## 4 Discussion  
In this section, we discuss the variation of the graphs for the beamformed output of the proposed AUAS swarm model with the ground-truth activation function as shown in Section 3. We sum up the discussion to obtain a low-cost AUAS swarm routing algorithm exploring the structures of the matrices and ML algorithms. For the purposes of understanding the variation of beamformed output of the proposed AUAS swarm model with the ground-truth activation function in Figure 3, we utilize min-max normalization following the formula \(y^{\prime}=\frac{y-y_{min}}{y_{max}-y_{min}}\), where \(y\) is a scalar value in \(\mathbf{y^{(u)}}\) or \(\gamma_{u,j,k}(t)\), \(y_{min}\) is the minimum scalar element of the tensor that contains each drone \(u\)'s beamformed signal \(\mathbf{y^{(u)}}\) for each time \(t\) or ground-truth signal \(\gamma_{u,j,k}(t)\) for each time, \(y_{max}\) is the maximum scalar value from this tensor, and \(y^{\prime}\) is the input value normalized to fall between the range \([0,1]\)[49]. Figure 7 shows the consistency between the beamformed output of the proposed model and the ground activation function. The magnitude of error depicted for higher values of \(k\) (i.e., number of elements in the antenna array) will not occur in practice because \(dk\) represents the positive horizontal offset of any antenna array element \(k\) from the center of the drone's spatial coordinate. We measure the difference between the imaginary part of the beamformed output and the activation function and the magnitude of it as shown in Figure 7c,d, respectively. The robust accuracy of the proposed algorithm shows small \(k\) values, which we mostly see in practice. Thus, the proposed AUAS swarm routing algorithm is useful as a heuristic, e.g., as a metric to sort a collection of neighboring drones according to their connection strength with any particular drone. The difference between a collection of ground-truth signals and the output beamformed signals of the AUAS swarm model would increase as the number of elements in the antenna arrays, i.e., \(N\) increases and/or the offset size \(dk\) increases but remains consistent even for large spatial drone arrangements. Thus, we will seek artificial-neural-network-based ML algorithms for the completion of the frequency Vandermonde matrices defined via spatial and temporal data to obtain a closed-form algorithm (while reducing the error as best as possible) and hence to navigate a massive collection of AUAS in real-time, in future. Figure 6: This shows the BER of the proposed AUAS swarm routing algorithm over a collection of signal-to-noise ratios (expressed as \(Eb/N0\), where \(Eb\) is the energy in one bit (Joules) and \(N0\) is the noise power spectral density (Watts per Hertz)). The x-axis and y-axis represent a collection of signal-to-noise ratios measured in decibels and BER on a logarithmic scale, respectively. Extending the future work, we have observed that in the AUAS swarm model (1), the most expensive calculation occurs in the brute-force computation of the matrix-vector products. Although \(\mathbf{S}_{i}^{(u)}\) is a diagonal matrix, \(\mathbf{V}_{ki}^{(u)}\) is a dense matrix. Fortunately, \(\mathbf{V}_{ki}^{(u)}\) is a Vandermonde structured matrix having nodes \(\{e^{-|\mu_{i}\sigma_{i}\tau_{i}\}\}_{i=1}^{M}\). Thus, in future, we will use the \(M\)-set of \(N\)-beamformed signal of \(M\) drones at time \(t\), i.e., \(\left[\mathbf{y}^{(1)}\,\mathbf{y}^{(2)}\cdots\mathbf{y}^{(M)}\right]\) as the columns of a throughput matrix and optimize the routing of AUAS swarm using low-complexity multi-beam beamforming algorithms described in [50; 51; 52]. For fast missing data inference, we will propose to explore the inherent data relation between vectors in consecutive time frames to quickly fill in the missing data in the new time slot using a vector autoregression method followed by a product of highly sparse and low-rank matrices to estimate the transition matrices \(\mathbf{\hat{V}}_{ki}^{(u)}\) characterizing the interaction among different components.