Mathematical analysis of crime dynamics in and out of prisons

Recently, there have been many attempts to develop a mathematical model that captures the nature of crime. One of the successful models has been based on diffusion‐type differential equations that describe how criminals spread in a specific area. Here, we propose a dynamic model that focuses on the effect of interactions between distinct types of criminals. The accumulated criminal records show that serious and minor crimes differ in many measures and are related in a complex way. While some of those who have committed minor crime spontaneously evolve into serious criminals, the transition from minor crime to major crime involves many social factors and has not been fully understood yet. In this work, we present a mathematical model to describe how minor criminals turn into major criminals inside and outside of prisons. The model assumes that a population can be divided into a set of compartments, according to the level of crime and whether arrested or not, and individuals have equal probability to change compartment. The model is design to implement two social effects which respectively have been conceptualized in popular terms “broken windows effect” and “prison as a crime school.” Analysis of the system shows how the crime‐related parameters such as the arrest rate, the period of imprisonment, and the in‐prison contact rate affect the criminal distribution at equilibrium. Without proper control of contact between prisoners, the longer imprisonment rather increases occurrence of serious crimes in society. An optimal allocation of the police resources to suppress crimes is also discussed.     

对警力与犯罪相互关系的实证研究

文章利用江苏省"十五"期间的面板数据对警力与犯罪问题进行实证研究。与以往文献不同,我们把一线民警从整体警力中剥离出来,认为一线民警的数量与犯罪率存在双向因果关系。通过对联立方程组进行三阶段最小二乘估计(3SLS),我们发现,刑事案件以及治安案件案发数对一线民警比重没有显著的影响;增加一线民警比重能够显著减少刑事案件相对数,而对治安案件却没有显著影响;此外,警察总量对刑事案件以及治安案件的发生都没有显著影响。进一步研究发现,在经济社会环境不同的地区(例如苏南、苏北),尽管案发绝对数存在显著差异,但是警力与犯罪的相互关系并没有表现出显著的差异。     

Key Player Policies When Contextual Effects Matter

We consider a model where the criminal decision of each individual is affected by not only her own characteristics, but also by the characteristics of her friends (contextual effects). We determine who the key player is, i.e., the criminal who once removed generates the highest reduction in total crime in the network. We propose a new measure, the contextual intercentrality measure, that generalizes the one proposed by Ballester, Calvó-Armengol, and Zenou (2006 Ballester , C. , Calvó-Armengol , A. , &; Zenou , Y. ( 2006 ). Who's who in networks. Wanted: The key player . Econometrica , 74 , 1403 – 1417 .[Crossref], [Web of Science ®] , [Google Scholar]) by taking into account the change in contextual effects following the removal of the key player. We also provide an example showing that the key player can be different whether contextual effects are taken into account or not. This means that the planner may target the wrong person if it ignores the effect of the “context” when removing a criminal from a network.     

融合多变量主成份与因子分析算法的犯罪率研究

基于犯罪过程的不确定性结构,融合多变量主成份分析与因子分析算法,分析了影响犯罪率的城市化率、失业率、客运量、人均GDP、城市基尼系数、农村基尼系数等多个相关变量因素,由MATLAB仿真建立了犯罪率的多元回归模型,并采用MINITAB数学软件对浙江省1988-2007二十年样本数据进行了实测,获取了影响犯罪率的潜在因子与方差贡献率,讨论了多元回归模型的统计检验及相对误差,解决了犯罪率各相关变量的权重分布及信息相互重叠问题.     

Generalized lot sentencing rule for estimating the value of production lot for sale

In the traditional lot sentencing rule, a buyer arrives to one of two decisions regarding lot disposition; either accept or reject a lot. However, it is more appropriate to consider choices between those two extreme decisions. A clear case where the traditional lot sentencing rule is not flexible is when a buyer purchases a lot from an English auction. In this paper, we propose a model that helps a buyer in estimating the value of a production lot. This model can be used by a bidder before the bidding process starts to estimate the value of an auctioned lot. The model provides an action plan that includes the estimated acquisition cost as a function of the number of defective items found in a random sample. Unlike the traditional lot sentencing rule, the proposed rule is more flexible and provides buyers with wider range of possible actions.     

On Greenberg's Two-Step Model of Criminal Recidivism

With more than two million people currently incarcerated in U.S. prisons and jails, many of them repeat offenders, there is widespread agreement that criminal recidivism is a serious and costly problem in both human and economic terms. In this article, a stochastic model, which is a somewhat simplified version of a model first proposed by Greenberg, is discussed. An argument for the model is made based on recent theoretical and empirical studies that demonstrate the importance of social bonds in both the commission of and the abstinence from crime. This argument provides a stronger theoretical basis for the model, since it suggests that the model's trichotomization of a cohort of released prisoners reflects truly discrete differences and not just a convenient simplification imposed on a continuous variable. The model is shown to fit trial data sets very well, enabling the measurement of asymptotic recidivism—the ultimate percentage of a release cohort who will recidivate.     

The Effect of Vehicular Flow Patterns on Crime and Emergency Services:

The use of cul-de-sacs and one-way streets has been proposed as a means to reduce crime in several locales. Closing streets or restricting flow patterns on them make travel through neighbourhoods more difficult and may also serve to define better the neighbourhood. Both of these effects it is hoped will reduce crime. A deleterious side-effect of cul-de-sacs is the increased difficulty of travel for all people who traverse the neighbourhood—specifically providers of essential services such as the fire department and the police department. A problem faced by city planners is where to most effectively place these impediments to travel so that they make travel difficult for the criminal, but not too difficult for residents and allow essential services to be provided expeditiously. This problem is modelled and its complexity explored, with several simplified variants studied. A prototype interactive design aid, that will enable city planners to use a computer to develop solutions, is described and its potential use is explored.     

ORGANIZED CRIMES AGAINST NATURE: ELEPHANTS IN SOUTHERN AFRICA

Poaching of elephants in Southern Africa is now dominated by international groups following a model of organized crime. This shift, from poaching conducted by small, local groups; with limited mobility, weapons, and technology, to individuals who organize, finance, equip, and transport well‐armed poaching units to previously scouted locations, has made the protection of elephants in Southern Africa much more difficult and dangerous. This paper develops a model of high‐tech criminal poaching. A poaching organization makes a decision on the number of “planned poaching expeditions.” If a poaching unit is intercepted the entire organization is destroyed, but is replaced by a new organization in the next year. The operating life of a poaching organization is a stochastic process, which in turn induces a stochastic evolution in the elephant population. Under plausible conditions, the number of planned poaching expeditions is highly sensitive to the probability of interception by anti‐poaching patrols, but is nonresponsive to reductions in the black‐market ivory price. Thereby it might be better to focus conservation efforts on increasing the probability of intercepting poaching units rather than trying to control black market ivory prices. A benchmark value of poaching expeditions is identified—above which elephants may slowly decline to extinction.     

Studying crime trends in the USA over the years 2000–2012

Studying crime trends and tendencies is an important problem that helps to identify socioeconomic patterns and relationships of crucial significance. Finite mixture models are famous for their flexibility in modeling heterogeneity in data. A novel approach designed for accounting for skewness in the distributions of matrix observations is proposed and applied to the United States crime data collected between 2000 and 2012 years. Then, the model is further extended by incorporating explanatory variables. A step-by-step model development demonstrates differences and improvements associated with every stage of the process. Results obtained by the final model are illustrated and thoroughly discussed. Multiple interesting conclusions have been drawn based on the developed model and obtained model-based clustering partition.

     

The social impact in a high-risk community: A cellular automata model

This research examines the spread of criminal behavior and hard drug consumption using a mathematical approach called cellular automata (CA). This CA model is based on two behavioral concepts. Firstly, peer association impacts criminal involvement. Secondly, addiction can heighten criminal activity. The model incorporates four types of actors who interact in a high-risk social community and one intervention method. The actors exert a social influence on each other by encouraging or discouraging drug use and criminal behavior. The intervention method called Incapacitation has a probabilistic impact on the individuals in the model. The results identify the threshold where positive influences on a population reduce the number of high-rate offenders in the community. These results are discussed to further the knowledge about the social influences in a high-risk community and how these influences can effect decisions on offender management.     

Analysis of complex network performance and heuristic node removal strategies

Removing important nodes from complex networks is a great challenge in fighting against criminal organizations and preventing disease outbreaks. Six network performance metrics, including four new metrics, are applied to quantify networks’ diffusion speed, diffusion scale, homogeneity, and diameter. In order to efficiently identify nodes whose removal maximally destroys a network, i.e., minimizes network performance, ten structured heuristic node removal strategies are designed using different node centrality metrics including degree, betweenness, reciprocal closeness, complement-derived closeness, and eigenvector centrality. These strategies are applied to remove nodes from the September 11, 2001 hijackers’ network, and their performance are compared to that of a random strategy, which removes randomly selected nodes, and the locally optimal solution (LOS), which removes nodes to minimize network performance at each step. The computational complexity of the 11 strategies and LOS is also analyzed. Results show that the node removal strategies using degree and betweenness centralities are more efficient than other strategies.     

Analysis of Current Penalty Schemes for Violations of Antitrust Laws

The main feature of the penalty schemes described in current sentencing guidelines is that the fine is based on the accumulated gains from cartel activities or price-fixing activities for the firm. The regulations suggest modeling the penalty as an increasing function of the accumulated illegal gains from price fixing to the firm, so that the history of the violation is taken into account. We incorporate these features of the penalty scheme into an optimal control model of a profit-maximizing firm under antitrust enforcement. To determine the effect of taking into account the history of the violation, we compare the outcome of this model with a model where the penalty is fixed. The analysis of the latter model implies that complete deterrence can be achieved only at the cost of shutting down the firm. The proportional scheme improves upon the fixed penalty, since it can ensure complete deterrence in the long run, even when penalties are moderate. Phase-diagram analysis shows that, the higher the probability and severity of punishment, the sooner cartel formation is blocked. Further, a sensitivity analysis is provided to show which strategies are most successful in reducing the degree of price fixing. It turns out that, when the penalties are already high, the antitrust policy aiming at a further increase in the severity of punishment is less efficient than the policy that increases the probability of punishment. The authors thank Eric van Damme, Thomas Fent, and an anonymous referee for stimulating discussions and valuable comments.     

Deep Learning for Real-Time Crime Forecasting and Its Ternarization

Real-time crime forecasting is important. However, accurate prediction of when and where the next crime will happen is difficult. No known physical model provides a reasonable approximation to such a complex system. Historical crime data are sparse in both space and time and the signal of interests is weak. In this work, the authors first present a proper representation of crime data. The authors then adapt the spatial temporal residual network on the well represented data to predict the distribution of crime in Los Angeles at the scale of hours in neighborhood-sized parcels. These experiments as well as comparisons with several existing approaches to prediction demonstrate the superiority of the proposed model in terms of accuracy. Finally, the authors present a ternarization technique to address the resource consumption issue for its deployment in real world. This work is an extension of our short conference proceeding paper [Wang, B., Zhang, D., Zhang, D. H., et al., Deep learning for real time Crime forecasting, 2017, arXiv: 1707.03340].     

Variables sampling plans with PPM fraction of defectives and process loss consideration

Acceptance sampling plans provide the vendor and the buyer decision rules for lot sentencing to meet their product quality needs. A problem the quality practitioners have to deal with is the determination of the critical acceptance values and inspection sample sizes that provide the desired levels of protection to both vendors and buyers. As today's modern quality improvement philosophy, reduction of variation from the target value is the guiding principle as well as reducing the fraction of defectives. The C_pm index adopts the concept of product loss, which distinguishes the product quality by setting increased penalty to products deviating from the target. In this paper, a variables sampling plan based on C_pm index is proposed to handle processes requiring very low parts per million (PPM) fraction of defectives with process loss consideration. We develop an effective method for obtaining the required sample sizes n and the critical acceptance value C₀ by solving simultaneously two nonlinear equations. Based on the designed sampling plan, the practitioners can determine the number of production items to be sampled for inspection and the corresponding critical acceptance value for lot sentencing.     

Demand estimation with failure and capacity constraints: An application to prisons

A stochastic model for projecting demand for a population-driven, input-output facility that incorporates demographic changes, facility returns representing “failures”, and capacity constraints is proposed and demonstrated. The model is applied to the problem of prison population projection. The approach models the flow of inmates through the prison system, exploits the differences in the incarceration hazard rates of individuals in the general population and those who have been incarcerated previously, and explicitly considers the impact of constrained prison capacity on release policy and future admissions. First-time arrivals to prison are modeled as a Poisson process arising from the general population; recidivist arrivals are modeled using a failure model, where the reincarceration hazard rate is a function of age and race. The model is demonstrated for the State of North Carolina. The effect of limited prison capacity on the average time served is shown. Further, the results demonstrate that an early release policy will generate an increase in prison admissions through t'he return to prison of former inmates. The implications for current “get tough” sentencing policy initiatives relative to the prison crowding problem, the length of stay for offenders not included in the new policies, and the recursive effect of these policies on the input-output dynamics are considered. The results suggest the tradeoffs that exist between early release policies and capacity limitations.     

Explicitly Solvable Nonlocal Eigenvalue Problems and the Stability of Localized Stripes in Reaction‐Diffusion Systems

The transverse stability of localized stripe patterns for certain singularly perturbed two‐component reaction‐diffusion (RD) systems in the asymptotic limit of a large diffusivity ratio is analyzed. In this semi‐strong interaction regime, the cross‐sectional profile of the stripe is well‐approximated by a homoclinic pulse solution of the corresponding 1‐D problem. The linear instability of such homoclinic stripes to transverse perturbations is well known from numerical simulations to be a key mechanism for the creation of localized spot patterns. However, in general, owing to the difficulty in analyzing the associated nonlocal and nonself‐adjoint spectral problem governing stripe stability for these systems, it has not previously been possible to provide an explicit analytical characterization of these instabilities, including determining the growth rate and the most unstable mode within the band of unstable transverse wave numbers. Our focus is to show that such an explicit characterization of the transverse instability of a homoclinic stripe is possible for a subclass of RD system for which the analysis of the underlying spectral problem reduces to the study of a rather simple algebraic equation in the eigenvalue parameter. Although our simplified theory for stripe stability can be applied to a class of RD system, it is illustrated only for homoclinic stripe and ring solutions for a subclass of the Gierer–Meinhardt model and for a three‐component RD system modeling patterns of criminal activity in urban crime.     

Using the Viable Systems Model to examine multi-agency arrangements for combatting transnational organised crime

Traditionally, most operational research applications in the area of law enforcement and criminal justice have used quantitative techniques to forecast patterns and levels of offending and to optimise various police, judicial and prison processes. This paper seeks to examine the role that more qualitative modelling approaches might contribute in this important application area. Specifically, the paper examines the role that cybernetic theory and viable systems modelling can contribute in helping government bodies and law enforcement agencies to think through how, organisationally, they can best respond to the increasingly complex problem of transnational organised crime. Focussing on new multi-agency collaborative arrangements, the paper argues that there is a need for more informed debate that can account for the complexity of the challenge and point towards more holistic and integrated solutions. Against this background, the paper argues that the viable systems modelling approach has much to offer because it is theoretically grounded in the management of complexity; because its flexibility allows it to accommodate the trend for law enforcement agencies to work more closely with partner agencies, and to do so at multiple organisational levels; and because its user-friendly annotated diagramming conventions allow stakeholders to focus discussion and work towards outcomes that make sense in particular settings.     

Dispersion effects in unreplicated factorial designs

Methods for estimation of dispersion effects in two‐level unreplicated factorial designs are studied. The consequences of non‐constant variance are discussed. A natural assumption concerning the form of the covariance of location effects leads to a particular normal model. Some linear combinations of the response variables are constructed in order to make a simple structure for inference. The precision of point estimators of dispersion effects, where one is based on experiments with replicates, are compared. A numerical example is given as an illustration of a test. Finally, estimations in fractional designs are described and discussed. Copyright © 2002 John Wiley & Sons, Ltd.