Normal approximation of subgraph counts in the random-connection model (Q6589590)

The paper treats the asymptotic behavior of random subgraph counts in the random-connection model, which is used to model physical systems. To be more precise, the results of normal approximation are derived for subgraph counts written as multiparameter stochastic integrals in a random-connection model based on a Poisson point process. The authors' approach relies on the study of cumulant growth rates as the intensity of the underlying Poisson point process tends to infinity. By combinatorial arguments the authors express the cumulants of general subgraph counts using sums over connected partition diagrams, after cancellation of terms obtained by Möbius inversion. Using the Statulevičius condition, they deduce convergence rates in the Kolmogorov distance by studying the growth of subgraph count cumulants. Normal approximation rates are obtained under a mild condition on the connection function of the random-connection model, by deriving growth rates of cumulants written as sums over connected partitions. Despite the fact that the related cumulant bounds have been obtained in the Erdős-Rényi model, this is the first time that the normal approximation of subgraph counts with convergence rates is established in the random-connection model. The results are applicable to general subgraphs in the dilute and full random graph regimes, and to tree-like subgraphs in the sparse random graph regime.