The concentration of edge betweenness in the evolution of planar graphs and street networks

The centrality measures of the nodes and edges of the street networks are related to various urban phenomena. In particular, betweenness centrality correlates with the spatial distribution of economic activities, the levels of congestion, and the structural changes in cities. In this work, we study how betweenness tends to concentrate in a small set of edges and develop a model to analyse this concentration throughout the growth of graphs. We show that random planar graphs tend to betweenness concentration as the number of nodes increases. The evolution of Paris and Tijuana street networks shows the same behaviour but at a higher rate. A set of 300 street networks worldwide follows a similar relationship between the number of nodes and the betweenness concentration. We find a significant correlation between congestion ranks and betweenness concentration.