Suppose we have an instance of Erdős-Renyi $G(n,p)$ graph with $p = d/n$. Thus the expected node degree is $d$ which we will fix, while letting $n \to \infty$. Then, there will be more than one connected component (CC) for large $n$. 


Suppose that we pick the largest CC and relabel the nodes randomly from $1,\dots,s$ where $s$ is the size of the component. Is there a way to describe the distribution of these connected graphs as $n \to \infty$? The edges are no longer independent, but can we approximate the whole distribution with another Erdős-Renyi graph with modified parameters (say increased $p$)?



Erdős-Renyi graph restricted to largest connected component
If $d>1$, the paper [Anatomy of the giant component: the strictly supercritical regime](http://research.microsoft.com/en-us/um/people/eyal/papers/giant_super.pdf) by Ding, Lubetzky and Peres might be what you are looking for. The paper also has references for the cases $d<1$ and $d=1$.


