Connecting the Random Connection Model

Consider the random graph $G({\mathcal{P}}_n,r)$ whose vertex set ${\mathcal P}{n}$isaPoissonpointprocessofintensity$n$on$(- \frac{1}{2}, \frac{1}{2}]^d$,$d \geq 2$\mathrm{.}Anytwovertices$X_i,X_j \in {\mathcal P}{n}$areconnectedbyanedgewithprobability$g\left( \frac{d(X_i,X_j)}{r} \right)$, independently of all other edges, and independent of the other points of ${\mathcal{P}}_n$. $d$ is the toroidal metric, $r>0$ and $g:[0,\infty) \to [0,1]$isnon-increasingand$\alpha = \int_{$\mathbb{R}$^d} g(|x|) dx < \infty$\mathrm{.}Undersuitableconditionson$g$, almost surely, the critical parameter $d_n$ for which $G({\mathcal{P}}_n,\cdot )$ does not have any isolated nodes satisfies $\lim_{n \to \infty} \frac{\alpha n d_n^d}{\log n} = 1$\mathrm{.}Let$\beta = \inf{x > 0: x g\left( \frac{\alpha}{x \theta} \right) > 1 }$,and$\theta$bethevolumeoftheunitballin$$\mathbb{R}$^d$. Then for all $\gamma >\beta$, $G\left({\mathcal P}_{n}, \left( \frac{\gamma \log n}{\alpha n} \right)^{\frac{1}{d}}\right)$ is connected with probability approaching one as $n\to \mathrm{\infty }$. The bound can be seen to be tight for the usual random geometric graph obtained by setting $g=1_{[0,1]}$. We also prove some useful results on the asymptotic behaviour of the length of the edges and the degree distribution in the {\it connectivity regime}.