Path-based metrics

Measurement of the graph-theoretical properties of street networks is challenging since, as noted above, many traditional metrics are not appropriate. Because of this, recent approaches have explored the use of other metrics that are better aligned to the main issues of interest for street networks. Many of these relate, in some sense, to journeys through the network, and three will be described here: betweenness, closeness and straightness.

To introduce definite notation, the network is represented by an undirected graph G = (V, E), which is composed of a set of N vertices, V = _{vi}, and a set of M

links, E = _{ei}. This is constructed according to the primal representation of the

network: in terms of streets, N is the number of junctions and M the number of street segments. Without subscripts, v and e are taken to refer to generic vertices and links, respectively.

A path in a network is any ordered sequence of vertices such that every consec- utive pair of vertices is connected by a link (i.e. a sequence of vertices which can be traversed by following links). The length of such a path can be defined in either metric or topological terms. The metric length of a path is the sum of the physical length of all constituent links, whereas the topological length is simply the number of links involved (the number of ‘hops’, which is 1 fewer than the number of vertices in the path). Although metric distance is intuitively the more meaningful form - its availability is one of the main motivations for using the primal representation -

topological distance does have relevance, and may correspond to notions of ‘mental distance’. The definitions of the majority of metrics are agnostic to this distinction.

For any pair of vertices vi and vj, a path between the two may or may not exist,

and indeed there may be more than one. When a path does exist, a shortest path between vi and vj is one such path of minimal length (though, again, there may be

more than one if there are multiple paths of equal length), and this length is denoted dij. By convention, dij is taken to be infinite when no path exists. To recapitulate in

real-world terms, d_ij is the shortest distance (either in metric or topological terms) one would have to cover to travel between two junctions vi and vj through the street

network. Efficient algorithms for the calculation of shortest paths can be found, along with many others related to graphs, in the book by Cormen (2009).

Betweenness The metric betweenness was proposed independently by Anthonisse (1971) and Freeman (1977), in the context of social networks, as a measure of the extent to which vertices occupy a position of ‘brokerage’. Rather than measuring how near they are to others, it measures the extent to which vertices act as an intermediary in communications between others. Two quantities are required for its calculation: σ_jk, which is the total number of shortest paths between two vertices vj and vk; and σjk(vi), which is the number of shortest paths between vj and vk

which pass through vi. Using these, the betweenness centrality of a given vertex

vi is defined as C_iB = X vj,vk∈V, vj∼vk σjk(vi) σjk , (3.1)

where _{∼ here represents the relation ‘there exists a path between v}j and vk’ (this

simply restricts the sum to pairs for which σ_{jk 6= 0). The essence of the metric} is to count the number of times that vi features on paths through the network,

assuming that a path exists between every possible pair of vertices. A more intuitive understanding can be gained by considering how it can be calculated:

2) consider all pairs of vertices vj and vk;

3) for each pair, find the shortest path(s) between them;

4) every time a vertex appears in one of these shortest path(s), incre- ment its CB by _w1, where w is the number of shortest paths between vj and vk (so if there is only one shortest path, add 1 to the CB of

each intermediate vertex).

In many applications, CB is normalised by dividing through by its maximum possi- ble value: N (N_{− 1), if the graph is fully-connected. This maximum is meaningless,} however, if the graph is not connected and, since the primary interest is usually only in the relative values of vertices, no normalisation is performed here.

Since paths are composed of both vertices and links, betweenness can be defined equally well for links. If σjk(ei) is analogously defined as the number of shortest

paths between vj and vk which pass through ei, the link betweenness for a link ei is

defined similarly as C_iB = X vj,vk∈V, vj∼vk σjk(ei) σjk . (3.2)

In the same way as for vertices, this is the frequency with which each link appears in paths through the network. Importantly, the betweenness of a link is not equal to the average betweenness of its two vertices: a rarely-used link can connect two highly-used vertices, for example. Whenever used, the distinction between link and vertex betweenness will be made clear from the context. The paper by Brandes (2008) presents algorithms for the efficient calculation of both forms, as well as other variations.

In terms of street networks, betweenness has a relatively clear interpretation as an estimate of the use of any given feature (junction or segment) by traffic passing through the network. Although the premise of single trips between all junctions is crude, it nevertheless represents a well-motivated first-order heuristic for urban movements. Of particular note is the fact that the value for every feature depends

entirely on its role in the wider network. The stylised example of Figure 3.3, showing link betweenness, is illustrative of its discriminatory value: the two links identified - one peripheral and one highly central - are those with values at each extreme.

e

e

v

Figure 3.3: Stylised illustration of link betweenness: link e1 (shown red) fea- tures in any path between one of the 7 vertices on the ‘left’ of the network and the 7 on the ‘right’, and therefore has a relatively high betweenness value of 49. Link e2 (green), on the other hand, is only traversed by paths starting/ending at v; there are 13 such paths and it therefore has a relatively low value of 13.

Closeness The closeness of a vertex (Sabidussi, 1966; Freeman, 1978) is a measure of the extent to which it is near to all others, on average. It is defined for a vertex vi as C_iC = NP− 1 vj∈V, vi6=vj dij , (3.3)

so that it is simply the inverse of the average length of all paths starting at v_i. It is important to note that CC is only meaningful for connected graphs (since otherwise dij = ∞ for at least one vj), but this can generally be assumed to be the case for

street networks.

In terms of street networks, closeness can be most naturally interpreted as rep- resenting the ‘accessibility’ of a vertex, in some sense. Locations with high closeness are those which are easiest to reach (in terms of distance) from all other locations, on average.

Straightness The concept of straightness (Vragovi´c et al., 2005; Porta et al., 2006b) examines the extent to which paths in the network deviate from a perfectly linear path in space. As such, it is only valid when dij is defined metrically. When

this is the case, if lij is defined as the Euclidean (as-the-crow-flies) distance between

vertices vi and vj, the straightness of a vertex vi is given by

C_iS = 1 N _{− 1} X vj∈V, vi6=vj l_ij dij . (3.4)

When vi is thought of as the terminus of the paths in question, this value represents

the extent to which it can be reached by straight, non-convoluted paths (regardless of their length). As such, it can be regarded as representing the prominence of a location; its ‘presence’ for those travelling around the space (Conroy Dalton, 2003).

Radial limits One problem which can arise with the three metrics described above is that of ‘edge effects’, whereby misleading results can be found near the spatial extremities of the network. This is particularly acute for closeness, since vertices will tend to receive higher values the closer they are to the centre of mass of the junctions (Porta et al., 2006b). To alleviate this problem, the definitions can be modified to local forms, in which only vertices within a certain distance of the vertex in question are considered in the calculation. For example, closeness is calcu- lated as the inverse of the average distance to all vertices within some given radius, r.

In the case of betweenness, only paths of length less than or equal to r are taken into account when calculating scores (in effect, the relation_{∼ in equations (3.1) and} (3.2) is replaced by ‘there exists a path of length at most r between vj and vk’).

These radially-limited variants are denoted by superscripts thus:



C_iB(r), C_iC(r), C_iS(r)=C_iB, C_iC, C_iS for only paths such that d_{≤ r} (3.5) The rationale for considering local measures is not purely technical. For many applications, it may be more appropriate to only consider paths of limited distance:

journeys across the breadth of a large city, for example, are scarcely relevant for pedestrian travel. The use of local measures ensures that these indices can be defined consistently and that they correspond properly to the issue of interest.