A Route-Length Efficiency Statistic for Road Networks

This note compares some current theoretical mathematical work on spatial networks with data on inter-city road networks within States. In designing a network, a natural constraint is the total length, and a natural objective is to want the route length `(i, j) between typical cities i, j to be not much longer than straight line distance d(i, j). Write r(i, j) = `(i,j) d(i,j) − 1 for the relative excess route-length. With an n-city network there are ( n 2 ) such numbers r(i, j). A recent theoretical insight is that, from a mathematical viewpoint, a good way to combine these into a single “route-length efficiency” statistic R∗ is to define R∗ as the maximum over d of the average of r(i, j) over city-pairs with d(i, j) ≈ d. The optimal tradeoff between total network length and R∗ is discussed elsewhere in a theoretical model of randomly-placed cities. What about real networks? For each of 10 U.S. States we studied the road network linking the 20 largest cities. In this note we present the data and discuss the relationship between data and the predictions from theory. ∗Department of Statistics, 367 Evans Hall # 3860, U.C. Berkeley CA 94720; [email protected]; www.stat.berkeley.edu/users/aldous. Aldous’s research supported by N.S.F Grant DMS-0704159.