# 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
It can be seen from the comparison that the time to death of the first node becomes shorter when the number of nodes increases. Compared with the other three algorithms, this algorithm has a longer network lifetime with a different number of nodes. It proves that our algorithm can effectively delete redundant links, reduce energy consumption, and prolong the network lifetime. Figure 8: Changes of energy variance in network simulation. Figure 7: Changes of alive nodes in network simulation. Figure 9: Changes of total energy in network simulation. ## 6 Conclusions

In order to meet the reliable transmission requirements of WSN in a complex agricultural environment, this paper studies the 2-connected fault-tolerant topology construction method based on clustering topology. By dividing the network into single hop effective clusters, similar sensed data are integrated to reduce redundant energy consumption. Considering the residual energy, node density, and expected lifetime, the double price function for CHs is constructed. When the residual energy is greater than the threshold, the price function \(u_{1}\) that can comprehensively optimize the network is used. Otherwise, the price function \(u_{2}\) with the goal of reducing the transmit power is adopted. Based on the double price function, this paper utilizes the potential game theory to initialize the topology between clusters. Then, cut vertex detection is used to eliminate the redundant links in the initial-inter cluster topology, which also ensures that the inter-cluster network is at least 2-connected. The fault-tolerant topology between clusters is conducive to the reliable transmission of important data. By comparing with different topology control methods, it is verified that this algorithm can reduce the average transmit power, balance the energy consumption and effectively prolong the network lifetime. However, there is still much work to be done in agricultural WSN topology. In future work, we plan to study wireless signal transmission characteristics within an agricultural dynamic environment, and further optimize the network performance by comprehensively considering the idle time of the cluster heads, cluster head percentage, channel access, and so on. **Author Contributions**: Conceptualization, H.W. and H.Z. ; methodology, H.W., X.H. and B.Y. ; validation, H.Z., X.H. and H.Z. ; formal analysis, B.Y. and Y.M. ; writing--original draft, all authors; funding acquisition, H.W. and Y.M. All authors have read and agreed to the published version of the manuscript. **Funding**: This research was funded by the National Natural Science Foundation of China, grant number 61871041; the China Agriculture Research System of MOF and MARA, grant number CARS-23-D07; and the Youth Fund of Beijing Academy of Agriculture and Forestry Sciences, grant number QNJJ202030. **Institutional Review Board Statement**: Not applicable. **Informed Consent Statement**: Not applicable. **Data Availability Statement**: Not applicable. **Acknowledgments**: The author would like to thank anonymous commentators and editors for their valuable comments and suggestions. **Conflicts of Interest**: The authors declare no conflict of interest. Figure 10: Network lifetime comparison. ## References

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