Directionality

That space-time clustering can be demonstrated for networks is a relatively small advance on previously-established results, essentially showing that the phenomenon of clustering can be understood in different terms. The particular value of the ap- proach, however, becomes apparent when considering the logical next step in the analysis and modelling of events of this kind. Given that it has been shown that space-time clustering (and the more specific concept of (near-)repeat victimisation) does occur, it is natural to examine what can be said about where such events occur.

Several approaches have been suggested for the prediction of areas at risk from near-repeat victimisation; these include traditional methods based on Kernel Den- sity Estimation (see Chainey & Ratcliffe, 2005) and the ProMap method proposed

by Bowers et al. (2004). All of these, however, operate on the principle that any boost effect acts uniformly in all directions; this is unsurprising, given that nothing more can be said on the basis of, for example, the Knox test. These techniques can be refined by adjusting predictions according to opportunity (i.e. the distribution of targets), but this is done under an assumption of uniform risk across targets, and takes no account of their relative risk. Again, this may be due to an absence of suitable explanatory factors which could be incorporated into such a calculation.

The properties of the street network, however, do indeed provide a means by which the inherent characteristics of space (in a sense which is relevant to crime) can be quantified. Areas surrounding the location of a crime can be distinguished from each other on the basis of their network properties, and these therefore represent a po- tential guide for prioritisation. If such a relationship can be found, in that network properties can be seen to influence the directionality of crime spread, then repeat victimisation effects can no longer be considered to be uniform in all directions.

4.3.3.1 The effect of the network

There is indeed good reason, based on criminological theory, to expect that the street network should influence (near-)repeat victimisation. This can be seen by ap- pealing to the key theoretical concepts of the topic: the flag and boost hypotheses, and the optimal forager principle.

The first of these - the flag hypothesis - is relatively uninteresting in this regard, since its assertion that (near-)repeat victimisation can be explained by fundamental heterogeneity of risk implies that all insights into the issue are simply corollaries of results for the static case. There is therefore little scope to develop hypotheses specific to dynamic effects.

In the case of the boost hypothesis, and related notion of the optimally-foraging offender, however, specific links can be identified. This account of (near-)repeat victimisation is based on the principle that the victimisation of a property actively

increases its risk of victimisation and that of those around it for a short period; that is, it causes them to be more likely or appealing targets. This can be reconciled with the expectation that the surrounding properties are likely to be in the activity space of the initial offender and are therefore more likely to be considered as possible targets for a prospective follow-up offence.

The implication of this argument is that, as with the static case, imbalances in the risk of victimisation can be explained by the tendency of some parts of the street network to feature more often or more prominently in awareness spaces. In the static case, this was estimated by considering the accumulation of all possible journeys through the network, encapsulated by the metric betweenness. Since all journeys were treated equally in these calculations, this was effectively an estimate of the travel activity of the pedestrian population at large: because the outcome of interest was victimisation by any offender, this was an appropriate choice.

When estimating awareness spaces in the context of (near-)repeat victimisation, though, there is a crucial difference: the fact of the initial victimisation determines the perspective from which the awareness space should be evaluated. If the prospec- tive follow-up offence is assumed to be the work of the same offender, the task changes from estimating awareness space in general to that of estimating the aware- ness space of one offender (or group of offenders) in particular. Importantly, one piece of information about this awareness space is known: it includes the location of the initial offence. Of course, this argument relies on simplifying assumptions - the same-offender principle, and a relatively literal and rigid interpretation of the role of awareness space in target choice - but both are crystallisations of concepts for which there is strong support within the literature (e.g. Bowers & Johnson, 2004; Bernasco, 2008).

The strategy implied by the above argument can be encapsulated by one ques- tion: given that an individual’s awareness space includes a certain street, which

other streets are likely to also be in the same awareness space? In terms of travel, this corresponds to asking which streets an individual is likely to travel along, if he or she is known to travel (or recently to have travelled) along a particular street. If this can be estimated, it provides a means of determining, after an initial offence, which surrounding streets the offender is most likely to be aware of and therefore to consider for a follow-up offence. This line of reasoning provides the motivation for the definition of a suitable network metric.

4.3.3.2 Commonality

The analysis of networks is typically performed in terms of the properties or char- acteristics of individual features (i.e. vertices or links). Frequently-used metrics such as degree and betweenness are of this type, and numerous others have been proposed, emphasising various aspects of network structure. There are, however, few methods for the measurement of dyadic relationships; that is, the association between pairs of features. ‘Pairs of features’ is taken here to refer to arbitrary pairs: the presence of a link is, of course, a dyadic relationship between the two terminal vertices, but non-neighbouring features might also be meaningfully compared. Such comparison might, for example, be used to quantify some notion of similarity, or to infer the potential for interaction between two features. The spread of crime risk between streets is, of course, such an interaction.

The case of crime, when considered in detail, exemplifies the need for such a dyadic measure. The argument above suggests that there is value in asking, given a certain link in a network - denoted e, for concreteness - which other links also tend to feature in journeys which use e. Simply considering the individual properties of these links is insufficient, and might be misleading. To take the example of betweenness, two other links e0 and e00 could have the same betweenness value but relate very differently to e: e0 might be an immediate neighbour, and e00 might be in an entirely different part of the network. Immediate neighbours are not all equal, either: e may be con- nected to a cul-de-sac at one end and a main road at the other. These problems can only be addressed by considering the centrality of links from the perspective of e.

To this end, a new measure of the commonality of two links is introduced. Its defini- tion can best be understood as a derivative of betweenness: where the betweenness of a link e counts the number of shortest paths which include it, the commonality of e with another link e0 calculates the proportion of these paths which also feature e0. In essence, then, it measures the co-occurrence in paths of any pair of links on the network, normalised by the overall occurrence of one of those links. Expressed another way, it measures the proportion of shortest paths passing through e which also pass through e0.

Commonality can be defined formally using many of the same terms used to specify betweenness in equation (3.1). For generic vertices v and w, and for any pair of links ei and ej, σvw is defined as the number of shortest paths between v and w, and

σvw(ei) is the number of those shortest paths which pass through ei. If a further

definition is added for σvw(ei, ej) as the number of shortest paths between v and w

which feature both e_i and e_j, then the commonality of e_j relative to e_i, denoted Cij, is defined as C_ij = P v∼w σvw(ei, ej) σvw P v∼w σvw(ei) σvw , (4.2)

As before, _{∼ represents the relation ‘there exists a path between v and w’.}

A number of basic properties of commonality are immediately apparent. The de- nominator of Cij is simply the betweenness of link ei, and so Cij can be thought

of as the proportion of the journeys that contribute to the betweenness of e_i which also incorporate ej. As such, values lie in the range [0, 1], and the upper limit of 1

is realised at least once for every i: trivially, σvw(ei, ei) = σvw(ei) and so Cii = 1 for

all values of i.

It is also important to note that the definition of Cij is not symmetric in i and

evident in the real-world situations to which the measure responds: to cite another example from street networks, where a busy street and quiet street are adjacent, the dependence of the quiet road’s traffic on the busy road is likely to be greater than that in the opposite direction. This point will be illustrated using real-world examples in the following section.

Commonality has one further advantageous feature, which concerns the fact that a distance-decay relationship is implicit within its definition. Models and theories of interactions in space typically incorporate some notion of distance-decay, whereby the strength of a relationship varies inversely with spatial separation. This is ev- ident in journey-to-crime data for many criminal phenomena (Wiles & Costello, 2000), and is a ubiquitous feature of near-repeat victimisation (see Tables 4.1 and 4.2, for example). Often this must be accounted for explicitly (by including a ‘dis- tance’ variable, for example), but consideration of commonality reveals that it is an effect which arises spontaneously. For a given link ei, values of Cij will tend to

be lower for links e_j which are further removed: moving further away from e_i, the plurality of possible routes increases with every junction, so that the load from ei

will become more dispersed.

The effect can be seen most clearly by considering extreme cases. All but one of the journeys which use ei will feature at least one of its immediate neighbours

(the exception is the trivial journey from one end of ei to the other) and so Cij will

tend to be high for such an e_j. On the other hand, if e_j is a distant cul-de-sac then it will feature in a very small proportion of e_i’s journeys and give rise to a correspond- ingly low Cij. This property is an important feature of commonality. Although the

ability to compare links in a manner which accounts for their relative location in the network appears modest, it represents a significant improvement on approaches which rely exclusively on immediate adjacency.

commonality can be refined in order to consider only trips whose length is less than some maximum radius r. This is done in the natural way - by modifying the defi- nition of _{∼ in (4.2) to incorporate only pairs of vertices which lie within r of each} other - and the resulting measurement is denoted C_ij(r). As with betweenness, r can be measured in either metric or topological units, and the distinction will be made whenever it is used. Again, restricting the maximum path length can be interpreted as representing more localised phenomena (pedestrian travel, for example) and as a means of ameliorating edge effects.

4.3.3.3 Real-world examples

As with many network metrics, the meaning and value of commonality can perhaps best be understood by considering how it is applied to a real-world example. In this section, examples from the street network of Birmingham will be used to illustrate its main properties. Figure 4.1 illustrates, using one section of the street network of Birmingham, the extent to which the centralities of street segments vary depend- ing on the perspective from which they are viewed. In order to establish a basis, 4.1a shows the betweenness of the links: this can be considered to be a ‘global’ perspective, and represents the extent of understanding which can be gained by considering the properties of links individually. The section shown includes an in- tersection between two highly-between roads (which appear to be arterial routes flowing north-south and east-west) and a number of more isolated side-roads with low betweenness values.

Figures 4.1b to 4.1e show commonality from the perspective of four particular links - e1, e2, e3 and e4 - and demonstrate that the true nature of traffic flow is somewhat

more nuanced. In particular, comparison of 4.1b and 4.1c reveals the disparity in flow patterns at the crossroads: e1 and e2 play identical roles at the junction, and

have very similar betweenness, yet there is a clear difference in the routes which use them. On the basis of commonality, it can be seen that the majority of journeys which use e1 also feature other links on the main north-south route; that is, it is

(a) Betweenness

0

0.2

0.4

0.6

0.8

1

e1 (b) Commonality with e1 e2 (c) Commonality with e2 e3 (d) Commonality with e3 e4

(e) Commonality with e4

Figure 4.1: Betweenness and commonality for a small section of the street net- work of Birmingham, using measures based on a limited radius of 2,000 metres. Panel a) shows links coloured according to betweenness, the notion of centrality from which commonality is derived. In panels b) to e), links are coloured accord- ing to their commonality relative to a specific choice of link - respectively e1, e2, e3 and e4 - showing the degree to which they feature in the same set of journeys. Colours therefore reflect values of C_ij(r) for specific choices of i.

Relatively few of the journeys which use e1 involve a turn at the central junction,

and the same is true of e2. The notion of commonality is necessary to reveal this:

from the perspective of e1, all three adjacent links at the central junction have the

same betweenness and are therefore indistinguishable on that basis.

Figures 4.1d and 4.1e show further examples of the added insight afforded by com- monality. Link e3 is a low-betweenness segment which is immediately adjacent to a

highly-between route; in addition, because it is a cul-de-sac, all journeys which use it must also include one of the two segments to which it is adjacent. Commonality reveals, however, that the load is not evenly distributed between the two, as it can

be seen that the more southerly of the two features in a greater proportion of trips. The most obvious reason for this is that the segment connects e3 to the main local

crossroad, and so will be used as journeys are ‘funnelled’ towards the main roads that meet there. This demonstrates another sense in which the commonality of two links is not simply a function of the distance between them.

Figure 4.1e shows a similar, though more extreme, case. Link e4 is, again, a cul-de-

sac, but is not immediately adjacent to a high-betweenness link. The commonality of one neighbour is only negligibly less than 1, since almost all journeys which use e4

originate/terminate somewhere in the main part of the network and must therefore incorporate that link. The other neighbour of e4 has only very small commonality,

since only one shortest path uses both links. This can be reconciled with human travel patterns: a pedestrian who traverses e4 is virtually certain to have also used

the high-commonality neighbour, whereas only one particular circumstance would involve the use of the other neighbour.

Each of the other situations can be translated straightforwardly into the language of pedestrian movement and awareness spaces. If a pedestrian is known to use e1,

for example, it can be said with high probability that his or her awareness space also includes other parts of the main north-south route. Other segments, however, are only used in a small number of circumstances, and are therefore less likely to be in the awareness space of a randomly-selected user of e1.

4.3.3.4 Analysis via discrete choice

The commonality metric introduced above provides a means of estimating the ex- tent to which network links tend to co-occur in travel patterns and, by extension, in the awareness spaces of pedestrians. According to the theoretical argument outlined in Section 4.3.3.1, this should correspond, to some extent, to the elevation in risk experienced by nearby locations in the aftermath of an initial victimisation. There are a number of methods by which this hypothesis can be tested.

The aspect of (near-)repeat victimisation over which commonality is expected to exert influence is the location at which the follow-up incident occurs (in practical terms, the street segment on which it occurs). It is, therefore, natural to take the occurrence of such a (near-)repeat pair as the starting point for analysis; that is, to examine whether, among all such pairs of incidents, there is a trend in the location of the second incident. Implicit in this approach is that the occurrence of the secondary offence is a fait accompli ; that is, that only the location of the second offence is in question, rather than its occurrence. This echoes the use of discrete choice models elsewhere in criminology, where target choice is treated independently of the initial decision to offend (e.g. Bernasco, 2009). It should be noted that, at this stage, the definition of a (near-)repeat is generic: it refers only to a pair of incidents which occur within D streets and T days of each other. For consistency, the two incidents will be referred to as ‘initial’ and ‘secondary’.

When these are identified in data, all pairs which meet the criteria are included; this allows the possibility, for example, that the same crime may be the secondary incident in more than one pair. Although this may be considered undesirable, the absence of a method by which crimes can definitively be linked means that some compromise must be made in this respect. Two arguments can be made in favour of using all pairs. On one hand, where an incident appears multiple times, it may be the case that all linked pairs in which it is involved are valid (in the sense of a common offender) and so all are the result of a common targeting process. In a more practical sense, this corresponds to what would be available in a predictive scenario: given an incident, it is possible that it may trigger multiple near-repeats in different locations.

Working with pairs of incidents, defined as above, as the fundamental unit, there are a number of ways in which the locations of the secondary incidents could be analysed. One possibility would be to examine the characteristics of the places at which they occur, either in terms of their network properties or otherwise. Although it may be possible, by this method, to identify features which tend to be shared by

secondary targets, it overlooks a crucial aspect of the situation; namely, that there

In document Spatio-temporal modelling for issues in crime and security (Page 162-187)