Mathematical representation

In order to understand the various metrics studied in the context of street networks, it is first necessary to introduce the methods by which such networks can be rep- resented mathematically. The natural approach for doing this is by encoding its structure in a graph; however, there is more than one way to do this. The various representations emphasise different aspects of the network structure, and their suit- ability depends on the perspective of the particular research application. The two most common of these - the primal and dual representations - will be introduced and discussed here. The empirical work presented in the later part of this chapter focusses on the primal representation, and this choice is discussed below.

3.1.4.1 Primal representation

The primal representation (Porta et al., 2006b) is the more intuitive of the two rep- resentations, and corresponds most closely to a traditional street network map. In such a graph, each junction in the street network is represented by a vertex (where ‘junction’ refers to any point at which a traveller through the network would have a choice of path). A link is then added between any pair of vertices for which the corresponding junctions are connected directly by a street; such a section of street, connecting two junctions, is defined as a street segment. Figure 3.1 shows the stages of construction: the identification of junctions and the addition of links between them.

The term ‘primal’ refers to the matching between dimensionalities of features: junc- tions (which are points, and therefore zero-dimensional) are represented by zero- dimensional vertices, and street segments (linear and one-dimensional in space) are encoded as links, which are also one-dimensional. Because of this, each graph fea- ture can be given meaningful spatial attributes: vertices have a definite geographical

(a) (b) (c)

Figure 3.1: The construction of the primal representation of a street network: a) the original map; b) vertices placed at each junction; and c) links added between any pair of junctions connected by a street segment.

position, and links have a physical length. The latter of these means that this can be regarded as a weighted graph, and the physical length of any route through the network can be found by reference to the graph.

3.1.4.2 Dual representation

The dual representation is that which arises when the roles of vertices and links are inverted, relative to the primal case. Streets are represented by vertices, and two vertices are linked if the associated streets intersect at some point. That ver- tices represent ‘streets’ rather than ‘street segments’ is significant: in this context, a street is a section of the network (possibly comprising multiple segments) which can be regarded as a coherent single entity. The intersections in question are therefore those between these larger unified streets.

There are several options for the process by which unified streets are identified. In the ‘space syntax’ approach (Hillier & Hanson, 1984), which pioneered represen- tations of this type, streets are defined on the basis of ‘axial lines’: straight lines which can be interpreted as lines-of-sight. As noted by Porta et al. (2006b), this is only obliquely related to the street network configuration itself, since the intersec- tions of axial lines do not necessarily coincide with junctions. A further problem arises because such representations may not be unique: for a given street network,

valid axial lines can be drawn in multiple ways.

An alternative is the ‘named street’ approach (Jiang & Claramunt, 2004), whereby contiguous elements of the network are unified if they share a common street name. A clear shortcoming of this nominalistic approach is that it depends on the relia- bility (and availability) of street names, which can be subjective and arbitrary. As an alternative, Porta et al. (2006a) proposed an algorithm in which street segments are grouped according to their geometric linearity at junctions, so that a street is a relatively continuous sequence of segments.

(a) (b) (c)

Figure 3.2: The construction of the dual representation of a street network: a) the original map, with black lines placed along streets; b) streets identified and coloured on the basis of street name; and c) the dual network derived on the basis of street intersections. The latter network is aspatial: other than indicat- ing adjacency, the location and form of vertices and links has no geographical meaning.

Figure 3.2 shows the construction of a dual graph in which, for ease of presen- tation, the named street approach is taken. Segments are coloured according to street name and those which intersect are linked in the derived graph. Its aspatial nature - a result of the dimensional mis-match - is immediately apparent: streets are collapsed to vertices, for example, which have no property analogous to length.

3.1.4.3 Choice of representation

Arguments in favour of the dual approach typically focus on its relevance to issues of wayfinding in an urban environment. Rosvall et al. (2005) express their relevance in informational terms, as corresponding to the way in which travellers perceive and encode journeys through the network. The information required to navigate from one point to another essentially comprises a list of important ‘turns’ (changes from one street to another), rather than a complete enumeration of all street segments. The number of such turns corresponds precisely to the notion of path length in the dual network (which is a topological, rather than metric, measure): each link traversed corresponds to a turn. When streets are based on some form of linearity, this is also well-aligned with the observed preference of wayfinders to go straight at intersections (Conroy Dalton, 2003).

There are, however, several shortcomings of the dual approach. Although concerned specifically with inconsistencies in space syntax, Ratti (2004) makes several points related to dual representations more generally, such as their susceptibility to edge effects and the loss of important geographical information. This is reaffirmed by Porta et al. (2006b), who suggest that the loss of metric information is simply too big a price to pay in the context of geographical analysis.

The issue of street unification is also problematic when the ultimate aim of the analysis concerns other processes, such as crime. A single street (represented by a vertex) in the dual representation can be very long and comprise many street seg- ments. Treating such a large feature as a single geographical entity - which would imply distilling all crime to a single value - conflicts directly with the aim of carrying out granular analysis.

Although problems can also be found with the primal approach - it is at risk of placing undue emphasis on the role of junctions, for example - the arguments above have led to it being favoured in recent studies (Crucitti et al., 2006b; Porta et al.,

2009; Chan et al., 2011). In the remainder of the analysis presented here, the primal representation will be used, with similar justification: namely, its granularity and explicitly geographical perspective. Though the dual representation will not be con- sidered further here, its properties have been investigated in a number of empirical studies (Jiang & Claramunt, 2004; Porta et al., 2006a; Kalapala et al., 2006; Jiang, 2007).