What is the expected global clustering coefficient $\mathbb{E}[C_{GC}]$ for the Erdős–Rényi random graph (ER-graph) $\mathcal{G}(n,p)$ (expectation is over the ensemble of all ER-graphs) as $n \rightarrow \infty$ and $p$ fixed?
The global clustering coefficient $C_{GC}$ is defined as
$C_{GC}={\frac {3\times {\mbox{number of triangles}}}{{\mbox{number of connected triplets of vertices}}}}={\frac {{\mbox{number of closed triplets}}}{{\mbox{number of connected triplets of vertices}}}}$.
A connected triplet is defined to be a connected subgraph consisting of three vertices and two edges. A closed triplet is a connected triplet that induces a triangle.
While it is easy to see that the expected mean local clustering coefficient is $p$ (see next section), 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 $n\rightarrow\infty$?
My current theory is to use Chebyshev's inequality on this, but I haven't tried it out yet.
Expected local clustering coefficient = p
In contrast, it is easy to see that the expected local clustering coefficient $\mathbb{E}[C_i]$ for any node $i$ is $p$.
The local clustering coefficient $C_i$ of node $i$ (for an undirected network) is defined as
$C_i = \frac{\text{number of triangles that contain $i$}}{k_i (k_i-1)/2}$,
where $k_i$ is the degree of $i$.
In other words, it is the proportion of links between the vertices within its neighbourhood divided by the number of links that could possibly exist between them.
We have $\mathbb{E}[C_i]=p$ in an ER-graph because 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 $n$.
Best Answer
Unless I'm missing something, this is a standard application of the probabilistic method: just show that the expected number of closed triplets is ${n \choose 3}p^3$ the expected number of connected triplets is ${n \choose 3}(3p(1-p)+p^3)$ and then use a Chebyshev bound to show that as $n \rightarrow \infty$ each converges to its mean so that $C_{GC} \rightarrow 3p^3/(3p^2-p^3) = p/(1-p/3)$.