Year: 2021
Type: article
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= −1 2,1 2d, d≥2 . Each vertex is endowed with an independent random weight distributed as W, where P(W>w)=w−β1[1,∞)(w), β>0. Given the vertex set and the weights an edge exists between x,y∈Ps with probability 1−exp−ηWxWyd(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=1 c0slogs+(k−1)loglogs+ξ+logαβ k!d, ξ∈R, the number of vertices of degree k converges in total variation distance to a Poisson random variable with mean e−ξ 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.
DOI: 10.1214/21-ejp651
Authors: Srikanth K. Iyer, Sanjoy Kr Jhawar
Institution: Indian Institute of Science Bangalore
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $675
OpenAlex ID: W3008565343