We report about universality of rank-integration distributions of open spaces in city space syntax similar to the famous rank-size distributions of cities (Zipf's law). We also demonstrate that the degree of choice an open space represents for other spaces directly linked to it in a city follows a power law statistic. Universal statistical behavior of space syntax measures uncovers the universality of the city creation mechanism. We suggest that the observed universality may help to establish the international definition of a city as a specific land use pattern.
Deep Dive into Scaling and Universality in City Space Syntax: between Zipf and Matthew.
We report about universality of rank-integration distributions of open spaces in city space syntax similar to the famous rank-size distributions of cities (Zipf’s law). We also demonstrate that the degree of choice an open space represents for other spaces directly linked to it in a city follows a power law statistic. Universal statistical behavior of space syntax measures uncovers the universality of the city creation mechanism. We suggest that the observed universality may help to establish the international definition of a city as a specific land use pattern.
Urban space is of rather large scale to be seen from a single viewpoint; maps provide us with its representations by means of abstract symbols facilitating our perceiving and understanding of a city. The middle scale and small scale maps are usually based on Euclidean geometry providing spatial objects with precise coordinates along their edges and outlines.
The widespread use of graph theoretic analysis in geographic science had been reviewed in [1] establishing it as central to spatial analysis of urban environments. In [2], the basic graph theory methods had been applied to the measurements of transportation networks.
Network analysis has long been a basic function of geographic information systems (GIS) for a variety of applications, in which computational modelling of an urban network is based on a graph view in which the intersections of linear features are regarded as nodes, and connections between pairs of nodes are represented as edges [3]. Similarly, urban forms are usually represented as the patterns of identifiable urban elements such as locations or areas (forming nodes in a graph) whose relationships to one another are often associated with linear transport routes such as streets within cities [4]. Such planar graph representations define locations or points in Euclidean plane as nodes or vertices {i}, i = 1, . . . , N, and the edges linking them together as i ∼ j, in which {i, j} = 1, 2, . . . , N. The value of a link can either be binary, with the value 1 as i ∼ j, and 0 otherwise, or be equal to actual physical distance between nodes, dist(i, j), or to some weight w ij > 0 quantifying a certain characteristic property of the link. We shall call a planar graph representing the Euclidean space embedding of an urban network as its primary graph. Once a spatial system has been identified and represented by a graph in this way, it can be subjected to the graph theoretic analysis.
A spatial network of a city is a network of the spatial elements of urban environments. They are derived from maps of open spaces (streets, places, and roundabouts). Open spaces may be broken down into components; most simply, these might be street segments, which can be linked into a network via their intersections and analyzed as a networks of movement choices. The study of spatial configuration is instrumental in predicting human behavior, for instance, pedestrian movements in urban environments [6]. A set of theories and techniques for the analysis of spatial configurations is called space syntax [7]. Space syntax is established on a quite sophisticated speculation that the evolution of built form can be explained in analogy to the way biological forms unravel [5]. It has been developed as a method for analyzing space in an urban environment capturing its quality as being comprehendible and easily navigable [6]. Although, in its initial form, space syntax was focused mainly on patterns of pedestrian movement in cities, later the various space syntax measures of urban configuration had been found to be correlated with the different aspects of social life, [8].
Decomposition of a space map into a complete set of intersecting axial lines, the fewest and longest lines of sight that pass through every open space comprising any system, produces an axial map or an overlapping convex map respectively. Axial lines and convex spaces may be treated as the spatial elements (nodes of a morphological graph), while either the junctions of axial lines or the overlaps of convex spaces may be considered as the edges linking spatial elements into a single graph unveiling the topological relationships between all open elements of the urban space. In what follows, we shall call this morphological representation of urban network as a dual graph.
The encoding of cities into non-planar dual graphs reveals their complex structure. The transition to a dual graph is a topologically non-trivial transformation of a planar primary graph into a nonplanar one which encapsulates the hierarchy and structure of the urban area and also corresponds to perception of space that people experience when travelling along routes through the environment. The dual transformation replaces the 1D open segments (streets) by the zero-dimensional nodes. The sprawl like developments consisting of a number of blind passes branching off a main route are changed to the star subgraphs having a hub and a number of client nodes. Junctions and crossroads are replaced with edges connecting the corresponding nodes of the dual graph. Places and roundabouts are considered as the independent topological objects and acquire the individual IDs being nodes in the dual graph. Cycles are converted into cycles of the same lengthes. The dual graph representation of a regular grid pattern is a complete bipartite graph, where the set of vertices can be divided into two disjoint subsets such that no edge has both end-points in the same subset, and every line joining the two subsets i
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