From the betweenness centrality in street networks to structural invariants in random planar graphs

This is the final version. Available on open access from Nature Research via the DOI in this recordData availability. All data needed to evaluate the conclusions are present in the paper and/or the Supplementary Information. The street networks were constructed from open access data. Any additional data related to this paper are available from the authors on reasonable request.The betweenness centrality, a path-based global measure of flow, is a static predictor of congestion and load on networks. Here we demonstrate that its statistical distribution is invariant for planar networks, that are used to model many infrastructural and biological systems. 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. Furthermore, the high betweenness nodes display a non-trivial spatial clustering with increasing spatial correlation as a function of the edge-density. 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.S Army Research OfficeCity of Pari