The online power-of-choices geometric graph giant-component conjecture

Consider the online power-of-choices geometric graph process on $n$ vertices, with radius

$$r = \sqrt[3]{\frac{c}{n\log\log n}}.$$

Here $c>0$, $f$ is a function from $(0,\infty)$ to $(0,1)$, and w.h.p. means with high probability. Online giant-component conjecture. There is a function $f\colon (0,\infty)\to(0,1)$ such that, assuming optimal play, the largest component has size

$$(1+o(1))\,f(c)\,n$$

with high probability. This conjecture proposes a limiting giant-component fraction for the online process at the critical scale where the radius is of order $(n\log\log n)^{-1/3}$; the paper establishes delay of the giant to this scale but does not determine the limiting fraction.