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

Abstract

We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process Ps of intensity s>0 on the unit cube S=12,12d, d2 . Each vertex is endowed with an independent random weight distributed as W, where P(W>w)=wβ1[1,)(w), β>0. Given the vertex set and the weights an edge exists between x,yPs with probability 1expηWxWyd(x,y)rα, independent of everything else, where η,α>0, d(,) is the toroidal metric on S and r>0 is a scaling parameter. We derive conditions on α,β such that under the scaling rs(ξ)d=1c0slogs+(k1)loglogs+ξ+logαβk!d, ξR, the number of vertices of degree k converges in total variation distance to a Poisson random variable with mean eξ as s, where c0 is an explicitly specified constant that depends on α,β,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

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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

Information

Received: 30 March 2020; Accepted: 22 May 2021; Published: 2021
First available in Project Euclid: 22 June 2021

Digital Object Identifier: 10.1214/21-EJP651

Subjects:
Primary: 60D05, 60G70
Secondary: 05C80, 05C82, 60G55

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