On a large finite connected graph let edges $e$ become “open” at independent random Exponential times of arbitrary rates $w_e$. Under minimal assumptions, the time at which a giant component starts to emerge is weakly concentrated around its mean.
"The incipient giant component in bond percolation on general finite weighted graphs." Electron. Commun. Probab. 21 1 - 9, 2016. https://doi.org/10.1214/16-ECP21