Random geometric graph

In graph theory, a random geometric graph is an random undirected graph drawn on a bounded region, eg. the unit torus $[0,1)^{2}$. It is generated by

1. Placing vertices at random uniformally and independently on the region
2. Connecting two vertices, u,v if and only if the distance between them is at most a threshold $r$, ie. $d(u,v)\leq r$

Several probabilistic results are known about the number of components in the graph as a function of the threshold $r$ and the number of vertices $n$.

References

Penrose, Mathew: Random Geometric Graphs (Oxford Studies in Probability, 5), 2003.