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

Goal

• Model the behavior of crime hotspots

• Focus on house burglaries

Assumptions

• Criminals prowl close to home

• Repeat and near-repeat victimization

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

Criminal Movement

• Modeled as a biased random walk

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

Computer Simulations

Three Behavioral Regimes are Observed:

• Spatial Homogeneity

• Dynamic Hotspots

• Stationary Hotspots

SpatialHomogeneity

DynamicHotspots

StationaryHotspots

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

Applications

• House burglaries

• Assault with a lethal weapon

• Muggings

• Terrorist attacks in Iraq

• Lootings