I would grateful to learn of work mixing random geometric graphs with random graphs under the Erdős-Renyi model, and in particular concerning spanners.

Select $n$ points uniformly at random from the unit square, and then form a graph $G=G(n,p)$ by connecting points by adding edges with probablity $p$. If $p_1$ is the threshhold for the formation of the giant component $C$, is $C$ almost surely a geometric spanner for the points it connects for $p=p_1 + \epsilon$? (My guess is: No.) One could ask a similar question about the threshold $p_2$ for complete connection of the point set. (Here perhaps the answer is: Yes?)

A geometric spanner has the property that, between any pair of points, there is path whose length is not much longer than (no more than some constant times) the Euclidean distance between those points.