  This paper derives normal approximation results for subgraph counts written
as multiparameter stochastic integrals in a random-connection model based on a
Poisson point process. By combinatorial arguments we express the cumulants of
general subgraph counts using sums over connected partition diagrams, after
cancellation of terms obtained by M\"obius inversion. Using the
Statulevi\v{c}ius condition, we deduce convergence rates in the Kolmogorov
distance by studying the growth of subgraph count cumulants as the intensity of
the underlying Poisson point process tends to infinity. Our analysis covers
general subgraphs in the dilute and full random graph regimes, and tree-like
subgraphs in the sparse random graph regime.
