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 $P(W>w)=w^{-\beta }{1}_{[1,\infty )}(w)$, $\beta >0$. Given the vertex set and the weights an edge exists between $x,y\in {\mathcal{P}}_s$ with probability $\left(1-\exp \left(-\frac{\eta W_x{W}_y}{{\left(d(x,y)/r\right)}^{\alpha }}\right)\right),$ 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.

21 pages, calculations are simplified significantly and results are proved under much weaker conditions

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