Erdős-Rényi model, network reliability
When $$ p=\frac{\ln n}{n}+\frac{s+o(1)}{n} $$ for some fixed $s\in \mathbb{R}$, $$ \mathsf{P}(G(n,p)\text{ is connected})\to e^{-\exp(-s)}. $$ (see, e.g., Theorem 7.3 here).
Erdős-Rényi model, network reliability
When $$ p=\frac{\ln n}{n}+\frac{s+o(1)}{n} $$ for some fixed $s\in \mathbb{R}$, $$ \mathsf{P}(G(n,p)\text{ is connected})\to e^{-\exp(-s)}. $$ (see, e.g., Theorem 7.3 here).