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A Stochastic-Statistical Residential Burglary Model with Finite Size Effects

  • Chuntian WangEmail author
  • Yuan Zhang
  • Andrea L. Bertozzi
  • Martin B. Short
Chapter
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Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

Transience of spatio-temporal clusters of residential burglary is well documented in empirical observations, and could be due to finite size effects anecdotally. However a theoretical understanding has been lacking. The existing agent-based statistical models of criminal behavior for residential burglary assume deterministic-time steps for arrivals of events. To incorporate random arrivals, this article introduces a Poisson clock into the model of residential burglaries, which could set time increments as independently exponentially distributed random variables. We apply the Poisson clock into the seminal deterministic-time-step model in Short et al. (Math Models Methods Appl Sci 18:1249–1267, 2008). Introduction of the Poisson clock not only produces similar simulation output, but also brings in theoretically the mathematical framework of the Markov pure jump processes, e.g., a martingale approach. The martingale formula leads to a continuum equation that coincides with a well-known mean-field continuum limit. Moreover, the martingale formulation together with statistics quantifying the relevant pattern formation leads to a theoretical explanation of the finite size effects. Our conjecture is supported by numerical simulations.

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Notes

Acknowledgements

We would like to thank the helpful discussions with Prof. A. Debussche, Prof. Andrea Montanari, Prof. Thomas Liggett, Prof. Carl Mueller, Prof. Wotao Yin, Mac Jugal Nankep Nguepedja, Da Kuang, Yifan Chen, Fangbo Zhang, Yu Gu, Jingyu Huang, Yatin Chow, Wuchen Li, and Kenneth Van. A. Bertozzi is supported by NSF grant DMS-1737770 and M. Short is supported by NSF grant DMS-1737925.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Chuntian Wang
    • 1
    Email author
  • Yuan Zhang
    • 2
  • Andrea L. Bertozzi
    • 1
  • Martin B. Short
    • 3
  1. 1.University of California, Los AngelesLos AngelesUSA
  2. 2.Peking UniversityBeijingPeople’s Republic of China
  3. 3.Georgia Institute of TechnologyAtlantaUSA

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