## 2 System Model and Problem Formulation  
We consider the network-assisted D2D WCS, where multiple users are distributed by uniform distribution over an area \(D\times D\), as shown in Figure 1. We assume that there is the interference from \(U\) cellular users. In addition, it is assumed that there are \(N\) single-antenna transceiver pairs, each of which comprises one transmitter and one receiver, and all transmissions are carried out simultaneously over the same frequency. Accordingly, each transmitter can interfere to other transceiver pairs, i.e., the receiver \(j\in\mathbb{J}\) receives data from the transmitter \(i\in\mathbb{I}\) and receives interference from other transmitters \(k\in\mathbb{I}\backslash\{i\}\), where \(\mathbb{I}=\{1,2,\cdots,N\}\) and \(\mathbb{J}=\{1,2,\cdots,N\}\) denote the set of the transmitters and receivers, respectively. The path loss model, which is denoted as \(G_{i,j}\), between transmitter \(i\) and receiver \(j\) is expressed, as follows:  
\[G_{i,j}=\beta d_{i,j}{}^{-\alpha}, \tag{1}\]  
where \(\beta\), \(\alpha\), and \(d_{i,j}\) are the path loss coefficient, path loss exponent, and distance between transmitter \(i\) and receiver \(j\), respectively. Using Equation (1), the wireless channel power between transmitter \(i\) and receiver \(j\) can be modelled, as follows:  
\[h_{i,j}=|g_{i,j}|^{2}G_{i,j}, \tag{2}\]  
where \(g_{i,j}\) is a Nakagami-\(m\) fading channel [17]. The Nakagami factor \(m\) means the ratio of line-of-sight and non-line-of-sight components. For instance, when \(m=1\), there is no line-of-sight component, which denotes the Rayleigh fading channel, whereas the line-of-sight component becomes stronger as \(m\) increases. Using Equation (2), the spectral efficiency (SE) for the \(i\)-th transceiver pair, which is used as a part of the loss function for deep learning, is expressed as  
\[SE_{i}=\log_{2}(1+\frac{h_{i,j}P_{i}}{N_{0}W+\sum_{k\in\mathbb{I}\backslash\{ i\}}h_{k,j}P_{k}+\sum_{u=0}^{U}z_{u,i}P_{c}}), \tag{3}\]  
where \(P_{i}\), \(N_{0}\), and \(W\) are the transmit power of transmitter \(i\), the noise spectral density and the bandwidth, respectively, \(0\leq P_{i}\leq P_{max}\), and \(P_{max}\) is the maximum transmit power at the transmitter. Moreover, \(P_{c}\) denotes the transmit power of the cellular users and \(z_{u,i}\) is the wireless channel power between cellular user \(u\) and receiver \(i\), which is obtained using Equation (2). In order to measure data-rate fairness for transceiver pairs, using Equation (3), the index of fairness (IF) is defined as [18]  
\[IF=\frac{(\sum_{i=1}^{N}S\bar{E}_{i})^{2}}{N\sum_{i=1}^{N}S\bar{E}_{i}^{2}}, \tag{4}\]  
where \(S\bar{E}_{i}\) represents the average SE for the \(i\)-th transceiver pair, and \(0\leq IF\leq 1\). As \(IF\) rises, the fairness performance becomes better and, thus, the perfect fairness is achieved when \(IF=1\).  
In order to consider the fairness performance in the transmit power control scheme, we use both SE and IF in Equations (3) and (4) in order to find the optimal transmit powers for each transmitter. Subsequently, in this paper, we determine the transmit powers that maximize the sum of SE for all the transceivers while satisfying the maximum IF. Thus, its optimization problem can be formulated, as follows:  
\[\begin{split}\max_{P_{i}}&\sum_{i\in\mathbb{J}}S \bar{E}_{i}\\ \text{s.t.}&\max_{P_{i}}IF,\\ 0&\leq P_{i}\leq P_{max}.\end{split} \tag{5}\]  
It is noted that the optimization problem that is given in Equation (5) is non-convex, and thus it is hard to solve it efficiently.