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 A i 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 (A 0 ) - overall attractiveness of the house Dynamic (B(t)) - based on repeat and near-repeat victimization

Dynamic Component When a house s is robbed, B s (t) increases When a neighboring house s’ is robbed, B s (t) increases B s (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: E s (t)

Computer Simulations

Three Behavioral Regimes are Observed: Spatial Homogeneity Dynamic Hotspots Stationary Hotspots

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

Applications House burglaries Assault with a lethal weapon Muggings Terrorist attacks in Iraq Lootings