We study centrality in urban street patterns of different world cities represented as networks in geographical space. The results indicate that a spatial analysis based on a set of four centrality indices allows an extended visualization and characterization of the city structure. A hierarchical clustering analysis based on the distributions of centrality has a certain capacity to distinguish different classes of cities. In particular, self-organized cities exhibit scale-free properties similar to those found in nonspatial networks, while planned cities do not. DOI: 10.1103/PhysRevE.73.036125 PACS number͑s͒: 89.75.Hc, 89.75.Fb, 87.23.Ge A particular class of complex networks ͓1-3͔ is composed of those embedded in real space-i.e., networks whose nodes occupy a precise position in two-or three-dimensional Euclidean space and whose edges are real physical connections. With a few exceptions ͓4-6͔, most of the works in the literature have focused on the characterization of the topological ͑relational͒ properties of spatial networks, while the spatial aspect has received less attention, when not neglected at all. However, it is not surprising that the topology of spatial networks is strongly constrained by their geographical embedding. For instance, the number of long-range connections ͓4-6͔ and the number of edges that can be connected to a single node ͓5͔ are limited by the spatial embedding. This is particularly evident in planar networks ͑e.g., those networks forming vertices whenever two edges cross͒ ͓7͔, as urban streets or ant networks of galleries ͓5͔, and has important consequences on the possibility to observe a small-world behavior or scale-free degree distributions ͓1-3͔. Consequently, spatial networks are different from relational networks, and as such they need to be treated. Centrality has remained a fundamental concept in network analysis since its introduction in structural sociology ͓8,9͔. The network approach has also a long tradition in economic geography and city planning, where it has been used to investigate the territorial relationships among communication flows, population, wealth, and land uses ͓10͔. However, when dealing with urban street patterns, centrality has been studied in relational networks only ͓11-13͔, neglecting a fundamental aspect such as the geography. In such an approach, known as the dual representation ͓12,13͔ or information city network ͓14͔, a city is transformed into a relational ͑topo-logical͒ graph by mapping the streets onto the graph nodes and the intersections between streets onto edges between the nodes. In the present paper, we study centrality in urban street patterns of different world cities represented as spatial networks. In our approach, which is opposite to the dual one, we work within a fully metric framework in which the distance has to be measured not just in topological terms ͑steps͒, like in the dual representation of a city ͓12-14͔ or in social ͓9͔ and other complex systems ͓1͔, but rather in properly spatial terms ͑meters, miles͒. The results indicate that a spatial analysis based on a set of different centrality measures ͑properly extended for spatial graphs͒ allows ͑1͒ a visual characterization of the structural properties of a city, ͑2͒ the evidence that planned and self-organized cities belong to two different classes, and ͑3͒ to find scale-free properties similar to those found in the degree distributions of relational ͑nonspatial͒ networks. We have selected 18 one-square-mile samples of different world cities from Ref. ͓15͔, imported them in a GIS ͑Geo-graphic Information System͒ environment and constructed spatial graphs of street networks. In our approach, each urban street sample is turned into an undirected, valued graph G, where intersections are nodes and streets are edges. We denote by N the number of nodes and by K the number of edges. The nodes are characterized by their position ͕x i
Centrality metrics have been used in the visualization and characterization of city structures and road networks as well REF .
Centrality measures in spatial networks of urban streets
Phys. Rev. E 73 (2006) 036125.
Medicine Physics