Characterization of spatial networklike patterns from junction geometry.

We propose a method for quantitative characterization of spatial networklike patterns with loops, such as surface fracture patterns, leaf vein networks, and patterns of urban streets. Such patterns are not well characterized by purely topological estimators: also patterns that both look different and result from different morphogenetic processes can have similar topology. A local geometric cue--the angles formed by the different branches at junctions--can complement topological information and allow the quantification of the large scale spatial coherence of the pattern. For patterns that grow over time, such as fracture lines on the surface of ceramics, the rank assigned by our method to each individual segment of the pattern approximates the order of appearance of that segment. We apply the method to various networklike patterns and find a continuous but sharp dichotomy between two classes of spatial networks: hierarchical and homogeneous. The former class results from a sequential growth process and presents large scale organization, and the latter presents local, but not global, organization.