Toby Davies

A mathematical model of the London riots and their policing

In August 2011, several areas of London experienced episodes of large-scale disorder, comprising looting, rioting and violence. Much subsequent discourse has questioned the adequacy of the police response, in terms of the resources available and strategies used. In this article, we present a mathematical model of the spatial development of the disorder, which can be used to examine the effect of varying policing arrangements. The model is capable of simulating the general emergent patterns of the events and focusses on three fundamental aspects: the apparently-contagious nature of participation; the distances travelled to riot locations; and the deterrent effect of policing. We demonstrate that the spatial configuration of London places some areas at naturally higher risk than others, highlighting the importance of spatial considerations when planning for such events. We also investigate the consequences of varying police numbers and reaction time, which has the potential to guide policy in this area.

Modelling patterns of burglary on street networks

A fundamental issue in crime prevention is the efficient deployment of resources and the effective targeting of interventions, both of which require some form of prediction of future crime. One crime for which this is feasible is burglary, the distinctive spatio-temporal signatures of which can be exploited to inform predictions. Mathematical models in particular are capable of both encoding concisely the theoretical foundations of criminal behaviour and allowing the quantitative analysis of specific scenarios, and their capacity to reproduce the general patterns of burglary suggests that the approach has considerable potential. Previous models, however, are situated on simplified representations of space and do not reflect realistically the built environment in which crime takes place; specifically, they do not incorporate urban street networks. Such networks are fundamental to situational theories of crime, in the sense that they determine the configuration of urban space and, therefore, shape those human activity patterns which are thought to give rise to crime. Furthermore, streets are the natural domain for many policing activities, and their structure is determined by planning decisions, so that insight into their relationship with crime is likely to be of immediate practical use. With this in mind, this paper presents a mathematical model of crime which is explicitly situated on a street network. After discussing theoretical considerations and specifying the model itself, examples of typical networks are explored.

A dynamic spatial model of conflict escalation

In both historical and modern conflicts, space plays a critical role in how interactions occur over time. Despite its importance, the spatial distribution of adversaries has often been neglected in mathematical models of conflict. In this paper, we propose an entropy-maximising spatial interaction method for disaggregating the impact of space, employing a general notion of ‘threat’ between two adversaries. This approach addresses a number of limitations that are associated with partial differential equation approaches to spatial disaggregation. We use this method to spatially disaggregate the Richardson model of conflict escalation, and then explore the resulting model with both analytical and numerical treatments. A bifurcation is identified that dramatically influences the resulting spatial distribution of conflict and is shown to persist under a range of model specifications. Implications of this finding for real-world conflicts are discussed.

STREET NETWORK STRUCTURE AND CRIME RISK: AN AGENT-BASED INVESTIGATION OF THE ENCOUNTER AND ENCLOSURE HYPOTHESES†

Street networks shape day-to-day activities in complex ways, dictating where, when, and in what contexts potential victims, offenders, and crime preventers interact with one another. Identifying generalizable principles of such influence offers considerable utility to theorists, policy makers, and practitioners. Unfortunately, key difficulties associated with the observation of these interactions, and control of the settings within which they take place, limit traditional empirical approaches that aim to uncover mechanisms linking street network structure with crime risk. By drawing on parallel advances in the formal analyses of street networks and the computational modeling of crime events interactions, we present a theoretically informed and empirically validated agent-based model of residential burglary that permits investigation of the relationship between street network structure and crime commission and prevention through guardianship. Through the use of this model, we explore the validity of competing theoretical accounts of street network permeability and crime risk—the encounter (eyes on the street) and enclosure (defensible space) hypotheses. The results of our analyses provide support for both hypotheses, but in doing so, they reveal that the relationship between street network permeability and crime is likely nonlinear. We discuss the ramifications of these findings for both criminological theory and crime prevention practice.

Event Networks and the Identification of Crime Pattern Motifs.

In this paper we demonstrate the use of network analysis to characterise patterns of clustering in spatio-temporal events. Such clustering is of both theoretical and practical importance in the study of crime, and forms the basis for a number of preventative strategies. However, existing analytical methods show only that clustering is present in data, while offering little insight into the nature of the patterns present. Here, we show how the classification of pairs of events as close in space and time can be used to define a network, thereby generalising previous approaches. The application of graph-theoretic techniques to these networks can then offer significantly deeper insight into the structure of the data than previously possible. In particular, we focus on the identification of network motifs, which have clear interpretation in terms of spatio-temporal behaviour. Statistical analysis is complicated by the nature of the underlying data, and we provide a method by which appropriate randomised graphs can be generated. Two datasets are used as case studies: maritime piracy at the global scale, and residential burglary in an urban area. In both cases, the same significant 3-vertex motif is found; this result suggests that incidents tend to occur not just in pairs, but in fact in larger groups within a restricted spatio-temporal domain. In the 4-vertex case, different motifs are found to be significant in each case, suggesting that this technique is capable of discriminating between clustering patterns at a finer granularity than previously possible.

Probabilistic Map-Matching for Low-Frequency GPS Trajectories

The ability to infer routes taken by vehicles from sparse and noisy GPS data is of crucial importance in many traffic applications. The task, known as map-matching, can be accurately approached by a popular technique known as ST-Matching. The algorithm is computationally efficient and has been shown to outperform more traditional map-matching approaches, especially on low-frequency GPS data. The major drawback of the algorithm is a lack of confidence scores associated with its outputs, which are particularly useful when GPS data quality is low. In this paper, we propose a probabilistic adaptation of ST-Matching that equips it with the ability to express map-matching certainty using probabilities. The adaptation, called probabilistic ST-Matching (PST-Matching) is inspired by similarities between ST-Matching and probabilistic approaches to map-matching based on a Hidden Markov Model. We validate the proposed algorithm on GPS trajectories of varied quality and show that it is similar to ST-Matching in terms of accuracy and computational efficiency, yet with the added benefit of having a measure of confidence associated with its outputs.