Fréchet distribution in geometric graphs for drone networks

In this paper, we focus on the link density in random geometric graphs (RGGs) with a distance-based connection function. After deriving the link density in $D$ dimensions, we focus on the two-dimensional (2D) and three-dimensional (3D) space and show that the link density is accurately approximated by the Fréchet distribution, for any rectangular space. We derive expressions, in terms of the link density, for the minimum number of nodes needed in the 2D and 3D spaces to ensure network connectivity. These results provide first-order estimates for, e.g., a swarm of drones to provide coverage in a disaster or crowded area.

Figure 1

Example realization of a RGG with $f\left(r\right)=e^{-0.07r^2}$ and $N=25$ in a 3D rectangular prism, where the link color refers to the connection probability between two nodes. For visualization purposes, only links with $f\left(r\right)>{10}^{-3}$ are shown.

Figure 2

Distance function $f\left(r\right)$ with respect to the normalized distance $\frac{r}{r_0}$ when $\beta =-127\text{dB}$, $\lambda =0.08\text{m,}$ and $r_c=1\text{m}$.

Figure 3

Error from approximate solution of link density in a square and a cube for $r_c=1\text{m}$, $K=4.65\times {10}^{-5}$, and $\beta =-103.3\text{dB}$.

Figure 4

Impact of the path loss exponent $\gamma$ on the link density $p$ in the 2D and 3D spaces for $\omega =0.75$, $\delta =0.5$, $r_c=1\text{m}$, $K=4.65\times {10}^{-5}$, and $\beta =-103.3\text{dB}$.

Figure 5

Link density comparison between the simulation of Eq. (11) (blue), the fitting of a Fréchet (green) and the polynomial solution from Eqs. (18) and (19) for a square of size $Z$ and $\gamma \to \infty$.

Figure 6

Difference $p_{\text{2D}}-p_{\text{3D}}$ of link density between the 2D and 3D spaces, for different values of $\gamma$ when $\omega =0.75$, $\delta =0.5$, $r_c=1\text{m}$, $K=4.65\times {10}^{-5}$, and $\beta =-103.3\text{dB}$.

Figure 7

Difference $p_{\text{2D}}-p_{\text{3D}}$ of link density between the 2D and 3D spaces for different combinations of $\omega$ and $\delta$ when $\gamma =2$, $r_c=1\text{m}$, $K=4.65\times {10}^{-5}$, and $\beta =-103.3\text{dB}$.

Figure 8

Minimum number of nodes $N_{\text{min}}$ needed for connectivity in respect to the link density $p$, for the scenarios shown in Table 2, as well as the fitting curves given by Eq. (22).

Figure 9

Panels (a), (b), and (c) illustrate the fitting parameters $a_{\text{2D}},b_{\text{2D}},\text{and}{c}_{\text{2D}}$, respectively, for different values of $\omega$ and $\gamma$, while panel (d) illustrates the RMSE of each fit. Panel (c) is in log-log scale.

Figure 10

Fitting parameters $(a_{\text{3D}},b_{\text{3D}},c_{\text{3D}})$ for different values of $\omega$ and $\delta$ and for $\gamma =5$.