On the metric, topological and functional structures of urban networks

Axial Graphs are networks whose nodes are linear axes in urban space, and whose edges represent intersections of such axes. These graphs are used in urban planning and urban morphology studies. In this paper we analyse distance distributions between nodes in axial graphs, and show that these distributions are well approximated by rescaled-Poisson distributions. We then demonstrate a correlation between the parameters governing the distance distribution and the degree of the polynomial distribution of metric lengths of linear axes in cities. This correlation provides ‘topological’ support to the metrically based categorisation of cities proposed in [R. Carvalho, A. Penn, Scaling and universality in the micro-structure of urban space, Physica A 332 (2004) 539–547]. Finally, we attempt to explain this topologico-metric categorisation in functional terms. To this end, we introduce a notion of attraction cores defined in terms of aggregations of random walk agents. We demonstrate that the number of attraction cores in cities correlates with the parameters governing their distance and line length distributions. The intersection of all the three points of view (topological, metric and agent based) yields a descriptive model of the structure of urban networks.