Theoretical Statistics and Mathematics Unit, ISI Delhi
In the classic random geometric graph (RGG) model G(n,r), n vertices are placed uniformly at random in the unit square and connected by an edge whenever distant at most r apart. We consider the following variants; first the random bipartite geometric graph where there are two types of vertex and connections only between opposite types, and second the `soft' RGG where vertices at most r apart are connected with probability p (for a further parameter p).
In both variants we describe asymptotic results on connectivity, both of which illustrate that the main obstacle to connectivity is often the presence of isolated vertices. We also give a result on percolation in the first of these variants.