In network data analysis, summary statistics of a network can provide us with meaningful insight into the structure of the network. The average clustering coefficient is one of the most popular and widely used network statistics. In this paper, we investigate the asymptotic distributions of the average clustering coefficient and its variant of a heterogeneous Erd\"{o}s-R\'{e}nyi random graph. We show that the standardized average clustering coefficient converges in distribution to the standard normal distribution. Interestingly, the variance of the average clustering coefficient exhibits a phase transition phenomenon. The sum of weighted triangles is a variant of the average clustering coefficient. It is recently introduced to detect geometry in a network. We also derive the asymptotic distribution of the sum weighted triangles, which does not exhibit a phase transition phenomenon as the average clustering coefficient. This result signifies the difference between the two summary statistics.

翻译：在网络数据分析中，网络的汇总统计量能提供关于网络结构的有意义洞见。平均聚类系数是最流行且广泛使用的网络统计量之一。本文研究了异质Erdős-Rényi随机图的平均聚类系数及其变体的渐近分布。我们证明标准化后的平均聚类系数依分布收敛于标准正态分布。有趣的是，平均聚类系数的方差表现出相变现象。加权三角形和是平均聚类系数的一种变体，最近被引入用于检测网络中的几何结构。我们还推导了加权三角形和的渐近分布，该分布并未像平均聚类系数那样表现出相变现象。这一结果揭示了这两种汇总统计量之间的差异。