What is the expected global clustering coefficient $ \mathbb{E}[C_{GC}]$ for the Erdős–Rényi random graph $ \mathcal{G}(n,p)$ (expectation is over the ensemble of all Erdős–Rényi random graph) as $ n \rightarrow \infty$ and $ p$ fixed?

The global clustering coefficient $ C_{GC}$ is defined as

$ C_{GC}={\frac {3\times {\mbox{number of triangles}}}{{\mbox{number of connected triplets of vertices}}}}={\frac {{\mbox{number of closed triplets}}}{{\mbox{number of connected triplets of vertices}}}}$ .

It is easy to see that the expected local clustering coefficient $ \mathbb{E}[C_i]$ for any node $ i$ is $ p$ , as the probability for an edge between any neighbours of the node is $ p$ , independent for any other edge. (For an alternative answer involving more algebra, see here).

Hence the expected mean local clustering coefficient $ \mathbb{E}[\sum_i C_i]$ is $ p$ for any $ n$ .

However, the expected global clustering coefficient is not identically $ p$ for any $ n$ .

For example, for $ n=3$ , $ C_{GC} = 1$ only when all edges are present (with probability $ p^3$ ) and is otherwise zero (with probability $ 1-p^3$ ). Hence the $ \mathbb{E}[E_{GC}] = p^3$ when $ n=3$ .

Computationally, I have found that $ \mathbb{E}[C_{GC}]\approx p$ for large $ n$ .

Is there a way to prove that $ \mathbb{E}[C_{GC}]= p$ as $ n\rightarrow\infty$ ?