Ah but that's a bit different. The giant component doesn't connect all the neurons, only a fraction. The wiki page doesn't say this but if you have c * n / 2 edges then the fraction of neurons in the giant component is 1 + W(-c * exp(-c))/c where W is the Lambert W function [0], also called the product logarithm.
As c tends to infinity this fraction tends to 1 but it's never 1.
[0] https://en.wikipedia.org/wiki/Lambert_W_function