rectangular area with A to B limits at the \(x\)-axis and C to D limits at the \(y\)-axis. Therefore, network area \([A,B]\times[C,D]\) is divided into \(i\) clusters along the \(x\)-axis and \(j\) along the \(y\)-axis. Every cluster position \((x_{i},y_{j})\) represents an independent sub-area. The central controller at the BS can compute the height of the surface, \(f(x_{i},y_{j})\), and the Average Surface Height (ASH) of all clusters is approximately equal to the height of the whole network, which can be calculated as:  
\[ASH=\frac{f(x_{o},y_{o})+...+f(x_{1},y_{1})+...+f(x_{i-1},y_{j-1})}{I\times J} \tag{1}\]  
Figure 2: Heterogeneous hybrid architecture of SDN-enabled Iteratively Reconfigurable WSNs (SDN-IRWSNs).  
Approximation sigma notation can be used to generalize this scalability:  
\[ASH\approx\frac{1}{I\dot{j}}\sum_{i=0}^{I-1}\sum_{j=0}^{J-1}f(x_{i},y_{j}) \tag{2}\]  
where the total length of \(i\) and \(j\) can be defined as \(I=B-A\) and \(J=D-C\). From this, the ASH scalability approximation can be derived as:  
\[ASH\approx\frac{1}{[B-A][D-C]}\sum_{i=0}^{I-1}\sum_{j=0}^{J-1}f(x_{i},y_{j}) \triangle x\triangle y \tag{3}\]  
where \(\triangle x\) and \(\triangle y\) is the variation at \(i\) and \(j\), respectively, and special notation for the limit of the double sum can be defined as:  
\[ASH\approx\lim_{I\rightarrow\infty}\sum_{i=0}^{I-1}\sum_{j=0}^{J-1}f(x_{i},y _{j})\triangle x\triangle y \tag{4}\]  
In this way, double integration optimization can be derived from the above expressions:  
\[\lim_{I\rightarrow\infty}\sum_{i=0}^{I-1}\sum_{j=0}^{J-1}f(x_{i},y_{j}) \triangle x\triangle y=\int_{A}^{B}\int_{C}^{D}f(x_{i},y_{j})dA \tag{5}\]  
\[ASH=\lim_{I\rightarrow\infty}\int_{A}^{B}\int_{C}^{D}f(x_{i},y_{j})dA \tag{6}\]  
Based on initial energy resources, sensor nodes are divided into three categories called normal nodes, advance nodes and super nodes, which contain different energy levels in ascending order, respectively. The central controller monitors the sensor layer-wise distribution according to energy level resources. The distribution and the iterative clustering of the sensor nodes are shown in Figures 3 and 4. Multi-application sensor nodes scattered in the case of the SACFIR protocol are able to report environmental parameters like temperature, humidity and pressure. This application-awareness of multiple sensing tasks of the SACFIR model is shown in Figure 5. This figure shows the application-awareness of the proposed model by sensing temperature, pressure and humidity. More importantly, the SACFIR model can adjust the information index preference decided by the application layer. In the case of SAMCFIR, the central controller adjusts the application sensitivity for monitoring of the temperature and pressure according to a regular pattern. To deal with emergency situations, this model considers this as critical data and directly transmits the data to the central controller. These iterative application-aware sensing outputs are shown in Figure 6. Furthermore, SAMCFIR has the flexibility to adjust the sampling rate of the sensing environment and can collect information according to the required application.  
Figure 3: Iterative clustering topology in the case of the SDN-based Application-aware Centralized adaptive Iterative Flow Reconfiguring (SACFIR) routing protocol.