Instead, more information can be gleaned from non-local higher-level metrics such as those based on network centralities, which while strongly correlated with degree in non-spatial networks 29, display non-trivial behavior in planar networks 30. Degree-based network measures, while well-studied on such systems, lead to rather uninteresting results the degree distribution is strongly peaked, and related metrics such as clustering and assortativity are high 2. The geographical embedding leads to strong effects on network topology with limitations on the number of long-range connections and the number of edges incident on a single node (its degree k) 27, 28. Street networks fall into the category of planar graphs 25 and their edges constitute a physical connection, as opposed to relational connections found in many complex networks 26. Different street structures result in varying levels of efficiency, accessibility, and usage of transportation infrastructure 11, 12, 13, 14, 15, 16, 17 consequently structural characteristics of roads have been of great interest in the literature 18, 19, 20, 21, 22, 23, 24. Patterns of streets and roads are particularly important, allowing residents to navigate the different functional components of a city. These networks are quite relevant in the context of urban systems 3, 4, 5, 6, 7, where analysis of their structural properties has uncovered unique characteristics of individual cities, as well as surprising statistical commonalities across different urban contexts 8, 9, 10. Recent years have witnessed unprecedented progress in our understanding of spatial networks that are pervasive in biological, technological and infrastructural systems 1, 2. Our results suggest that the spatial distribution of betweenness is a more accurate discriminator than its statistics for comparing static congestion patterns and its evolution across cities as demonstrated by analyzing 200 years of street data for Paris. Furthermore, the high betweenness nodes display a non-trivial spatial clustering with increasing spatial correlation as a function of the edge-density. Empirical analysis of street networks from 97 cities worldwide, along with simulations of random planar graph models, indicates the observed invariance to be a consequence of a bimodal regime consisting of an underlying tree structure for high betweenness nodes, and a low betweenness regime corresponding to loops providing local path alternatives. Here we demonstrate that its statistical distribution is invariant for planar networks, that are used to model many infrastructural and biological systems. The betweenness centrality, a path-based global measure of flow, is a static predictor of congestion and load on networks.
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