  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.
