For a given set of $N$ random points, distributed uniformly on a unit square, I construct its Delaunay triangulation. Taking the triangulation as an unweighted graph, I need to know the expected average path length and expected diameter of the graph.
I have conducted computer simulations and obtained a scaling for both quantities as $\sim N^{\alpha}$, with $\alpha \simeq 0.4$. I would like to corroborate the result analytically, but I cannot figure out how to do it.
The only similar result I have found is that the ratio between the (weighted) shortest-path distance in the Delaunay triangulation and the Euclidean distance is a constant [1], but I don't know if it is of help for my problem.
[1] D. P. Dobkin, S. J. Friedman and K. J. Supowit, Delaunay graphs are almost as good as complete graphs, Discr. Comput. Geom. 5 (1990) 399–407.