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Poisson approximation and connectivity in a scale-free random connection model

Iyer, SK and Jhawar, SK (2021) Poisson approximation and connectivity in a scale-free random connection model. In: Electronic Journal of Probability, 26 . pp. 1-23.

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Official URL: https://doi.org/10.1214/21-EJP651


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 (Formula Presented). Each vertex is endowed with an independent random weight distributed as W, where (Formula Presented). Given the vertex set and the weights an edge exists between x; y 2 Ps with probability (Formula Presented); 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 (Formula Presented), the number of vertices of degree k converges in total variation distance to a Poisson random variable with mean e-ξ as s � 1, 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. © 2021, Institute of Mathematical Statistics. All rights reserved.

Item Type: Journal Article
Publication: Electronic Journal of Probability
Publisher: Institute of Mathematical Statistics
Additional Information: The copyright for this article belongs to Authors
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 01 Dec 2021 14:41
Last Modified: 01 Dec 2021 14:41
URI: http://eprints.iisc.ac.in/id/eprint/70034

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