# Robust Beamforming Based on Graph Attention Networks for IRS-Assisted Satellite IoT Communications
## 4 Simulation and Numerical Results
### Numerical Results  
In this subsection, we present the numerical results of the proposed approach. We assume that the users' locations are fixed within a time slot, the time slot is small enough and the low-complexity RBF-GAT can implement active and passive beamforming within the time slot.  
To verify the convergence rate of the proposed RBF-GAT scheme, we plot the loss value during training versus the number of epochs with three different training parameters. Figure 3 shows that the proposed scheme converges to a locally optimal solution in less than 200 training epochs. In addition, the smaller the number of IRS reflection elements or users, the faster the convergence speed of the algorithm. This is because the number of users or IRS elements is positively correlated with the number of weight parameters to be trained.  
Figure 4 shows the sum-rate versus the number of users under different schemes. The sum-rates under all considered schemes increase with an increasing number of users \(K\), and the larger the value of \(K\) is, the slower the growth trend. This phenomenon can be explained by the concavity of the log function from the sum rate. It is seen that those methods aided with IRS observably exceed the one without IRS, and the performance gain by deploying IRS is inappreciable if the phase shift matrix is initialized by random value. Moreover, both RBF-GAT and GNN can achieve performance close to the upper boundary, but the proposed RBF-GAT consistently outperforms GNN, and the gap increases with \(K\). The reason for the increase is that the LEO satellite is insensitive to user's location information, in contrast to GNN directly merging the position coordinates of the user, RBF-GAT can effectively capture the dynamic network topology by GAT layers. Thus, the proposed RBF-GAT is more suitable for IRS-aided LEO satellite dynamic scenario communication.  
Figure 3: The convergence rate with different training parameters.  
Figure 5 illustrates the sum-rate of different schemes with respect to the transmit power \(P_{L}\) when \(N=64\). It is observed that the sum-rate increases for all considered schemes as the transmission power increases, and the random phase method still has only a weak gain. As we expected, the performance of RBF-GAT is always closest to the upper boundary under the condition of equal pilot length. In addition, as the pilot length increases, the rate sum also increases but never exceeds the upper boundary. This is because the longer the pilot signal received is, the richer the CSI contained. Thus, more features can be learned by the neural network. On the other hand, increasing the pilot length will also lead to a larger delay in data transmission. In practical applications, we should make trade-offs according to different requirements.  
Figure 6 shows the sum-rate versus the number of elements of the IRS in four different schemes. In Figure 6, we set the number of feeds at LEO satellite as \(M=8\), pilot length as \(L=20\), and the transmit power of LEO satellite as \(P_{L}=30\) dBW. We can find that the sum-rate of the without IRS method remains constant and the random phase method increase slightly as the numbers of IRS element grows. From Figures 5 and 6, we can calculate that the proposed RBF-GAT approach can achieve more than 95% of the performance provided by the upper bound. In addition, both the number of IRS elements and the increase in transmit power can improve the sum-rate. However, compared to increasing the transmit  
Figure 4: The sum-rate versus the number of users \(K\) with \(M=8\), \(N=64\), \(L=20\) and \(P_{L}=30\) dBW.  
Figure 5: The sum-rate versus the LEO satellite transmit power \(P_{L}\).  
power, increasing the IRS elements to improve the sum-rate performance is a more energy-efficient scheme due to the IRS elements being passive. The above numerical simulations further validate the robustness and effectiveness of our proposed RBF-GAT schemes.