# Fault-Tolerant Topology of Agricultural Wireless Sensor Networks Based on a Double Price Function
## 4 Hybrid WSN Topology
### Topology Initialization between Clusters Based on Potential Game  
As mentioned above, the CH is responsible for collecting data within the cluster and transmitting it to the sink, so the CHs are of high importance and consume energy quickly. We adopted non-cooperative game theory to construct the initial inter-cluster topology and to reduce the energy consumption on the basis of ensuring network connectivity. \(<M\), \(S\), \(u(S)\)\(>\) is an ordinal potential game. The ordinal potential functions of \(u_{1}\) and \(u_{2}\) are:  
\[V_{1}(p_{i},p_{-i})=\sum\limits_{i\in\mathcal{V}}\left(f(p_{i},p_{-i})(\alpha p _{i}^{\max}\frac{T_{i}^{\max}}{T_{i}^{\min}}+\beta(\frac{neb(p_{i},p_{-i})}{ neb_{i}^{\max}}+\frac{Er_{i}^{\min}}{E^{\max}})\right)-\alpha p_{i} \frac{T_{i}^{\max}}{T_{i}^{\min}}-\beta\frac{neb(p_{i},p_{-i})}{ neb_{i}^{\max}} \tag{8}\]  
\[V_{2}(p_{i},p_{-i})=\sum\limits_{i\in V}\left(f(p_{i},p_{-i})(\alpha p_{i}^{ \max}\frac{T_{i}^{\max}}{T_{i}^{\min}})-\alpha p_{i}\frac{T_{i}^{\max}}{T_{i}^ {\min}}\right) \tag{9}\]  
It is proved that \(p_{i}\) and \(q_{i}\) are the two optional transmit powers of the node \(i\). Assuming \(p_{i}>q_{i}\), \(f\) (\(p_{i}\), \(p_{-i}\)) \(\geq f\) (\(q_{i}\), \(p_{-i}\)). For \(u_{1i}\), when node \(i\) selects power \(p_{i}\) and \(q_{i}\) respectively, the price difference is calculated as follows:  
\[ \begin{split}\Delta u_{1i}&=u_{1i}(p_{i},p_{-i})-u _{1i}(q_{i},p_{-i})\\ &=f(p_{i},p_{-i})(\alpha p_{i}^{\max}\frac{T_{i}^{\max}}{T_{i}^{ \min}}+\beta(\frac{neb(p_{i})_{r}}{neb_{i}^{\max}}+\frac{Er_{i}^{\min}}{E^{ \max}}))-\alpha p_{i}\frac{T_{i}^{\max}}{T_{i}^{\min}}-\beta\frac{neb(p_{i})}{ neb_{i}^{\max}}\\ &-f(q_{i},p_{-i})(\alpha p_{i}^{\max}\frac{T_{i}^{\max}}{T_{i}^{ \min}}+\beta(\frac{neb(p_{i})}{neb_{i}^{\max}}+\frac{Er_{i}^{\min}}{E^{\max} }))-\alpha q_{i}\frac{T_{i}^{\max}}{T_{i}^{\min}}-\beta\frac{neb(p_{i})}{ neb_{i}^{\max}}\end{split}  \tag{10}\]  
Similarly:  
\[ \begin{split}\Delta V_{1}&=V_{1}(p_{i},p_{-i})-V_{ 1}(q_{i},p_{-i})\\ &=\Delta u_{1i}+\sum\limits_{j\in V_{j}\
eq i}(f(p_{i},p_{-i})( \alpha p_{j}^{\max}\frac{T_{j}^{\max}}{T_{j}^{\min}}+\beta(\frac{neb(p_{j})}{ neb_{j}^{\max}}+\frac{Er_{j}^{\min}}{E^{\max}}))-\alpha p_{j}\frac{T_{j}^{ \max}}{T_{j}^{\min}}-\beta\frac{neb(p_{j})}{neb^{\max}})\\ &-\sum\limits_{j\in V_{j}\
eq i}(f(q_{i},p_{-i})(\alpha p_{j}^{ \max}\frac{T_{j}^{\max}}{T_{j}^{\min}}+\beta(\frac{neb(p_{j})}{neb_{j}^{\max}}+ \frac{Er_{j}^{\min}}{E^{\max}}))-\alpha p_{j}\frac{T_{j}^{\max}}{T_{j}^{\min}}- \beta\frac{neb(p_{j})}{neb^{\max}})\\ &\text{Set }z_{j}=\alpha p_{j}^{\max}\frac{T_{j}^{\max}}{T_{j}^{\min}}+\beta( \frac{neb(p_{j})}{neb_{j}^{\max}}+\frac{Er_{j}^{\min}}{E^{\max}})-\alpha q_{i} \frac{T_{j}^{\max}}{T_{j}^{\min}}-\beta\frac{neb(p_{j})}{ne_{j}^{\max}}\\ &\Delta V_{1}=\left\{\begin{array}{ll}\Delta u_{1i}&f(p_{i},p_{-i} )=f(q_{i},p_{-i})\\ \Delta u_{1i}+\sum\limits_{j\in V_{j}\
eq i}z_{j}&f(p_{i},p_{-i})\
eq f(q_{i},p_ {-i})\end{array}\right.\end{split}  \tag{12}\]  
For Equation (12), obviously \(\Delta u_{1i}>0\) and \(z_{j}>0\), we get \(\Delta V_{1}>0\). Therefore, for Equation (12) above, \(\Delta V_{1}>0\Leftrightarrow\Delta u_{1i}>0\). Similarly, when transmit power \(p_{i}\) and \(qi\) are respectively selected at node \(i\), the price difference \(\triangle u_{2i}\) is:  
\[