Characteristics of disorder

In seeking insight into the behaviour of individuals during an episode of rioting, we consider both existing theoretical research into such incidents36 (and criminal activity in general) and specific observations from the London disorder. The latter takes the form of analysis of data provided by the Metropolitan Police, which contains the details of all individuals arrested in relation to the riots and matches the home addresses and offence locations of suspects. Since it is typically argued that individuals act rationally during a riot (i.e. that their decisions are based on some cost/benefit analysis)16,17,18 these observations can be used to inform a model of the actions of rioters.

A fundamental observation is the predominant targeting of retail sites, reflecting the acquisitive nature of much offending. Crimes against commercial premises, including both acquisitive crime and criminal damage, accounted for 51% of all offences in the UK as a whole37 and offences clustered in areas such as Clapham Junction, Croydon, Ealing and Brixton. This can be immediately reconciled with crime pattern theory23; the richness of opportunity at retail premises is likely to be common knowledge amongst riot participants and they therefore act as crime attractors. In line with this, for our model we adopt a system of retail centres as the sites of disorder.

We also consider the origins of offenders, i.e. the locations of their residences, and, therefore, the distances they travelled to the sites where they offended. As seen in Figure 1a, the flows of offenders follow a clear distance-decay relationship. Although statistical tests38 find that the distribution does not correspond to most common forms, the best fit is provided by an exponential distribution with parameter 0.274. An offender's perception of distance does not necessarily aggregate to an exponential distance decay, since other factors, some of which are temporally-varying, are likely to contribute and we nevertheless incorporate an exponential distance decay within in our model. Distributions such as these are reminiscent of those seen in the analysis of flows in retail systems39,40 and so, noting also the central role of commercial centres, we model the behaviour of rioters partly by analogy with this.

Figure 1 Observations from arrest data. (a) Log-linear plot of the complementary cumulative distribution function of D, the distance between residential and offence locations. The straight line shows a hypothetical exponential distribution with parameter 0.274 (±0.01 for a 95% confidence interval), for which the Kolmogorov-Smirnov distance statistic is 0.0246 (which compares with 0.332 for the equivalent fitted power-law). (b) Lorenz curve for the distribution of riot locations amongst Lower Super Output Areas (LSOAs; UK census units with average population approximately 1,500) ranked according to deprivation (where 1 is most deprived). The dashed line represents perfect equality. (c) Relationship between area-level deprivation and the proportion of residents involved in disorder, where the horizontal axis represents a score derived from IMD so that all values lie in [0, 1] and so that London's most deprived area is given a value of 1. d) Temporal distribution of recorded crime. Full size image

Analysing the riot locations further, we explore the relationship between deprivation and offending. Figure 1b shows that a disproportionately high number of offences occurred in more deprived areas (approximately 50% within the 20% most deprived), using the UK's Index of Multiple Deprivation (IMD) to rank census units. Looking instead at suspects' residences, Figure 1c shows the average proportion of riot suspects for groups of LSOAs ordered by deprivation, where in this case, anticipating its incorporation as a variable in the model, we use a deprivation score based on IMD ranking. A relationship between offending and deprivation has also been found elsewhere37 and youth unemployment and child poverty have also been identified41. That the most deprived areas acted disproportionately as both origins and destinations will clearly influence the distance distribution and vice versa, but work elsewhere shows that both effects persist when controlling for the other21.

With this notion in mind, we incorporate the deprivation score discussed above as a feature within our model, allowing for a higher probability of offending in deprived areas.

We also note distinctive temporal patterns in the riot data, as seen in Figure 1d. From the small initial disturbance, incidents escalated in volume and intensity on each successive day, with police response growing in line with this, from 3,480 on Saturday evening to 16,000 by Tuesday5. This may also be seen at the scale of individual days, where the majority of criminality took place at night and built to a peak in the early hours. Whilst various explanations for this have been put forward, a particularly compelling one suggests that awareness of disorder provided a self-reinforcing stimulus to rioter involvement6 and a contagion-based model is therefore appealing.

Model

We develop a mathematical model with the aim of exploring the spatial and temporal patterns of the events in London. Recognising that non-linearities inherent in the system imply a significant dependence on initial conditions (which are unknown) and that numerous factors not considered here are likely to play a material role, we do not seek to replicate exactly the London events. Rather, we aim to produce a ‘generative’ type model which can give rise to realistic patterns and macro-level behaviour that insight might plausibly be gained through analysis of the underlying dynamics.

Our model draws on elements of several existing ones; our contribution is in their combination and adaptation to produce an integrated spatial model of disorder and in the analysis of varying police strategies. The model can be divided into three components: an epidemiological model for riot participation, a spatial interaction model (SIM)42 for the spatial allocation of rioters and police and a model for interaction between rioters and police previously applied in the context of civil violence26.

General concepts

The model is defined across a discrete system of two entities: residential areas and retail centres. These are indexed by i and j respectively and embedded in space and we use LSOAs and defined ‘retail cores’ when considering London. Participating individuals are notionally tracked through the system via a logical sequence which involves a decision to participate taking place at their home, a choice of site at which to offend and possible removal due to arrest by police officers. These officers are active at all times but may move and be located according to different principles.

To model rioters' decisions, some concept of the attractiveness of a riot site is required. This is formulated using a ‘cost/benefit’ structure, as is normal for SIMs, where benefit represents the potential reward at a site and cost embodies both travel cost and the deterrent effect of police.

We assume that the benefit for site j is given by the logarithm of Z j , a non-dimensional measure of its relative value (e.g. the ratio of j's floorspace to the mean across the system), as is a standard assumption in retail models of this type43,44, reflecting diminishing returns to scale. For b ij , the benefit of site j as perceived by an individual in i, we therefore have

Turning to deterrence, we suggest that the primary gauge by which an individual assesses whether the situation at a site is conducive to riot is the probability of arrest, determined by the relative numbers of rioters and police: low perceived chance of capture encourages participation. Several such expressions for probability of arrest have been proposed; in this case we take an adapted version of the formulation of Epstein26 as our starting point:

where Q j is the number of police officers in j, D j the number of rioters in j and a the number of police officers required, on average, to ‘contain’ one rioter. The use of the floor function has empirical motivation; the Metropolitan Police review of the London disorder11 explicitly states that “decisions were made not to arrest due to the prioritisation of competing demands…specifically, the need to protect emergency services, prevent the spread of further disorder and hold ground until the arrival of more police resources”. Accordingly, when the police are ‘outnumbered’ at a site (i.e. Q j < aD j ), the situation is considered to be out of control and the police are unable to make any arrests without the addition of ‘backup’ (and thus the probability is 0). On the basis that increased probability corresponds to increased deterrence, we therefore express deterrence thus:

We also incorporate a linear function of the distance between residential areas and riot sites, as is typical for analogous retail systems. Taking this as proportional to d ij , the distance between the centroids of i and j, we can then combine with (2) and (3) to obtain the full expression for benefit - cost:

where the w n are constants. The associated attractiveness term W ij which appears in the terms of the spatial interaction model can, as described elsewhere44, then be written as follows:

where α r , β r and γ r (which itself absorbs a) are parameters to be obtained in calibration with real-world data (the subscript r denoting reference to riot participants). It is through the form of (5) that an exponential distance decay, discussed in the previous section, features in the model.

Riot participation

Motivated by the hypothesis, consistent with the temporal progression of the riots, that exposure to nearby disorder had the effect of inciting participation, we propose a Susceptible-Infected-Removed (SIR) model45; that is, a mechanism akin to infection by which individuals transfer to an active rioting state according to their level of exposure. Recalling the correlation between propensity to riot and deprivation, we also incorporate this and the function we propose is therefore:

where ρ i is a measure of the deprivation in i (which we take to be based upon the IMD) and μ an exponent to be calibrated. A logistic function is used here to represent the existence of a threshold at which rioting becomes appealing; any transition is likely to be localised rather than gradual. Intuitively, this probability will be small when the overall attractiveness of potential riot areas is low, whereas, when the ‘ambient’ level of rioting is high, the probability of offending tends towards . From another perspective, where two areas were equally exposed to disorder, greater participation would arise in the more deprived of the two.

Translating this to the macro-level for a residential area i, we therefore find an expression for N i (t), the rate at which individuals choose to participate at time t. Under the assumption that decisions are independent between individuals, this is given by the product of population size and decision probability,

where η is an infection rate and I i (t) the number of inactive individuals resident in area i. We can now formulate expressions for I i (t) and R i (t), the number of rioters whose residence is in a given zone i, as well as their change in a time period [t, t + δt). These, along with their initial conditions (I i (0) is the residential population of i and R i (0) a seed of participants, to be chosen) determine the numbers of individuals of each type, in each residential area, at all times. The choice to structure the model in this way is motivated by our focus on the residential origins of rioters, since it enables us to understand the composition of rioting groups in these terms. At this stage we also include an extra term C i (t), to be fully defined later, for the rate at which participants from i are arrested at time t:

Spatial assignment

We assign active rioters to sites of disorder using an entropy-maximising SIM; the purpose of these models is to estimate the most probable flows in a spatial system such as ours, given certain constraints42.

Rather than incorporating the attractiveness function, W ij , directly into the spatial interaction equations, we use its moving average over a number of previous time steps, for several reasons: to account for factors such as travel time on the part of rioters, to represent ‘lag’ in the spread of information through the system and to dampen the effect of sudden fluctuations in attractiveness. The values used to determine the assignments at a given time, referred to as effective attractiveness and denoted , are therefore the average values of W ij over the L r most recent time steps in our discretised temporal scheme (which has intervals δt; when t < (L r – 1)δt, we ‘pad’ with the t = 0 value):

Following the standard entropy maximising derivation of a SIM44, it can be shown that S ij , an estimate of the number of rioters from i who are participating in disorder in j at time t is given by:

An identical expression for S ij may be formulated using an alternative derivation: by considering (4) as a utility term in a conditional logit model46. In either case, summing over residential areas i yields the total number of rioters D j in j:

It should be noted here that each time unit is therefore implicitly defined as the mean time taken for each participant to travel from a home location to a chosen riot site.

The assignment of police resources to areas of disorder is also realised via a SIM, as for riot participants; there are, however, noteworthy differences. First, police units have no ‘home’ location and are active and situated at potential sites of disorder at all times. The response lag L p is also different to that for rioters (and intended to be higher); reflecting the delay in learning of the plans and movements of rioters and conferring upon the rioters a degree of ‘first-mover advantage’.

The main difference for police, however, is in the attractiveness function, analogous here to the requirement for officers at a given site. Following a similar argument to that of the rioters seen in (4), we assume the benefit - cost of police follows:

This expression (13) includes no spatial decay term, reflecting the fact that the police do not prioritise incidents on the basis of proximity10 and can travel to incidents rapidly. In addition, the second term is a function of rioter numbers only: given that their aim is to eliminate all disorder, the number of police already at a site is likely to be immaterial to the police. As in (5) and described elsewhere44, the attractiveness function V j representing police requirement, is therefore:

where α p and γ p are, as before, parameters to be calibrated which encode the relative importance of the two factors. Following the identical process seen with (5) above, we may first calculate effective requirement to take into account time lags in the system,

and, in conjunction with a SIM, as in (11) and (12), can derive an expression for the total number of police officers in location j at time t:

where P is the total number of police officers in the system.

Interaction between police and rioters

To model the interaction of police and rioters, we return to the mechanism of arrest and its associated probability described previously. This gives the probability of capture for an individual rioter and multiplying by the number of participants present therefore gives the expected number arrested. Since, for reasons explained previously, we classify participants by residential location, this is done separately for each area to give C i (t), the rate at which individuals who originated in i are arrested at time t:

where τ is an arrest rate parameter.

Demonstration case

As a step towards verification of the model and to establish a ‘base case’ for further investigation, a series of numerical simulations were run, representing the escalation of events during a typical evening. Individual simulations ran for 10 time units (where one unit is the time taken for a rioter to travel to their destination) and involved sequential iteration through the model equations in the order (12), (16), (8). The system was seeded with 100 riot participants, assigned to residential areas in proportion to population and allocated to sites of disorder according to the static component of attractiveness (i.e. ). Similarly, 5,000 police officers (the approximate number deployed on each of the first 3 days in London) were initially placed at retail sites according to . Given the high dimensionality of the model, many parameter sets were found to yield feasible results. To focus our discussion an example parameter set was chosen (Table 1) which gives rise to outcomes broadly in agreement with the features observed in the data, both in terms of borough level participants (Figure 2) and distance decay (Figure 3).

Table 1 Parameters used in base case simulation Full size table

Figure 2 Borough level choropleth of rioter residential locations from (a) data and (b) simulated results. Although the extreme dependence on initial conditions precludes our model from generating an exact replica of the observed incidents, the results show good qualitative agreement, with 26 of the 33 boroughs showing rioter percentages in the same or adjacent bands as the data. The remaining discrepancy may be accounted for by factors specific to the London disorder, such as communication between groups, other activity patterns occurring at the time, or social factors beyond the scope of this work. The labels 1,2,3,4 correspond to retail centres in Brixton, Croydon, Clapham Junction and Ealing respectively, which are considered individually in our later simulations. Full size image

Figure 3 Log-linear plot of the complementary cumulative distribution function for D, the distances between residences and offence locations within the demonstration simulation. Full size image

Since the riots occurred over 5 days, with incidents initialised in various locations across that period, these aggregate results offer little validation other than to confirm that the model is capable of replicating the general characteristics of the data. By instead initialising small incidents at just two locations, rather than simultaneously across the city, we may explore the susceptibility of retail centres. Such initialisation is also reflective of the way in which real incidents are thought to arise: many of the outbreaks began as small local gatherings of unrest11,7.

Our analysis considers retail sites which were worst affected: Brixton, Croydon, Clapham Junction and Ealing. We ran four such simulations in each case, pairing the site of interest with each of its closest geographical neighbours. In all simulations (Figure 4) the centres which experienced widespread rioting in reality also saw substantial growth from the initial small disturbance in the model, while the vast majority of other retail locations saw incidents decay to zero. These results serve as further validation, but also, given the structure of the model, offer insight into why some sites were more susceptible than others, since the dynamics are based on a combination of factors: proximity to populous areas of high deprivation and the balance of centre size and police presence. These are important results, as such an approach might be applied as an indicator of future susceptibility.

Figure 4 The susceptibility of retail sites. For each of the four centres worst affected in the riots: (a) Brixton, (b) Croydon, (c) Clapham Junction and (d) Ealing, we ran four separate simulations, pairing the site of interest with each of its closest geographical neighbours in turn. An initial disturbance of one rioter was included at both sites and the model run to allow the incidents to evolve. Results shown for the sites of interest are the average of their four simulations and in each case substantial growth is seen, particularly in comparison to the neighbouring centres. Full size image

Police resources and response

To gain quantitative insight into the level of police resource required to maintain control in a situation such as London's, we used the results of our demonstration case to analyse the effect of policing configuration on the development of disorder. To meaningfully compare realisations of the system, we define a quantity severity to summarise the cumulative disorder, given by the overall extent to which police are outnumbered by rioters:

Two parameters were varied independently in our simulations - total police P and response lag L p - with parameters as in Table 1 otherwise and results are shown in Figure 5. Police numbers correspond to those seen in London and reflect what was seen in data: numbers above approximately 10,000 appear sufficient to suppress disorder. In the case of speed of response, the difference in severity as L p increases, relative to a base case of L p = 0, is plotted. After a noisy stage at small values, the severity appears to increase with lag. Although the increase is small as a proportion of absolute value, it should be borne in mind that these simulations are run with parameter values chosen such that a certain level of severity is assumed. Any changes, therefore, are variations around a level which has been implied a priori by other factors, such as police and rioter numbers. As expected, the trend observed reflects the importance of delivering police to scenes of disorder before control is lost. The same simulations were also run for other police configurations - specifically where police are assigned to locations initially, either uniformly or proportionally with and remain static throughout - but results differ only slightly from the dynamic case and are not shown.