  This article analyzes the complex geometry of urban transportation networks
as a gateway to understanding their encompassing urban systems. Using a
proposed ring-buffer approach and applying it to 50 urban areas in the United
States, we measure road lengths in concentric rings from carefully-selected
urban centers and study how the trends evolve as we move away from these
centers. Overall, we find that the complexity of urban transportation networks
is naturally coupled, consisting of two distinct patterns: (1) a fractal
component (i.e., power law) that represent a uniform grid, and (2) a second
component that can be exponential, power law, or logarithmic that captures
changes in road density. From this second component, we introduce two new
indices, density index and decay index, which jointly capture essential
characteristics of urban systems and therefore can help us gain new insights
into how cities evolve.
