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

TL;DR

This study reveals that betweenness centrality distribution in planar graphs, especially street networks, remains invariant under various changes, driven by underlying structural properties and spatial correlations, with implications for urban planning and network analysis.

Contribution

It demonstrates the invariance of betweenness centrality distribution in planar graphs through empirical, simulation, and analytical methods, highlighting its dependence on planarity and network size.

Findings

BC distribution is invariant in most planar graphs.

High BC nodes cluster around the barycenter as density increases.

Betweenness centrality predicts congestion and bottlenecks in street networks.

Abstract

We demonstrate that the distribution of betweenness centrality (BC), a global structural metric based on network flow, is an invariant quantity in most planar graphs. We confirm this invariance through an empirical analysis of street networks from 97 of the most populous cities worldwide, at scales significantly larger than previous studies. We also find that the BC distribution is robust to major alterations in the network, including significant changes to its topology and edge weight structure, indicating that the only relevant factors shaping the distribution are the number of nodes and edges as well as the constraint of planarity. Through simulations of random planar graph models and analytical calculations on Cayley trees, this invariance is demonstrated to be a consequence of a bimodal regime consisting of an underlying tree structure for high BC nodes, and a low BC regime arising from the presence of loops providing local path alternatives. Furthermore, the high BC nodes display a non-trivial spatial dependence, with increasing spatial correlation as a function of the number of edges, leading them to cluster around the barycenter at large densities. Our results suggest that the spatial distribution of the BC is a more accurate discriminator when comparing patterns across cities. Moreover, the BC being a static predictor of congestion in planar graphs, the observed invariance and spatial dependence has practical implications for infrastructural and biological networks. In particular, for the case of street networks, as long as planarity is conserved, bottlenecks continue to persist, and the effect of planned interventions to alleviate structural congestion will be limited primarily to load redistribution, a feature confirmed by analyzing 200 years of data for central Paris.