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Self-organised critical hot spots of criminal activity

  • H. BERESTYCKI (a1) and J.-P. NADAL (a1) (a2)

In this paper1 we introduce a family of models to describe the spatio-temporal dynamics of criminal activity. It is argued here that with a minimal set of mechanisms corresponding to elements that are basic in the study of crime, one can observe the formation of hot spots. By analysing the simplest versions of our model, we exhibit a self-organised critical state of illegal activities that we propose to call a warm spot or a tepid milieu2 depending on the context. It is characterised by a positive level of illegal or uncivil activity that maintains itself without exploding, in contrast with genuine hot spots where localised high level or peaks are being formed. Within our framework, we further investigate optimal policy issues under the constraint of limited resources in law enforcement and deterrence. We also introduce extensions of our model that take into account repeated victimisation effects, local and long range interactions, and briefly discuss some of the resulting effects such as hysteresis phenomena.

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Bak, P. (1997) How Nature Works: The Science of Self Organised Criticality, Oxford University Press.
Becker, G. (1968) Crime and punishment: An economic approach. J. Polit. Econ. 76, 169217.
Berk, R. (2008) How you can tell if the simulations in computational criminology are any good. J. Exp. Criminol. 4 (3), 289308.
Bernasco, W. (2009) Modeling micro-level crime location choice: Application of the discrete choice framework to crime at places. J. Quant. Criminol. 26 (1), 113138.
Bourguignon, F., Nunez, J. & Sanchez, F. (2003 a) A structural model of crime and inequality in Colombia. J. Eur Econ. Assoc. 12 (2–3), 440449.
Bourguignon, F., Nunez, J. & Sanchez, F. (2003 b) What part of the income distribution does matter for explaining crime? The case of Colombia. J. Eur. Econ. Assoc. 1 (2–3), 440449.
Brunel, N. & van Rossum, M. C. W. (2007) Lapicques 1907 paper: From frogs to integrate-and-fire. Bio. Cybern. 97 (5–6), 337339.
Calvó-Armengol, A. & Zenou, Y. (2004) Social networks and crime decisions: The role of social structure in facilitating delinquent behavior. Int. Econ. Rev. 45, 939958.
Campbell, M. & Ormerod, P. (1998) Social interactions and the dynamics of crime. Volterra Consulting Preprint. URL:
Chow, C., Gutkin, B., Hansel, D., Meunier, C. & Dalibard, J. (editors) (2004) Methods and Models in Neurophysics. Session LXXX, Lecture Notes of the Les Houches Summer School 2003, Elsevier.
Clarke, R. V. & Felson, M. (editors) (1993) Routine Activity and Rational Choice. Advances in Criminological Theory, Vol. 5, Transaction Books, New Brunswick, NJ.
Cohen, L. E. & Felson, M. (1979) Social change and crime rate trends: A routine activity approach. Am. Sociol. Rev. 44, 588608.
Crane, J. (1991) The epidemic theory of ghettos and neighborhood effects on dropping out and teenage childbearing. Am. J. Sociol. 96 (5), 12261259.
Eck, J. & Liu, L. (2008) Contrasting simulated and empirical experiments in crime. J. Exp. Criminol. 4, 195213.
Glaeser, E. L., Sacerdote, B. & Scheinkman, J. A. (1996) Crime and social interactions. Q. J. Econ. 111, 507548.
Gordon, M. B. (2010) A random walk in the literature on criminality: A partial and critical view on some statistical analyses and modeling approaches. Eur. J. Appl. Math. 21, 283306.
Gordon, M. B., Iglesias, J. R., Semeshenko, V. & Nadal, J.-P. (2009a) Crime and punishment: The economic burden of impunity. Eur. Phys. J. B – Condens. Matter Complex Syst. 68 (1), 133144.
Gordon, M. B., Nadal, J.-P., Phan, D. & Semeshenko, V. (2009b) Discrete choices under social influence: Generic properties. Math. Models Methods Appl. Sci. (M3AS) 19 (Supplementary Issue 1), 14411481.
Groff, L. (2007) Simulation for theory testing and experimentation: An example using routine activity theory and street robbery. J. Quant. Criminol. 23 (2), 75103.
Johnson, S. D., Summers, L. & Pease, K. (2009) Offender as forager? A direct test of the boost account of victimization. J. Quant. Criminol. 25 (2), 181200.
Nadal, J.-P., Gordon, M. B., Semeshenko, V. & Iglesias, J. R. (2010) Modelling the individual and collective dynamics of the propensity to offend. Eur. J. Appl. Math. 21, 421440.
Nadal, J.-P., Phan, D., Gordon, M. B. & Vannimenus, J. (2006) Multiple equilibria in a monopoly market with heterogeneous agents and externalities. Quant. Finance 5 (6), 557568.
Nagin, D. & Paternoster, R. (1993) Enduring individual differences and rational choice theories of crime. Law Soc. Rev. 467.
Nuño, J. C., Herrero, M. A. & Primicerio, M. (2008) A triangle model of criminality. Physic. A: Stat. Mech. Appl. 387 (12), 29262936.
Ormerod, P. (2005) Crime: Economic Incentives and Social Networks, Insitute of Economic Affairs, London.
Pease, K. (1998) Repeat victimization: Taking stock. Crime Detection and Prevention Series Paper 90. The Home Office: Police Research Group.
Pitcher, A. B. (2010) Adding police to a mathematical model of burglary. Eur. J. Appl. Math. 21, 401419.
Sethna, J. P. (2009) Statistical Mechanics. Entropy, Order Parameters and Complexity, Oxford University Press.
Short, M. B., Brantingham, P. J., Bertozzi, A. L. & Tita, G. E. (2010) Dissipation and displacement of hotspots in reaction-diffusion models of crime. Proc. Natl. Acad. Sci. (PNAS) 107 (9), 39613965.
Short, M., D'Orsogna, M., Pasour, V., Tita, G., Brantingham, P., Bertozzi, A. & Chayes, L. (2008) A statistical model of criminal behavior. Math. Models Methods Appl. Sci. (M3AS) 18 (Suppl.), 12491267.
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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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