Poisson Approximation and Connectivity in a Scale-free Random Connection Model

We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process ${\mathcal{P}}_s$ of intensity $s>0$ on the unit cube $S={(-\frac{1}{2},\frac{1}{2}]}^d,$ $d\ge 2$ . Each vertex is endowed with an independent random weight distributed as $W$, where $P(W>w)=w^{-\beta }{1}_{[1,\mathrm{\infty })}(w)$, $\beta >0$. Given the vertex set and the weights an edge exists between $x,y\in {\mathcal{P}}_s$ with probability $\left(1

• \exp\left( - \frac{\eta W_xW_y}{\left(d(x,y)/r\right)^{\alpha}} \right)\right),$independentofeverythingelse,where$\eta, \alpha > 0$,$d(\cdot, \cdot)$isthetoroidalmetricon$S$and$r > 0$ is a scaling parameter. We derive conditions on $\alpha ,\beta$ such that under the scaling $r_s(\xi)^d= \frac{1}{c_0 s} \left( \log s +(k-1) \log\log s +\xi+\log\left(\frac{\alpha\beta}{k!d} \right)\right),$ $\xi \in \mathbb{R}$, the number of vertices of degree $k$ converges in total variation distance to a Poisson random variable with mean $e^{-\xi }$ as $s\to \mathrm{\infty }$, where $c_0$ is an explicitly specified constant that depends on $\alpha ,\beta ,d$ and $\eta$ but not on $k$. In particular, for $k=0$ we obtain the regime in which the number of isolated nodes stabilizes, a precursor to establishing a threshold for connectivity. We also derive a sufficient condition for the graph to be connected with high probability for large $s$. The Poisson approximation result is derived using the Stein’s method.