graph: given the parameters $n$ and $r$, consider a Poisson point process of intensity $n$ on the unit torus, equipped with some norm $\| \cdot \|$, and declare an edge between any two vertices that are distance $\le r$ from each other.

Unlike the well-known Erdős–Rényi random graph, the random geometric graph’s definition leads to strong dependence between edges: if three vertices form a “V” shaped graph, they are far more likely to have the third edge of the triangle than if no assumption were made on the other edges, as a consequence of the triangle inequality.

Many properties of this graph model have been studied. The classic monograph of Mathew Penrose [@Penrose2] studies results pertaining to many graph-theoretical functions of random geometric graphs, including (but not limited to) laws of large numbers and central limit theorems for subgraph counts, independence number, and chromatic number, as well as many properties connected to the giant component. Many of the results presented in this monograph have been improved and generalized by Penrose and his coauthors in the 12 years since its initial publication. Besides this, there have been investigations into other probabilistic features, such as threshold functions for cover times and mixing 