Poisson approximation of fixed-degree nodes in weighted random connection models

Abstract

We present a process-level Poisson-approximation result for the degree-$k$ vertices in a high-density weighted random connection model with preferential-attachment kernel in a finite-volume Borel set. Our main focus lies on the impact of the left tails of the weight distribution for which we establish general criteria based on their small-weight quantiles. To illustrate that our conditions are broadly applicable, we verify them for weight distributions with polynomial and stretched exponential left tails. The proofs rest on truncation arguments and a recently established quantitative Poisson approximation result for functionals of Poisson point processes.