I have a question regarding the existence of a closed-form solution of the connectivity in terms of the radius of vertices (disks) in a two-dimensional ($d=2$) random geometric graph (RGG) with open boundary conditions. I know, from Dall & Christensen, 2002 that for continuous boundary conditions (i.e. periodic boundary conditions) one can express the connectivity $\kappa$ in terms of the radius of the disks:
$$ \kappa = NV_{\text{ex}} = 4\pi N r^2 $$ when $V_\text{ex} = \pi\cdot(2r)^2$ the excluded volume of a disk with radius $R=2r$. I have performed simulations that verify this.
However, when the boundaries are open instead (i.e. edges do not "wrap around") the connectivity is lower; $\kappa_{\text{open}} < \kappa$. I have been searching for a closed-form solution for the connectivity of RGGs with open boundaries, but I was not able to find it. However, it seems that the critical connectivity, i.e. the connectivity after which a giant component with $\mathcal{O}(N)$ vertices appears, is independent of the boundary conditions as $N\rightarrow \infty$. Yet, one often wishes to plot quantities of the RGG versus its connectivity $\kappa$ (see the figures in the mentioned paper), however in order to do this one needs to be able to compute $\kappa$ from the given radius $r$, which I am clearly unable to do. I can do this numerically, but this do not enable me to generate graphs with a specific connectivity.
So, is there a closed form solution for the connectivity in terms of the disks' radius $r$ in two-dimensional systems with open boundary conditions?
