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Kinetic and Moment Models for Cell Motion in Fiber Structures Read chapter Read first chapter

2019 | OriginalPaper | Chapter

A Stochastic-Statistical Residential Burglary Model with Finite Size Effects

Authors : Chuntian Wang, Yuan Zhang, Andrea L. Bertozzi, Martin B. Short

Published in: Active Particles, Volume 2

Publisher: Springer International Publishing

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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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previous chapter Singular Cucker–Smale Dynamics
next chapter Correction to: A Stochastic-Statistical Residential Burglary Model with Finite Size Effects
Appendix
Available only for authorised users
Footnotes
1
With minor changes we can also consider, e.g., the Dirichlet boundary conditions, which is more realistic.
 
2
More precisely, the criminal agents are assumed to be uniformly (randomly) distributed over the 128 × 128 grids, while \( \sum _{\mathbf {s}\in \mathcal S ^\ell } n0^\ell _{\mathbf {s}} = 128^2 \bar {n}^\ell \).
 
3
In [64], one can directly compute Γf and obtain that when f = Id then the infinitesimal variance comes up.
 
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Metadata
Title
A Stochastic-Statistical Residential Burglary Model with Finite Size Effects
Authors
Chuntian Wang
Yuan Zhang
Andrea L. Bertozzi
Martin B. Short
Copyright Year
2019
Book
Active Particles, Volume 2

Print ISBN: 978-3-030-20296-5

Electronic ISBN: 978-3-030-20297-2

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-20297-2_8

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