Limit behaviors of random connected graphs driven by a Poisson process

We consider a class of random connected graphs with random vertices and random edges with the random distribution of vertices given by a Poisson point process with the intensity $n$ localized at the vertices and the random distribution of the edges given by a connection function. Using the Avram–Bertsimas method constructed in 1992 for the central limit theorem on Euclidean functionals, we find the convergence rate of the central limit theorem process, the moderate deviation, and an upper bound for large deviations depending on the total length of all edges of the random connected graph.