Asymptotic distribution of the global clustering coefficient in a random annulus graph
Mingao Yuan
TL;DR
The paper tackles the problem of the asymptotic distribution of the global clustering coefficient in the random annulus graph (RAG), a variant of the Erdős-Rényi model that captures intermediate proximity in networks. It develops a central limit theorem for the standardized global clustering coefficient using degenerate U-statistics with a sample-size-dependent kernel, yielding an explicit asymptotic mean $\frac{3}{4}\frac{(r_1-2r_2)^2}{(r_1-r_2)^2}$ and a convergence to the standard normal law. A key contribution is the exact CLT scaling $\frac{2\sqrt{2}(r_1-r_2)^2 n}{3\sigma_{n2}}$ with $\sigma_{n2}^2=\Theta(r_1^3)$, together with the result $\mathcal{C}_n=\frac{3}{4}\frac{(r_1-2r_2)^2}{(r_1-r_2)^2}+O_P(1/n)$, which shows the limit can vary between 0 and 3/4 depending on $\lambda=r_1/r_2$. This differentiates RAG from the random geometric graph and suggests RAG as a flexible model for networks with tunable clustering properties; the methodology extends degenerate U-statistics to size-dependent kernels for network statistics.
Abstract
The global clustering coefficient is an effective measure for analyzing and comparing the structures of complex networks. The random annulus graph is a modified version of the well-known Erdős-Rényi random graph. It has been recently proposed in modeling network communities. This paper investigates the asymptotic distribution of the global clustering coefficient in a random annulus graph. It is demonstrated that the standardized global clustering coefficient converges in law to the standard normal distribution. The result is established using the asymptotic theory of degenerate U-statistics with a sample-size dependent kernel. As far as we know, this method is different from established approaches for deriving asymptotic distributions of network statistics. Moreover, we get the explicit expression of the limit of the global clustering coefficient.