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 $S=left(-frac{1}{2},frac{1}{2}right]^{d},$ $d geq 2$ . Each vertex is endowed with an independent random weight distributed as $W$, where $P(W>w)=w^{-beta}1_{[1,infty)}(w)$, $beta>0$. Given the vertex set and the weights an edge exists between $x,yin mathcal{P}_s$ with probability $left(1 - expleft( - frac{eta W_xW_y}{left(d(x,y)/rright)^{alpha}} right)right),$ independent of everything else, where $eta, alpha > 0$, $d(cdot, cdot)$ is the toroidal metric on $S$ and $r > 0$ is a scaling parameter. We derive conditions on $alpha, beta$ such that under the scaling $r_s(xi)^d= frac{1}{c_0 s} left( log s +(k-1) loglog s +xi+logleft(frac{alphabeta}{k!d} right)right),$ $xi in mathbb{R}$, the number of vertices of degree $k$ converges in total variation distance to a Poisson random variable with mean $e^{-xi}$ as $s to infty$, 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 Steins method.