Efficient algorithm to generate undirected graph edges from 3D distribution of nodes based on distance
This is the intersection graph of a set of spheres.  The case of 2D disks, all of the same size, is well-studied; its intersection graph is known as a unit disk graph, and it has been studied in some depth in the literature.  I don't know if your case has been studied in the literature, but those are some search terms you could use to try to find related work.