# Fault-Tolerant Topology of Agricultural Wireless Sensor Networks Based on a Double Price Function
## 4 Hybrid WSN Topology
### Two-Connected Fault-Tolerant Topology between Clusters
The detailed parameters are shown in Table 1.

### Comparison of Algorithm Robustness

According to Equations (1) and (2), \(\alpha\) and \(\beta\) parameters will affect the results of the potential game algorithm. Therefore, we repeated 20 times, randomly distributed 50 nodes in the farmland, and adjusted the parameters \(\alpha\) and \(\beta\) respectively to verify the influence of the weight factor on the network topology performance. Figure 3 shows the influence of the algorithm on network performance when \(\alpha\in\) {20,30,40,50,60,70,80,90,100} and \(\beta=1\) is fixed. Figure 4 shows the influence of the algorithm on network performance when \(\beta\in\) {1,2,3,4,5} and \(\alpha=50\) is fixed. Average transmit power and average node degree are used to analyze the network performance. \begin{table}
\begin{tabular}{c c} \hline
**Parameter Description** & **Value** \\ \hline Total number of nodes & 50–150 \\ Size of monitoring area & 100 \(\times\) 100 m\({}^{2}\) \\ Node initial energy & 5 ] \\ Transmit power & [10\({}^{-5}\), 5] W \\ \(c_{1}\) & 50 nJ/bit \\ \(c_{2}\) & 100 pJ/bit/m\({}^{2}\) \\ \(r\) & 3.5 \\ \(\eta\) & 0.5 \\ \(v\) & 0.5 \\ Received power & \(p_{i}(1+d_{ij})^{-r}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Experimental parameters. As can be seen from Figure 3, with the increase of \(\alpha\), the transmit power and average node degree increase. When \(\alpha\) is 50, the adjustment effect is no longer obvious. In the Figure, with the increase of \(\beta\), the average transmit power and average node degree decrease accordingly. There is a proportional relationship between weight factor and network performance. According to the general theory of network topology, when the transmit power is low, network energy consumption can be reduced. Moderate node degree can improve the robustness of the network. Therefore, considering the network transmission power and average node degree, \(\alpha=30\), \(\beta=3\). After determining the algorithm parameters, when the number of nodes is 50, we compared four algorithms, namely our algorithm, the double price function based potential game algorithm (DPOG), EFTCG algorithm [14], and the 3 DK-RNG algorithm [17]. The DPOG algorithm only constructs the network topology based on the game theory, and the effectiveness of the proposed algorithm in redundant link elimination was verified by comparison with DPOG algorithm. The EFTCG algorithm introduces the influence factor of network 2-connectivity into the price function, and constructs a fault-tolerant topology based on game theory. The 3 DK-RNG algorithm first adopted game theory to initialize the whole network topology, and then eliminated network redundant links based on a rigid graph, which ensured that the network was at least 2-connected. Figure 5 shows the generated network topology. With maximum transmit power, the network communication link is too complex and has redundant links as shown in Figure 5a. Original network topology increases the energy consumption of the node and causes the nodes to die prematurely. In order to reduce energy consumption on the basis of ensuring network transmission performance, the network was clustered as shown in Figure 5b. It can be seen that the network clusters were evenly distributed and the number of nodes in the cluster were balanced. Figure 5c shows the network topology generated by the potential game. Figure 5d shows the network topology optimized by the cut vertex detection algorithm. Although the potential game algorithm effectively reduces the number of links, there are still redundant links, and the instability of a wireless signal in agricultural scenarios was not considered. Figure 4: When \(\alpha=30\), \(\beta\) impact on network performance: (**a**) Change of average transmit power; (**b**) Change of average node degree. Figure 3: When \(\beta=1\), \(\alpha\) impact on network performance: (**a**) Change of average transmit power; (**b**) Change of average node degree.