## 4 Numerical Evaluations
### Theoretical Discussion  
From the results, we saw that there is a fundamental relationship between the number of nodes, the wireless transmission range and reachability. In this subsection, we will briefly discuss some results on random geometric graphs [13, 14] that provide theoretical insights into this relationship. Random geometric graphs have become a useful theoretical tool for modeling connectivity in static networks [15] or dynamic networks [16], i.e., where nodes move with a random walk, and they are defined as follows.  
Let \(N\) nodes be placed at independent and identically distributed random positions in the unit torus \([0,1)^{2}\). We define the random geometric graph over this set of \(N\) nodes and the set of edges connecting each pair of nodes, \(x_{i}\) and \(x_{j}\), if \(\|x_{i}-x_{j}\|\leq r\), where \(r\) is a positive real-valued number and \(\|\cdot\|\) denotes the Euclidean distance in the torus. This definition resembles our static network configuration in a MANET for nodes having the wireless transmission range, \(r\).  
Figure 10: Reachability improvement _versus_ firework threshold shows that a firework threshold of 0.5 reveals a good tradeoff between reachability improvement and delay/overhead (\(N=4000\)). **(a)** Reachability difference; **(b)** delay difference; **(c)** overhead difference.  
It was shown that for such a random geometric graph, a critical radius,  
\[r_{c}=\sqrt{\frac{\ln(N)}{\pi N}}\]  
exists for which connectivity occurs with high probability as \(N\rightarrow\infty\). By rescaling \(r_{c}\) from the unit torus to our modeled area side \(s=10^{4}\,\mathrm{m}\), we obtain the scaled radius, \(\hat{r}_{c}\), which we can compare with our simulated results (Figure 11). For the considered number of nodes in this paper of \(N=2000\), \(3000\) and \(4000\), we obtain numerical values of \(\hat{r}_{c}=347.81\,\mathrm{m}\), 291.46 m and 256.91 m, respectively. It should be noted that these values can only be roughly compared, since the theoretical values only consider static nodes, while our simulation includes an RWK mobility model. However, they show that for our considered number of nodes, the critical transmission ranges lie around 250-350 m, as we could also see from the simulation results.  
Furthermore, [PERSON] _et al_.[16] investigated the analytical properties of random geometric graphs with mobility. Similar to our simulations, each node performs a random walk with a speed, \(s\). The expected length of the connectivity period is derived in [16] as:  
\[\lambda_{C}\sim\frac{1}{1-\exp\left(-\mu\left(1-\exp\left(-4 srN/\pi\right) \right)\right)}\]  
where \(\mu:=N\exp\left(-\pi r^{2}N\right)\). By rescaling \(\lambda_{C}\) from the unit torus to our real-world values as \(\hat{\lambda}_{C}\), we obtain Figure 12, where \(\hat{\lambda}_{C}\) is plotted over \(r\). We limited the y-axis to 100, which is the length of our simulation time, and can see that the radius for which \(\hat{\lambda}_{C}\) reaches 100 matches quite well to the ranges in which we have a reachability probability of nearly one, as shown in Figure 3.  
Figure 11: Scaled critical radius \(\hat{r}_{c}\) for the investigated number of nodes, \(N\), in this study.