## 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 $\left(1-exp\left(-\frac{\mathrm{\eta}{W}_{x}{W}_{y}}{{\left(d(x,y)\u2215r\right)}^{\mathit{\alpha}}}\right)\right),$ 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.

## Funding Statement

SKI has been supported in part from SERB Matrics grant MTR/2018/000496 and DST-CAS. SKJ has been supported by DST-INSPIRE Fellowship.

## Acknowledgments

We would like to thank an anonymous referee for a careful reading of the paper and suggesting numerous improvements.

## Citation

Srikanth K. Iyer. Sanjoy Kr Jhawar. "Poisson approximation and connectivity in a scale-free random connection model." Electron. J. Probab. 26 1 - 23, 2021. https://doi.org/10.1214/21-EJP651

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