# Fault-Tolerant Topology of Agricultural Wireless Sensor Networks Based on a Double Price Function
## 2 Related Works  
Recently, a large number of topological control algorithms have been proposed, which mainly focus on power control [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21] and network clustering [22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. By adjusting the transmit power to affect the communication range, the residual energy consumption can be reduced, and the communication quality can be guaranteed [10]. Cluster topology control uses data fusion to cluster nodes to reduce transmission load. The wireless network topology can be constructed by controlling the transmit power. Game theory is an effective method to control the transmit power. Players interact and cooperate to obtain the best strategy. Reference [11] used this feature to optimize the transmit power. [PERSON] et al., proposed a topology control algorithm based on energy consumption, which made the node obtain the minimum hop under the maximum power and selected the power with the maximum price through the optimal response strategy [12]. However, the calculation of the model is complex. In addition, [PERSON] et al., proposed a distributed WSN topology control method based on the Lagrange duality method, sub-gradient method, which combined network coding and topology control to establish an automatic dynamic structure for wireless sensors [13]. The above research focussed on constructing the network topology with balanced energy consumption, without considering the fault-tolerant characteristics of the network. [PERSON] et al., studied an energy-efficient and fault-tolerant topology control game algorithm (EFTCG), designed the price function under the constraint of the 2-connected network and obtained the Nash equilibrium by the potential game [14]. The model did not fully consider the energy balancing of the network. Considering the link disconnection caused by dead nodes, a connectivity reconstruction scheme was proposed to establish a more reliable path according to energy efficiency [15]. In [16], used a rigid graph obtained network topology, which ensures the 2-connected network. The network topology generated by the algorithm had low overhead and effectively prolonged the network lifetime. For reducing redundant links and network energy consumption, reference [17] integrated the minimum rigid graph and potential game to design a 3 DK-RNG topology control method for an underwater WSN. The potential game was adopted to control the transmit power to form the optimal network topology, and then the redundant links were eliminated based on the rigid graph. In agricultural environments, sensors need to be densely deployed for accurate environmental sensing. The existing fault-tolerant topologies provided multiple links for each node in the network, which causes a large amount of energy consumption and is not conducive to long-term monitoring of dense networks. In terms of network clustering, cluster head selection rules are the focus of existing studies. In [22], the fuzzy system was used to integrate the residual energy, the distance from sink and the node density, so as to calculate the cluster head (CH) selection probability of each node to ensure load balance. [PERSON] et al., studied the CH selection combining the dolphin echolocation algorithm and crow search algorithm [23]. [PERSON] et al., proposed an energy-based multi-path routing algorithm, which generated multiple paths from the routing topology between clusters [24]. In data transmission, the optimal path is dynamically selected considering path energy consumption, hop, and path energy [25]. A fitness function was designed under the influence of multiple indicators such as energy, distance, and link lifetime, and an intelligent optimization algorithm based on penguin fuzzy was used to select CHs [26] Reference [28] further provided the maximum weight for the residual energy and the minimum weight for the transmission distance, based on which the cluster head probability is calculated. Reference [29] proposed a distributed redundant awareness topology control protocol for WSN.