## 4 The Proposed Algorithm-TCEB
### Intra-Cluster Topology Construction  
After the non-cooperative-game-based cluster-head selection, the cluster-heads have been exactly determined as well as the probably partitioned region of the cluster. However, the current clusters constitute only the basic topology so that communication on intra-cluster and inter-cluster is further to build to carry out the data transmission. Usually, the underwater acoustic communication is affected by multipath and high end-to-end delay, while also depending on the distance and the time as well as the frequency. Therefore, topology construction will have a great impact on network communication performance. Initially, we take the delay with the multipath effects into consideration when the intra-cluster topology is built. Due to the non-uniformity of the underwater medium space, the acoustic channel has a multipath phenomenon. That means the signal from the source to the destination may pass through the different path under a certain transmission power. On the other hand, because of the difference of path length, the acoustic wave which is to reach the destination by different paths takes a different time, as well as signal attenuation. As shown in Figure 1, multiple paths exist from the source node \(i\) to the destination node \(j\). They can roughly be represented by \(L_{0}\), \(L_{1}\), and \(L_{2}\), respectively, where \(L_{0}\) indicates the direct path between the source and the destination, \(L_{1}\) denotes that a reflection arrives through the water surface, and \(L_{2}\) is a reflection as well but through the water bottom. Obviously, \(L_{0}\) is the shortest path for acoustic wave propagation, which costs the minimum duration with the less delay. The delay is complicated to correctly calculate in the practical underwater environment, thus we show the scenario in Figure 1. Inspired by [PERSON] et al. [27], the total delay between the node \(i\) to \(j\) can be written as:  
\[delay_{i\to j}=\frac{l}{C}+\frac{d\left(i,j\right)}{c}+\Delta\tau \tag{13}\]  
where \(l\) and \(c\) are the length of the data packet in bits and the propagation velocity of the acoustic wave in water, respectively. \(d(i,j)\) refers to the distance between the node \(i\) and \(j\), \(\Delta\tau\) is the delay caused by multipath propagation. \(C\) is the channel capacity in bits per second which can be expressed according to the [PERSON]'s theorem:  
\[C=B\cdot\log_{2}\left(1+SNR\right) \tag{14}\]  
where \(B\) is the bandwidth of the channel, and \(SNR\) refers to the signal to noise ratio. Assuming that the noise is Gaussian and the channel is time-invariant for some interval time, the capacity can divide the total bandwidth into many narrow sub-bands [28]. The \(i\)th sub-band is centered around the frequency \(f_{i}\) (\(i=1,2,...\)), and its width is \(\Delta f\). We introduce the power spectral density of the signal while considering the real scenario. Therefore, more general channel capacity [29] can be expressed as:  
\[C=\sum_{i}\Delta f\log_{2}^{\left[1+\frac{X(f)H(f)}{N(f)}\right]} \tag{15}\]  
where \(X(f)\) is the power spectral density of the transmitted signal from the source, \(H(f)\) denotes the channel transmission function, and \(N(f)\) is the noise power spectral density. Let \(delay_{i\to j}^{0}=\frac{l}{C}+\frac{d_{0}(i,j)}{c}+\tau_{0}\) denote the total delay when the signal is propagating on path \(L_{0}\). To reduce the complexity of delay analysis by multipath effects, the communication paths from the source to the destination would be restricted in this paper, which means data propagation through the reflection path is limited. Therefore, according to this rule, the path can be defined invalid if the delay for signal propagation from the node \(i\) to \(j\) on such path (any path except for \(L_{0}\)) is greater than \(delay_{i\to j}^{0}\). When the intra-cluster topology construction is beginning, all the non-cluster-head nodes wait to join one of the \(k_{opt}\) clusters.