## 3 A flexible Approach to a Reachable Network Topology with a Minimum Connection Degree
### A Full-Reachable WSN Based on an Adaptive Transmission Range with Minimum Node Degree  
An adaptive approach to support minimum node degree and energy efficiency of WSN is proposed. A flowchart of the proposed approach is shown in Figure 3. The details of the proposed approach will be presented in this subsection. This study mainly emphasizes topology control for power control based on a hierarchical network. Therefore, the optimal transmission range will be obtained according to node degree and node density through the proposed approach. The architecture of the proposed framework is initially composed of the network environment, construction of the hierarchical network, control of the topology network and construction of the transmission route. First, the environmental parameters are set to imitate a real application. Next, the cluster technique [43] is applied to cluster sensor nodes of the target network. The hierarchical network is composed of node clusters to manage the transmission range in a cluster. The transmission range is obtained through the proposed adjustment model of the transmission range to connect adjacent nodes and form a network topology. Thus the proposed approach will contribute an optimal network topology as indicated in Figure 3.  
**Figure 3.** The flowchart of an adaptive optimization transmission with a minimum node degree. In cooperation with the authors' previous work of the smart cluster algorithm [44] and the routing algorithm [45], a high energy efficiency WSN is obtained through the steps as follows. Step 1: Initialization of the network environment  
To imitate a real application, the required parameters, including those of the network environment and node specification, are carefully determined. The deployment space is delimited. The maximum transmission powers, power consumption per watt and default node degree for each node are established. Step 2: Construction of the network architecture  
A smart cluster technique is applied to cluster the network based on the evolutionary computation technique of this framework. To develop the network architecture, information on position and clusters is not known in advance. For the hierarchical network in this study, there are three layers: data sink, routers, and sensor nodes. The data sink is in the first layer. All of the cluster heads in the second layer are included to form a cluster and function as a router. The third layer is composed of groups from clustered sensor nodes. The sensor nodes in a cluster deliver data to the data sink through the corresponding cluster head. The cluster heads transmit or forward data to the data sink. Step 3: Control of the network topology  
A hierarchical network with nodes organized in clusters is constructed in step 2. In this procedure, an adjustment model of the transmission range is produced to determine an optimal transmission range through the node density, expected node degree, and expected probability of k-degree nodes. The optimal network with energy efficiency and full reachability will be constructed through the derived transmission range. In the deployment of a wireless sensor network, transmission power should depend on the deployed node density and expected node degree. In other words, transmission power should be adapted according to node sparseness. Thus, [PERSON] [46] proposed a probability of finding an expected node degree (\(n_{0}\)) in two dimensions with uniform distribution and a known node density and transmission radius.