Crime Modeling with Lévy Flights

Abstract

Sci., 18 (2008Sci., 18 ( ), pp. 1249Sci., 18 ( -1267, models the formation of hotspots of criminal activity. In this paper, we extend the UCLA model to incorporate a more realistic model of human locomotion. The movement of the criminal agents follows a biased Lévy flight with step sizes distributed according to a power-law distribution. The biased Brownian motion of the original model is then derived as a special case. Starting with an agent-based model, we derive its continuum limit. This consists of two equations and involves the fractional Laplacian operator. A numerical method based on the fast Fourier transform is used to simulate the continuum model; these simulations compare favorably with the direct numerical simulations of the agent-based model. A Turing-type analysis is performed to estimate how the instability of the homogeneous steady state, as well as the expected number of hotspots, depends on the system parameters and especially the exponent of the underlying power law. The assumptions of the underlying agent-based model naturally lead to a separation of scales of the diffusion coefficients in the continuum limit. Using these assumptions, we asymptotically construct the leading-order profile of the localized hotspot of criminal activity.

Deter and protect: crime modeling with multi-agent learning

“…The presence of guardian agents was used to deter offenders from committing crime. Chaturapruek et al [29] adjusted Short et al [27] by changing how offenders moved around the grid environment. Instead of using local information to their current location, offenders could survey the whole grid and move following a stochastic process called Lévy flights.…”

A Decision Support System for predictive police patrolling

“…Also, they mathematically proved that there were different types of hotspots, even though they seemed similar at first sight. This breakthrough was further developed using Levy Flight models by Chaturapruek et al [14]. More recently, Zipkin et al [58] introduced a police behavior component aiming at suppressing hotspots of criminal activity.…”

Nonlinear Stability through Algebraically Decaying Point Spectrum: Applications to Nonlocal Interaction Equations

“…The stability theory for this broad class of equations is quite large. The stability of both stationary and traveling wave patterns for nonlocal reaction-diffusion has been extensively studied [12,25,8,13,28,26,22,27,30,29,23,50], especially in the plasma physics community. Usually the nonlocal operator is fractional diffusion in these settings; however, more exotic nonlocal equations also have been studied in the kinetic setting [14,20,49,19,21,15].…”