The random connection model and functions of edge-marked Poisson processes: second order properties and normal approximation

TL;DR

This paper investigates the second order properties of the random connection model, a Poisson-based random graph, providing central limit theorems for component counts and introducing general results for functions of edge-marked Poisson processes.

Contribution

It introduces new second order analysis and CLTs for the random connection model, along with general results for functions of edge-marked Poisson processes.

Findings

Proves multivariate and univariate CLTs for component counts.

Establishes a CLT for the total number of finite components.

Develops general results for functions of edge-marked Poisson processes.

Abstract

The random connection model is a random graph whose vertices are given by the points of a Poisson process and whose edges are obtained by randomly connecting pairs of Poisson points in a position dependent but independent way. We study first and second order properties of the numbers of components isomorphic to given finite connected graphs. For increasing observation windows in an Euclidean setting we prove qualitative multivariate and quantitative univariate central limit theorems for these component counts as well as a qualitative central limit theorem for the total number of finite components. To this end we first derive general results for functions of edge marked Poisson processes, which we believe to be of independent interest.