Poisson approximation and connectivity in a scale-free random connection model

Abstract

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 $S={\left(-\frac{1}{2},\frac{1}{2}\right]}^d$, $d\ge 2$ . Each vertex is endowed with an independent random weight distributed as W, where $P(W>w)=w^{-\mathit{\beta }}{1}_{[1,\mathrm{\infty })}(w)$, $\mathit{\beta }>0$. Given the vertex set and the weights an edge exists between $x,y\in {\mathcal{P}}_s$ with probability $\left(1-exp\left(-\frac{\mathrm{\eta }{W}_x{W}_y}{{\left(d(x,y)∕r\right)}^{\mathit{\alpha }}}\right)\right),$ independent of everything else, where $\mathrm{\eta },\mathit{\alpha }>0$, $d(\cdot ,\cdot )$ is the toroidal metric on S and $r>0$ is a scaling parameter. We derive conditions on $\mathit{\alpha },\mathit{\beta }$ such that under the scaling $r_s{(\mathrm{\xi })}^d=\frac{1}{c_{0}s}\left(logs+(k-1)loglogs+\mathrm{\xi }+log\left(\frac{\mathit{\alpha }\mathit{\beta }}{k!d}\right)\right),$ $\mathrm{\xi }\in \mathbb{R}$, the number of vertices of degree k converges in total variation distance to a Poisson random variable with mean $e^{-\mathrm{\xi }}$ as $s\to \mathrm{\infty }$, where $c_0$ is an explicitly specified constant that depends on $\mathit{\alpha },\mathit{\beta },d$ and η 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.