Random Geometric Graph in unit disk

Question

According to the Wikipedia article, to define a random geometric graph, one needs a metric space.

In the examples they give for 2D random geometric graphs, they say that "an RGG can be constructed by choosing a flat unit square $[0, 1]$ or a torus of unit circumferences $[0, 1)^2$ as the embedding space."

Are those the only two choices available? Why not have a unit disk as an embedding space?

Answer 1

No, you can really take any space you want. The two reasons these spaces are chosen most of the time are:

1. Conceptual simplicity. Before tackling weirdly defined spaces, how does the model behaves on the most natural spaces?
2. Easiness to get mathematical results. It's easy to compute the expected number of points, etc, which then allow to get results on more elaborate graph parameters.

By the way, usually the torus is chosen to avoid the annoying border effects of the unit disk, which simplifies reasonings and calculations. But you may want to consider border effects, so you have these two choices.

So if you want, you take any space and try to get equivalent results on the corresponding random geometric graphs.