Many real-world networks exhibit the phenomenon of edge clustering, which is typically measured by the average clustering coefficient. Recently, an alternative measure, the average closure coefficient, is proposed to quantify local clustering. It is shown that the average closure coefficient possesses a number of useful properties and can capture complementary information missed by the classical average clustering coefficient. In this paper, we study the asymptotic distribution of the average closure coefficient of a heterogeneous Erd\"{o}s-R\'{e}nyi random graph. We prove that the standardized average closure coefficient converges in distribution to the standard normal distribution. In the Erd\"{o}s-R\'{e}nyi random graph, the variance of the average closure coefficient exhibits the same phase transition phenomenon as the average clustering coefficient.

翻译：许多现实网络展现出边聚类现象，通常由平均聚类系数进行度量。近期，一种替代性度量——平均闭包系数被提出用于量化局部聚类。研究表明，平均闭包系数具有若干有用性质，并能捕捉经典平均聚类系数遗漏的互补信息。本文研究异质 Erdős-Rényi 随机图平均闭包系数的渐近分布。我们证明标准化后的平均闭包系数依分布收敛于标准正态分布。在 Erdős-Rényi 随机图中，平均闭包系数的方差呈现出与平均聚类系数相同的相变现象。