As this circle gets expanded further out, the number of nodes in the given area increases proportional to  as the probability of having a random link with any node remains proportional , meaning the probability of the original node having a weak tie with any node a given distance away is effectively independent of distance.
As this circle expands further outward, the number of nodes within a given area increases in proportion to  as the likelihood of forming a random connection with any node remains constant. This implies that the probability of the original node having a weak tie with any node at a certain distance is unaffected by distance.