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We present a process-level Poisson-approximation result for the degree-kvertices in a high-density weighted random connection model withpreferential-attachment kernel in the unit volume. Our main focus lies on theimpact of the left tails of the weight distribution for which we establishgeneral criteria based on their small-weight quantiles. To illustrate that ourconditions are broadly applicable, we verify them for weight distributions withpolynomial and stretched exponential left tails. The proofs rest on truncationarguments and a recently established quantitative Poisson approximation resultfor functionals of Poisson point processes.

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