Asymptotic distribution for the average closure coefficient of the random geometric graph

Edge clustering is a prominent feature observed in many real-world networks, often quantified through the average clustering coefficient. Recently, the average closure coefficient has been proposed as an alternative metric to assess local clustering, offering additional information that is not captured by the clustering coefficient. In this study, we explore the asymptotic behavior of the average closure coefficient in Geometric Random Graphs. We show that the average closure coefficient converges in distribution to a sum of functions of independent standard normal random variables. This result contrasts with the Erdős–Rényi random graph case, where the average closure coefficient converges to the standard normal distribution. This highlights the different asymptotic behaviors of the average closure coefficient in Geometric Random Graphs compared to the Erdős–Rényi graphs, offering unique insights into the structure of these networks.