Consider the $G(n,p)$ random graph model where $n$ is a ``large'' positive integer and $p\in (0,1)$. We may equip every realized random graph $G$ with its shortest path distance, making it into a (random) metric space $(G,d_G)$. Since $G$ is finite then $(G,d_G)$ is doubling.

Are there known estimates for the expected doubling constant of such a random graph?