  A geometric graph is a graph embedded in the plane with vertices at points
and edges drawn as curves (which are usually straight line segments) between
those points. The average transversal complexity of a geometric graph is the
number of edges of that graph that are crossed by random line or line segment.
In this paper, we study the average transversal complexity of road networks. By
viewing road networks as multiscale-dispersed graphs, we show that a random
line will cross the edges of such a graph O(sqrt(n)) times on average. In
addition, we provide by empirical evidence from experiments on the road
networks of the fifty states of United States and the District of Columbia that
this bound holds in practice and has a small constant factor. Combining this
result with data structuring techniques from computational geometry, allows us
to show that we can then do point location and ray-shooting navigational
queries with respect to road networks in O(sqrt(n) log n) expected time.
Finally, we provide empirical justification for this claim as well.
