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A Statistical Model of Criminal Behavior
M.B. Short, M.R. D’Orsogna, V.B. Pasour, G.E. Tita, P.J. Brantingham, A.L. Bertozzi,
L.B. Chayez
Maria Pavlovskaia
The Discrete Model
• A neighborhood is a 2d lattice
• Houses are vertices
• Vertices have attractiveness values Ai
• Criminals move around the lattice
Criminal Movement
A criminal can:
• Rob the house he is at
- or -• Move to an adjacent house
• Criminals regenerate at each node
Attractiveness Values
• Rate of burglary when a criminal is at that house
• Has a static and a dynamic component
• Static (A0) - overall attractiveness of the house• Dynamic (B(t)) - based on repeat and near-repeat victimization
Dynamic Component
• When a house s is robbed, Bs(t) increases
• When a neighboring house s’ is robbed, Bs(t) increases
• Bs(t) decays in time if no robberies occur
Dynamic Component
• The importance of neighboring effects:
• The importance of repeat victimization:
• When repeat victimization is most likely to occur:
• Number of burglaries between t and t: Es(t)
Computer Simulations
Three Behavioral Regimes are Observed:
• Spatial Homogeneity
• Dynamic Hotspots
• Stationary Hotspots
Computer Simulations
Three Behavioral Regimes are Observed:
• Spatial Homogeneity – Large number of criminals or burglaries
• Dynamic Hotspots– Low number of criminals and burglaries– Manifestation of the other two regimes due to finite size effects
• Stationary Hotspots– Large number of criminals or burglaries
Continuum Limit
In the limit as the time unit and the lattice spacing becomes small:
• The dynamic component of attractiveness:
• The criminal density:
Continuum Limit
• Reaction-diffusion system
• Dimensionless version is similar to:
– Chemotaxis models in biology (do not contain the time derivative)
– Population bioglogy studies of wolfe and coyote territories
Computer Simulations
• Dynamic Hotspots are never seen
• Spatial Homogeneity or Stationary Hotspots?
– Performed linear stability analysis
– Found an inequality to distinguish between the cases
Summary
• Discrete Model
• Computer Simulations
– Spatial Homogeneity, Dynamic Hotspots, Stationary Hotspots
• Continuum Limit
– Dynamic Hotspots are not observed: due to finite size effects– Inequality to distinguish between Homogeneity and Hotspots cases