## 3 Probabilistic Topology Control Algorithm
### PTC-Based FTC Algorithm  
Based on Sections 3.2 and 3.3, the transmission power adjustment probability of AUV can be calculated. After that, the AUVs will adjust their transmission power according to this probability. Since many transmission power allocation algorithms have been proposed in the past decades, the transmission power allocation algorithm will be not the main research topic of this paper. In this paper, the calculation of the optimal transmission power is based on the FTC algorithm which is proposed by [6]. The FTC algorithm is the learning-based fuzzy logic control algorithm for topology control. In the following, we will introduce this algorithm briefly. Figure 10 shows the system structure of FTC. Adjusting the communication power is a very common capability of many AUVs. The output of FTC is the transmission power (_TP_). The target of FTC is to reach a specific degree of AUVs. Therefore, the input is the desired AUV's degree, denoted by \(ND_{ref}\). On the other hand, according to the conclusion in [35], the probability that AUV's degree is \(n\) is shown in Equation (24), so the probability that an AUV has \(n\) neighbors is another fuzzy logic controller input, denoted by _Prob_. In practice, \(ND_{ref}\) is integer and the transmission power has an upper bound \(P_{\max}\), i.e., \(ND_{ref}>0\) and \(0\leq P\leq P_{\max}\):  
\[P(ND\geq k)=f(k,P)=1-\sum\limits_{n=0}^{k-1}\frac{\left(\rho\pi r^{2}\right)^{ n}}{n!}e^{-\rho\pi r^{2}}, \tag{24}\]The training data set is provided by Equation (24); the fuzzy controller can be obtained through the neuro-adaptive learning algorithm. In Equation (24), the transmission range can be calculated based on Equation (13). The parameters of the membership function are automatically tuned through a back propagation algorithm individually or in combination with a least squares method. The generation of the training data set can be shown as follows. As illustrated in Figure 10 and Equation (24), the inputs are \(ND_{ref}\) and \(Prob\), and the output is the transmission power. Given \(\rho\), \(ND\in\{k_{1},k_{2},\ldots,k_{m}\}\) and \(TP\in\{p_{1},p_{2},\ldots,p_{m}\}\), \(Prob=f(ND,TP)\) can be calculated from Equation (24). The training data set \(T\) is a \(s\times 3\) matrix in the form of \([ND,Prob,TP]\), where \(s=m\cdot j\). For instance, one element in the training data set is (3, 0.9, 0.25); this means that the transmission power is set to 0.25 if the probability that \(ND\geq 3\) is 0.9, where the transmission power is normalized (i.e., the maximum transmission power is (1). Since \(ND\) is characterized by probability, it is necessary to adjust the AUV's degree if an AUV does not reach \(ND\). For instance if \(TP=0.25\) cannot actually lead to \(k=ND\), then the next step is to adjust \(Prob\) to a higher value according to the AUV's degree error \(e_{ND}\). There is an integral controller outside the fuzzy control to adaptively change \(Prob\) (Figure 10). From the control theory point of view, the system properties are controlled by parameter \(Prob_{0}\) and \(K\). If \(e_{ND}\) is less than 0, \(K\) is configured to be half of its initial value.