Let $G'$ be a graph obtained from $G$ after contracting each edge with probability $p$. Let $n = |V(G)|, e = |E(G)|$.

I would like to compute (or at least obtain a lower bound) for $E[|V(G')|]$ in terms of some known  graph invariants (number of edges, degree sequence, connectivity,..)

I am sure I am not the first one that studied such a probabilistic space and since I couldn't  find any estimates for $E[|V(G')|]$ in my textbook I am asking: is there any simple identity/estimate for  $E[|V(G')|]$ ? Is there any reference to a paper studying this quantity?

**Edit**: I have removed the completely wrong attempt to estimate $E[|V(G')|]$.