Which Connected Spatial Networks on Random Points have Linear Route-Lengths?

In a model of a connected network on random points in the plane, one expects that the mean length of the shortest route between vertices at distance $r$ apart should grow only as $O(r)$ as $r \to \infty$, but this is not always easy to verify. We give a general sufficient condition for such linearity, in the setting of a Poisson point process. In a $L \times L$ square, define a subnetwork $\GG_L$ to have the edges which are present regardless of the configuration outside the square; the condition is that the largest component of $\GG_L$ should contain a proportion $1 - o(1)$ of the vertices, as $L \to \infty$. The proof is by comparison with oriented percolation. We show that the general result applies to the relative neighborhood graph, and establishing the linearity property for this network immediately implies it for a large family of proximity graphs.