## 3 Results
### Beamfomed Output of the AUAS Swarm Routing Model  
In order to understand the error threshold of the proposed AUAS swarm routing model, we compare the \(N\)-beamformed signals in antenna arrays of each drone at time \(t\) with a ground-truth activation function \(\gamma_{u,j,k}(t)=\sum\limits_{i=1}^{M}\sin(2\pi(\beta_{u,j,k}+t))\), where \(\beta_{u,i,k}\) is the distance from drone \(u\)'s \(k^{th}\) antenna element to each antenna element of drone \(i\) at time \(t\). As we have taken the activation function in terms of _sine_, and beamformed outputs are complex-valued vectors, for the generation of Figure 2, we compare the activation function with the magnitude of the beamformed output vector times the _sine_ value of the phase of the beamformed output of all drones in spatial arrangements of drones at a given time experienced by each element of the antenna array, i.e.,  
Figure 1: This flowchart depicts the execution of the proposed AUAS swarm routing algorithm followed by the model (1). \[\sum_{i=1}^{M}|\mathbf{y}^{(u)}|\cdot\sin\Bigl{(}\arctan\Bigl{(}Imag(\mathbf{V}_{ ki}^{(u)}\cdot\mathbf{S}_{i}^{(u)}\cdot\mathbf{x}^{(u)}),Real(\mathbf{V}_{ ki}^{(u)}\cdot\mathbf{S}_{i}^{(u)}\cdot\mathbf{x}^{(u)})\Bigr{)}\Bigr{)}\Bigr{)}.\]  
For the generation of Figure 2, we have assumed that each drone is spatially arranged equidistantly along the circumference of a circle whose radius is one spatial unit such that the angle \(\alpha\) from the center of the circular formation to drone \(u\) is \(\alpha_{i}^{(u)}=\frac{2\pi}{M}i\), where the center of the formation is the origin so that each drone \(u\)'s spatial coordinates are \(\left(x^{(u)},y^{(u)}\right)=\left(\cos\alpha^{(u)},\sin\alpha^{(u)}\right)\). Moreover, we assume that the amplitude of the received signal of drone \(i\) from drone \(u\) is \(a_{i}^{(u)}=1\) and each drone's linear \(N\)-element antenna array is oriented horizontally, such that an antenna \(k\)'s spatial coordinates of drone \(u\) are \(\left(x^{(u)}+d(k-1),y^{(u)}\right)\), where \(d=0.01\) is the inter-element distance of the antenna arrays. Thus, Figure 2 depicts the \(N\)-beamformed signals (color codes to show the strength of the signals) in antenna arrays of \(M\) number of drones (without having base station communication) at time steps \(t=0,0.1,0.2,...0.9\), i.e., \(\mathbf{y}^{(u)}(\omega_{i},a_{i},\theta_{i},t)\) when \(u\) and \(i\) runs from \(1,2,\cdots,M\) and \(M=N=8\). Next in Figure 3; (a)-(l), we show the 2D projections of 3D views of the \(N\)-beamformed signals in antenna arrays of \(M\) number of drones at time \(t\) and the ground-truth signals of the activation function at a time \(t\) extending the illustration in Figure 2 with different \(M\) and \(N\) values. As shown in Figure 3; (a)-(h), the beamformed output of the proposed AUAS swarm algorithm shows similar results as the ground-truth signals of the activation function except for a few beamformed signals, and the effect becomes minor when the number of drones increases. When we fix the number of drones and increase the number of elements in the antenna arrays at \(t=0\), say \(M=128\) or \(256\) as in Figure 3; (g)-(l), the beamformed output of the proposed AUAS swarm algorithm does not show favorable results with the ground-truth signals of the activation function as in the \(N=8\) case, but not significantly. Thus, we will explore the deviations of the beamformed output of the proposed model and activation function as in Section 4. Figure 2: (a) depicts the beamformed signals of the AUAS swarm routing algorithm \(\mathbf{y}^{(u)}(\omega_{i},a_{i},\theta_{i},t)\) over 10 time steps with 0.1 per time step, where each drone \(u\) in the circular spatial arrangement is represented along the Drone Index axis, and its \(N\)-element antenna array is represented along the Antenna Element Index axis.