Abstract Crime is a major threat to society’s well-being but lacks a statistical characterization that could lead to uncovering some of its underlying mechanisms. Evidence of nonlinear scaling of urban indicators in cities, such as wages and serious crime, has motivated the understanding of cities as complex systems—a perspective that offers insights into resources limits and sustainability, but that usually neglects details of the indicators themselves. Notably, since the nineteenth century, criminal activities have been known to occur unevenly within a city; crime concentrates in such way that most of the offenses take place in few regions of the city. Though confirmed by different studies, this concentration lacks broad analyses on its characteristics, which hinders not only the comprehension of crime dynamics but also the proposal of sounding counter-measures. Here, we developed a framework to characterize crime concentration which divides cities into regions with the same population size. We used disaggregated criminal data from 25 locations in the U.S. and the U.K., spanning from 2 to 15 years of longitudinal data. Our results confirmed that crime concentrates regardless of city and revealed that the level of concentration does not scale with city size. We found that the distribution of crime in a city can be approximated by a power-law distribution with exponent α that depends on the type of crime. In particular, our results showed that thefts tend to concentrate more than robberies, and robberies more than burglaries. Though criminal activities present regularities of concentration, we found that criminal ranks have the tendency to change continuously over time—features that support the perspective of crime as a complex system and demand analyses and evolving urban policies covering the city as a whole.

Citation: Oliveira M, Bastos-Filho C, Menezes R (2017) The scaling of crime concentration in cities. PLoS ONE 12(8): e0183110. https://doi.org/10.1371/journal.pone.0183110 Editor: Eduardo G. Altmann, University of Sydney, AUSTRALIA Received: April 7, 2017; Accepted: July 28, 2017; Published: August 11, 2017 Copyright: © 2017 Oliveira et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All crime data are official open data sets that are available as described in the Supporting Information file. Funding: Marcos Oliveira received funding from CAPES Foundation under grant 1032/13-5. This work was partially supported by the Army Research Office under grant W911NF-17-1-0127-P00001. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

Introduction Cities are the fundamental drivers of human societies; their capability to bring individuals together fosters innovation, wealth creation, and economic growth, but unfortunately they suffer from problems such as pollution, disease spread, and more pervasively, crime. Yet, even though crime is a danger to the development of cities, and counter-measures are greatly desired, we still fail to understand its structure and dynamics [1, 2]. Notably, the interconnected dimensions in cities, such as social and infrastructural, coupled with their natural dynamics, requires an understanding of cities not as static objects or locations but as complex systems [3, 4]. This point of view has provided the means to comprehend the growth of cities and its impact on urban indicators, such as employment, patent, wage, and crime [5–15]. Still, only a few studies have taken into account the intricacies of these indicators when analyzing allometric relationships of cities [13–16]. For almost two centuries, however, crime in cities has been known to be unevenly distributed [17, 18]. Criminal events concentrate in such way that most of the offenses happen in very few regions [19]. Still, this aspect of crime has never been objectively characterized, albeit confirmed in different locations. Such characterization has the potential to help researchers to create realistic models of crime and to present the grounds to understand the impact of local activities on global patterns of cities. The very notion of a city bringing people together to interact comprises the idea of emergence of self-organized coordination derived from local activities [20–23]. Despite the apparent individual disorder in the decisions and processes at local levels, cities exhibit several regularities that are argued to be a result of the need to expand and to develop [4, 23–32]. These findings support the perspective of cities as complex systems and have helped to understand various aspects of cities [6–11]. Several urban indicators have been found to scale with the population size N of the city according to a law of the form: (1) where the exponent β relates to the class of the indicator [6]. For aspects associated with infrastructure (e.g., roads, gasoline stations), the quantities scale sub-linearly, while sociological dimensions, such as innovation, wealth, or crime, present superlinear scalings—though the scaling depends on the city definition and the model for Pr(Y|N) (see S1 Text) [33–35]. In the case of sublinear scalings, cities utilize resources more efficiently as they grow, while superlinear relationships imply more accumulation in larger cities. The superlinear scaling is claimed to be associated with population density and human interactions in cities [8–11]. As individuals meet in space and time, simple principles on the formation of ties can explain the existence of regularities in urban indicators, despite idiosyncrasies of each city [8]. Such models and analyses disregard, however, details of urban indicators such as variations across the city, likely due to the lack of high-granularity data. Still, social media and mobile phone data have been used to demonstrate that human interactions scale super-linearly with city size while the probability 〈p c 〉 that two peers of an individual interact presents scale-invariance with 〈p c 〉 ≈ 0.25 [14, 16]. Such features imply efficient spreading processes in the social network when cities grow and suggest the emergence of regularities in urban indicators as an outcome of patterns in human interactions [14]. Accordingly, human dynamics also play a major role in criminal activities, which are likely to drive patterns in crime activity [2, 36–38]. In fact, empirical evidence has shown that crime presents a remarkable regularity of concentration in several dimensions that relate to context (e.g., target, location, offender) and to features (e.g., spatial, temporal, type of crime) [39]. In particular, the spatial concentration of crime exists in such way that, regardless of granularity level, some areas have disproportionately more crime than others—popularly called hotspots [19]. The phenomenon has been confirmed in different cities using various spatial aggregation units including street and area level (e.g., street segments, census tracts, blocks) [40–42]. Such ubiquity motivated the proposition of the law of crime concentration which states that a small number of micro-geographic units account for most of the offenses in a neighborhood or city [19]. Yet, the use of distinct approaches to aggregate criminal events hinders an objective definition of crime concentration—though necessary to confirm the existence of the phenomenon. Even when the same type of aggregation unit is used, analyses might be biased due to particularities in the units of the cities (e.g., street segments). The lack of a more general framework for analyzing the spatial distribution of offenses prevents the general characterization of crime concentration. Such framework enables the examination of allometric scaling in cities regarding the clustering of crime and its dynamics as well as to assess signatures in different types of crime. The characterization of crime concentration paves the way for unveiling the very mechanisms that underlie the phenomenon in cities. Yet, an unbiased assessment of any regularity in crime needs to consider the relationship between population and crime, and thus an ideal framework must employ aggregation units that take into account the population in each unit [6, 12, 43–45]. Here we develop a framework to assess the distribution of criminal activities in cities by dividing the area of a city into regions with equal population size and aggregating offenses that happened within the same regions. This general framework allows us to perform a comprehensive analysis of the allometric relationship between crime distribution and city population. We examined criminal data from locations in the United States and the United Kingdom, and found that not only crime concentrates regardless of city, but also population size does not have an influence on the levels of concentration—despite the relationship between crime and total population. Crime concentration manifests in the probability distribution of crime across a city which can be described by a power law (2) where the exponent α relates to the type of crime. From the perspective of cities as complex systems, our results indicate cities, and thus crime, growing in such a way to maintain the concentration of crime. To evaluate the dynamics of crime we measured the entropies of the ranks of criminal regions in the cities. We found that the certainty about the region in a position of the rank decreases exponentially with the position rank, which implies that we have only confidence about few of the most criminal regions of a city. The high fluctuation of crime across the city suggests that crime in cities is not in a state of equilibrium, despite the regularity in the concentration of offenses; such features support the viewpoint of crime as a complex system. This perspective encourages crime analyses that cover the whole city, instead of the focus on criminal hot spots. Our work sheds light on the challenges posed by the increasing number of people in cities which demands strategies towards sustainable development.

Discussion Crime is ubiquitous in cities but needs still quantitative understanding. To characterize crime in cities, we examined criminal activities in 25 locations from two different countries using longitudinal data sets spanning 2 to 15 years. We developed a method to assess the spatial concentration of crime which divides a city into regions based on the resident population; then analyzed the distribution of crime in the regions. In all considered cities, we were able to confirm previous studies and identified that offenses take place in few regions of a city. Here we performed a comprehensive statistical characterization of the phenomenon in cities and showed that not only crime concentrates but also presents concentration level that depends on the type of crime and exhibits independence of the size of the city—despite the relationship between population and number of crimes. Yet, though cities have such regularity in the concentration of crime, our results revealed that criminal ranks in the cities have the tendency to change over time. The regularities in the concentration of crime coupled with the constant displacement of crime suggest an understanding of crime as a complex system. Criminal activities flow continuously across the city while maintaining the organization of the system in such way that its dynamics and regularities appear to be scale-invariant. Different types of crime exhibit particular dynamics that lead to distinct levels of concentration and allometric scaling laws. Our results revealed thefts presenting a well-behaved concentration over cities which indicates invariance with city size and with idiosyncrasies of cities; while burglaries and robberies are more dependent on the city. These findings are particularly intriguing in light of the superlinear scaling found in thefts in contrast to the linearity in burglaries—though we are still in need of more conclusive analyses on the scaling laws of robberies (see S1 Text). Such regularities in crime concentration might be linked to the way crime scales in cities. The characterization of crime paves the way for a better understanding of crime dynamics and provides the means to create and validate models. Though the proposal of a generative mechanism is beyond the scope of the present study, our framework can be employed for modeling given its implicit network of regions which can be used to represent a city. A theory or model attempting to explain this complex phenomenon have to conform to the skewed distribution of crime and the existence of distinct concentrations of offenses for different types of crime. For instance, models for burglary are expected to be more dependent on features of the city such as the layout of the streets or demographics. One should not conclude that we argue for any universality of power laws here, but instead we present statistical characteristics in criminal activities which we systematically found in different locations [47]. The perspective of crime as a complex system demands analyses that need to cover the system as a whole in order to assess crime. The connectedness of the city suggests that one should resist to neglect the “cold” areas by studying solely the hotspots of crime. Moreover, our results suggest that areas of high concentration of crime are expected to exist as the city grows—finding that urges for proper government policies. Still, the notion of the city as a process implies that developing static policies is likely to fail and, as such, policy-makers should pursue evolving strategies based on real-time data [48]. Urban planners may take advantage of our framework to analyze different types of criminal regions and categories of crime dynamics. Such objective analyses of the city have the potential to assist sustainable urban development, not only regarding crime but also with respect to other demographics.

Methods Data sources Since police departments employ different nomenclature for types of crime as well as different subcategory of offenses, we preprocessed the records in order to group together thefts, burglaries, and robberies (as described in S1 Text). For the spatial analysis, we considered the bounding box of the U.S. cities and the jurisdiction of the U.K. constabularies. In the case of the temporal analysis, we analyzed only the U.S. cities because the U.K. data include solely the month when offenses occurred. The sources for all the criminal data sets and census bases are further described in S1 Text. Splitting cities To split a city into regions with same population size, we use census data in order to build a graph with nodes that represents roughly the same number of people and divide this graph into R partitions. To construct the graph, a set s i of p i random coordinates is created for each census block b i of a place L, where p i is the number of people in b i and each x–y coordinate is uniformly generated within the geographical shape of the block. The nodes of the graph are created based on the cells of each Voronoi diagram v i that is constructed from each s i , and the edges between nodes exist if their respective cells are neighbors of each other. Finally, this graph can be partitioned using a graph partitioning algorithm in order to generate regions (i.e., partitions) with approximately the same population size [49]. Still, to properly analyze crime in a given city c with this method, a value for R c has to be chosen to allow us to examine crime distribution. In all data sets we analyzed, we found that the number of regions that contain at least one offense Rn ≥ 1 increases with the total number of regions R, until Rn ≥ 1 saturates at a point Rn ≥ 1(r u ) = u in which new regions do not have any crime occurring within them. A plausible reason for such behavior is the accuracy level used in police offices as offenses are registered in the criminal systems. In order not to bias our results with any particularity of such procedures, we have to set R c = ρr u with ρ lesser than the unit and sufficiently high to avoid any averaging problem [50], thus for all data sets, we define ρ = 0.9 (see S1 Text).