  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.
