Where to look for power Laws in urban road networks?

Spatial embeddedness and planarity of urban road networks limit the range of their node degree values. Therefore, pursuing analysis based on the distribution of node degrees e.g. scale free aspect could not be accomplished in urban road networks. We have inspected the distribution of degree, betweenness centrality, weighted degree (based on incident link capacities), and alpha weighted degree for eight urban road networks across the world. These networks are abstracted from Philadelphia (USA), Berlin (Germany), Chicago (USA), Anaheim (USA), Gold Coast (Australia), Birmingham (UK), and Isfahan (Iran). Our results show that although the degree (weighted and unweighted) distributions of these networks are totally different, they all show power law distributions in betweenness centrality. Thus, scale free aspect could be observed in the betweenness centrality distribution. We then analyzed the collapse of network as a result of node removals. The collapse patterns suggest that critical nodes of urban road networks could not be detected solely based on betweenness centrality. Therefore, we conclude that the concept of betweenness centrality in urban road networks is more of functional merit than topological merit. In other words, central nodes play an important role in transmitting the flow but their loss would not harm the connectivity of urban networks. This claim is supported by analyzing the correlation among node flow and node betweenness in Isfahan and Anaheim.