# 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
Figure 5d further reduces the number of communication links on the basis of Figure 5c, while ensuring that the network is at least 2-connected and has fault tolerance. Comparing Figure 5c with Figure 5d, the network is connected as a whole, and the redundant link can be eliminated. Clustering changes the logical network topology. The links generated in this paper are not very regular due to clustering. However, compared with the network topology in Figure 5e,f, the network topology in Figure 5d is more concise. In addition, we further compare the performance of this algorithm with that of the comparison algorithm as follows. ### Comparison of Topology Performance with Different Number of Nodes

In this paper, the number of nodes is increased from 50 to 150. Figure 6a,b show the comparison of average transmit power and average node degree. With the increase in the number of nodes, the average transmit power shows a downward trend. The average transmission power of our algorithm is 0.024 W with 50 nodes. When the number of nodes is 150, the average transmit power of our algorithm is 0.020 W. Among them, the average transmit energy obtained by EFTCG is the lowest, because it uses the potential game algorithm for selfish power control while ensuring a 2-connected network in the construction process. Nodes only select the transmit power according to their own needs. Figure 5: Network topology comparison: (**a**) Initial network topology; (**b**) Network clustering; (**c**) Inter-cluster topology constructed by DPOG; (**d**) Inter-cluster topology constructed by cut vertex detection; (**e**) Network topology constructed by EFTCG; (**f**) Network topology constructed by 3 DK-RNG. Our algorithm and the DPOG algorithm are affected by clustering, and the average transmit power obtained is slightly larger than EFTCG. However, due to the combination of single-hop effective clusters, our algorithm has the lowest average node degree. While the transmit power is only lower than that of EFTCG, the node degree is reduced to ensure that the inter-cluster topology is fault-tolerant. ### Comparison of Network Performance under Data Transmission Simulation

In order to better verify the network performance, when the number of nodes is 50, the opportunistic routing protocol enables the network to transmit data with an increase in the number of simulation rounds, and each data transmission reduces the node energy. To facilitate comparison, we take the first node dead time as the network lifetime, which reflects the essence of the lifetime optimization problem, that is, to balance the node energy consumption in the whole wireless sensor network. The network is no longer working when there are dead nodes in the network. Figure 7 shows changes in alive nodes. Figure 8 shows the standard deviation of the residual energy and Figure 9 shows the average residual energy. Obviously, our algorithm maintains a small standard deviation of residual energy and a large average residual energy. The death time of the first node in our algorithm is the latest. The 3 DK-RNG network has the highest energy consumption and the shortest network lifetime. EFTCG has the lowest energy consumption and the lowest standard deviation of energy consumption in the early stages, but it does not take into account the energy balance of the whole network. The imbalance of energy consumption leads to the premature death of some nodes and the sharp decline of network performance. Although the DPOG considers energy balance, there are still redundant links, which consume more energy. However, our algorithm introduces a cluster structure and adopts a double price function, so the node transport energy consumption is smaller, which effectively guarantees the energy balance of the network and prolongs the network lifetime. Figure 6: Comparison of network performance with different number of nodes: (**a**) Comparison of average transmit power; (**b**) Comparison of average node degree. Figure 10 shows the simulation round number when the first node dies.