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 $S=\left(-\frac{1}{2},\frac{1}{2}\right]^{d},$ $d \geq 2$ . 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 $x,y\in \mathcal{P}_s$ with probability $\left(1 - \exp\left( - \frac{\eta W_xW_y}{\left(d(x,y)/r\right)^{\alpha}} \right)\right),$ 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 $r_s(\xi)^d= \frac{1}{c_0 s} \left( \log s +(k-1) \log\log s +\xi+\log\left(\frac{\alpha\beta}{k!d} \right)\right),$ $\xi \in \mathbb{R}$, the number of vertices of degree $k$ converges in total variation distance to a Poisson random variable with mean $e^{-\xi}$ as $s \to \infty$, 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.