We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process of intensity on the unit cube , . Each vertex is endowed with an independent random weight distributed as W, where , . Given the vertex set and the weights an edge exists between with probability independent of everything else, where , is the toroidal metric on S and is a scaling parameter. We derive conditions on such that under the scaling , the number of vertices of degree k converges in total variation distance to a Poisson random variable with mean as , where is an explicitly specified constant that depends on and η but not on k. In particular, for 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.
SKI has been supported in part from SERB Matrics grant MTR/2018/000496 and DST-CAS. SKJ has been supported by DST-INSPIRE Fellowship.
We would like to thank an anonymous referee for a careful reading of the paper and suggesting numerous improvements.
"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