## 1 Introduction
However, for the UCAN, in which the network resources are limited and the network topology changes frequently, this approach is not as efficient as in a static network. The heavy control cost caused by AUVs' mobility will deteriorate the network performance greatly. Therefore, it is necessary to investigate how to reduce the control cost in order to improve the network performance of underwater cooperating AUV networks. Additionally, for an underwater cooperating AUV network, once the current transmission power of an AUV is not equal to the optimal transmission power which is calculated based on optimal algorithms (such as interference-based, transmission delay-based, energy consumption- based, etc. ), considering the other parameters of the AUV, adjusting the transmission power may not be a good strategy to improve the network performance. This means it is necessary to accept a tradeoff between improving the network performance in one aspect and keeping the functionality of network (such as, network connectivity). For instance, when the optimal transmission power of AUV is \(P^{*}\) and the current transmission power is \(P\), if \(P\approx P^{*}\) (which means the difference between \(P\) and \(P^{*}\) is small enough), then even if the current transmission power does not equal the optimal one, optimization rules do not need to be applied when energy consumption and network congestion are taken into account. This tradeoff is useful for an underwater cooperating AUV network. Moreover, in an underwater cooperating AUV network, whether the transmission power needs to be adjusted or not also relates to other network parameters. For example, if \(P\ll P^{*}\), but the residual energy of an AUV is small, in this case, it is better to not increase the transmission power in order to prolong the lifetime of this AUV. Motivated by these facts, in this paper, we propose a new topology control algorithm called the probabilistic topology control (PTC) algorithm for underwater cooperating AUV networks, which is based on the value of an AUV's residual energy, queue length, current transmission power, and number of neighbors to determine the transmission power adjustment probability of the AUV. In PTC, when the transmission power is not equal to the optimal one, the AUVs do not adjust their transmission power immediately. The deviations of each AUV's residual energy, current transmission power, queue length, and number of neighbors are used to calculate each parameter's adjustment probability through a fuzzy logic algorithm. The larger the deviation, the larger the adjustment probability is. The probabilities are \(\rho_{P}\), \(\rho_{O}\), \(\rho_{E}\), and \(\rho_{n}\), respectively. The maximum adjustment probability will be chosen as the transmission power adjustment probability of an AUV. Based on these innovations, the PTC algorithm can improve the network performance greatly; In particular it can reduce the transmission power adjustment ratio of the network. Note that the PTC algorithm can combine with any power control algorithm to reduce the transmission power adjustment ratio. The main contributions of this paper are as follows:  
* We propose the definition of transmission power adjustment probability. Based on this definition, we propose the probabilistic topology control (PTC) algorithm for underwater cooperating AUV networks. In PTC, when the current transmission power does not equal the optimal one, whether an AUVs needs to adjust its transmission power or not will be decided based on the parameters of this AUV;
* We propose the definition of transmission power adjustment ratio for topology control algorithm. Based on this definition, we analyze the properties of PTC algorithm on reducing the transmission power ratio of the network;
* Combining with the fuzzy-logic topology control (FTC) algorithm, in this paper, we compare the performance of the PTC-based FTC algorithm and the standalone FTC algorithm. The simulation results demonstrate the effectiveness of the PTC algorithm in improving the network performance.