## 3 Probabilistic Topology Control Algorithm
### Transmission Power Adjustment Probability Calculation  
When the parameter deviations have been determined, the transmission range adjustment probability can be calculated based on these deviations. The transmission range adjustment probability is defined in Definition 2. **Definition 2**. : _In an underwater cooperating AUVs network, considering the tradeoff between improving the network performance as one aspect and keeping the function of AUVs, the AUVs, in which the current transmission power does not equal the optimal transmission power that is calculated based on optimal algorithms, do not need to adjust their transmission power; rather the AUVs change their transmission power probability. This probability is called the transmission power adjustment probability._  
The calculation of the transmission power adjustment probability is based on the value of the parameter deviations. The larger the deviation, the larger the probability is. Since the mathematical relationship between the transmission power adjustment probability and the parameter deviation cannot be defined clearly, in this paper, we use the fuzzy logic algorithm to calculate the transmission power adjustment probability. The input of the fuzzy logic system is the value of parameter deviation, and output is the transmission power adjustment probability of each parameters. As introduced in [34], the core part of fuzzy logic system is the fuzzy rules design, which decides the accuracy of the output. The more fuzzy rules are applied, the more accurate outputs are. Therefore, similarly to [34], the number of fuzzy rules used in this paper is set to 7, which are shown in Table 3.  
The membership functions of inputs and outputs are shown in Figure 9. \begin{table}
\begin{tabular}{c c} \hline \hline
**INPUT (If): Parameter Deviation** & **OUTPUT (Then): Adjustment Probability** \\ \hline Very small & Very small \\ Medium small & Medium small \\ Small & Small \\ Medium & Medium \\ Large & Large \\ Medium large & Medium large \\ Very large & Very large \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The Fuzzy rules. The outputs of the fuzzy logic system are the transmission power adjustment probabilities of different parameters, which are \(\rho_{P}\), \(\rho_{Q}\), \(\rho_{E}\), and \(\rho_{n}\), respectively. However, considering the fact that for one AUV, there is only one transmission power adjustment probability, therefore, the actual transmission power adjustment probability should be determined based on these four probabilities. Moreover, for guaranteeing the network performance, the transmission power adjustment probability should be determined by the parameter in which the performance is the worst (i.e., the parameter which the deviation is the largest); this is called the Cask Principle. For instance, for \(\rho_{P}\), \(\rho_{Q}\), \(\rho_{E}\), and \(\rho_{n}\), assuming that \(\rho_{Q}\) is the largest in these four probabilities, which means that the queue length is long in AUV; if the AUV chooses a probability which is smaller than \(\rho_{Q}\) as the transmission power adjustment probability, then the performance of queue length cannot be guaranteed. Therefore, the approach used to decide the transmission power adjustment probability in this paper is setting the maximum probability of these four probabilities as the actual transmission power adjustment probability of AUV, which can be expressed as:  
\[\rho_{s}=\max\bigl{\{}\rho_{P},\rho_{Q},\rho_{E},\rho_{n}\bigr{\}}, \tag{19}\]  
Equation (19) means that the probability will be decided by the parameter which has the worst performance of all the parameters that are concerned. This approach is efficient. On the one hand, the ratio of the AUVs which need to adjust their transmission power is reduced, so the control cost of the network reduces, too; on the other hand, since the transmission power adjustment probability is decided by the parameter which has the worst performance, the network performance can be guaranteed.