Watch crime events cluster in space and time on a grid governed by a simplified self-exciting (Hawkes) process: every incident briefly raises the risk of another nearby, producing hotspots that flare up, spread, and fade — then try hotspot patrol to see targeted policing suppress them.
A self-exciting spatiotemporal point process, the same mathematical structure used in earthquake aftershock modeling and applied to crime by researchers since 2011. Each event injects excitation into a local Gaussian neighbourhood, which decays exponentially over time, exactly mirroring "near-repeat victimization" and the "broken windows" idea that disorder invites more disorder.
Raise Excitation strength to intensify near-repeat triggering, raise Background rate for more spontaneous crime everywhere, and adjust Decay time to control how long a hotspot's "heat" lingers. Toggle Patrol to see a moving enforcement zone suppress whichever cluster is currently hottest.
Burglary at one house has been shown in real crime data to elevate burglary risk at neighbouring houses for several weeks — a phenomenon called near-repeat victimization that directly inspired the mathematics used in predictive-policing software such as PredPol.
This simulation models urban crime as a self-exciting spatiotemporal point process, the framework introduced to criminology by George Mohler and colleagues in their influential 2011 paper "Self-Exciting Point Process Modeling of Crime," building on the mathematical theory of Hawkes processes developed by Alan Hawkes in 1971 to describe earthquake aftershock sequences. In this model, crime does not occur independently at random locations; instead, every incident temporarily raises the probability of another incident occurring nearby, so that a background rate of "spontaneous" crime is amplified into visible clusters — hotspots — that flare up, drift, and decay over time as excitation from past events fades exponentially.
The mechanism mirrors a well-documented empirical pattern called near-repeat victimization: after a burglary, houses within roughly 400 metres face measurably elevated risk for several weeks, an effect criminologists attribute partly to repeat offenders returning to a familiar, vulnerable target and partly to the "broken windows" theory, which holds that visible signs of disorder and unaddressed crime signal that an area is unmonitored and invite further offending. These ideas underpin real-world predictive policing systems such as PredPol (now Geolitica), which allocate patrol resources to computer-forecasted hotspots. The approach remains controversial: while proponents argue it improves resource efficiency, researchers and journalists have documented feedback loops in which increased patrols in already over-policed neighbourhoods generate more recorded crime there, reinforcing the very predictions that sent officers back, and have raised concerns about racial and socioeconomic bias in historical training data.
A Hawkes process is a mathematical model of random events in which each occurrence temporarily increases the probability of further events happening soon afterward and nearby, rather than events arriving independently as in a simple random (Poisson) process. First developed by Alan Hawkes in 1971 to model clustered phenomena like earthquake aftershocks, it is defined by a base intensity plus a sum of decaying "excitation" contributions from every past event, making clustering an emergent, quantifiable property of the model.
Near-repeat victimization is the empirical finding, confirmed across many cities and crime types, that a property near a recent crime (especially burglary) faces significantly elevated risk for a period of days to weeks afterward, with risk decaying as both distance and time from the original event increase. It is one of the most consistent patterns in environmental criminology and is the direct real-world analogue of the spatial excitation kernel used in this simulation.
Excitation strength scales how much each crime event raises the risk field in its local neighbourhood, roughly analogous to the "branching ratio" in Hawkes-process terminology — the expected number of follow-on events directly triggered by one parent event. Low values keep events close to independent (Poisson-like) and scattered; high values push the branching ratio toward 1, producing dramatic, self-sustaining hotspot clusters, similar to a nearly critical epidemic.
Broken windows theory, proposed by James Q. Wilson and George Kelling in 1982, argues that visible disorder — broken windows, graffiti, litter — signals a lack of social control and invites escalating crime, so aggressively addressing minor disorder can prevent more serious offending. It inspired "zero tolerance" and order-maintenance policing strategies in cities including New York. Critics argue the empirical evidence for the causal chain is weak, that it has been used to justify aggressive policing of minor offenses in minority and low-income neighbourhoods, and that observed crime reductions may stem from unrelated factors such as economic change or general policing increases rather than disorder enforcement specifically.
Hotspot policing concentrates patrol resources on the small number of places responsible for a disproportionate share of crime, based on research showing that crime is highly concentrated spatially — often "the law of crime concentration" describes roughly half of a city's crime occurring on under 5% of street segments. Randomized controlled trials, including the influential Minneapolis and Jersey City hotspot experiments, generally find modest but real reductions in crime at treated hotspots with limited "displacement" of crime to nearby areas, making it one of the better-evidenced policing strategies, though effects vary by tactic and context.
Predictive policing uses statistical or machine-learning models, often built on Hawkes-process foundations like the one in this simulation, to forecast where and when crime is likely to occur and to direct patrols accordingly; PredPol (rebranded Geolitica) and HunchLab were prominent commercial examples. Documented criticisms include feedback loops where predictions concentrate patrols in already over-policed neighbourhoods, generating more recorded crime there and reinforcing future predictions regardless of true underlying crime rates; concerns that historical arrest data embeds racial and socioeconomic bias; and limited independent evidence that the forecasts meaningfully outperform simpler hotspot maps. Several U.S. cities, including Los Angeles, discontinued PredPol contracts following these concerns.
In a homogeneous Poisson process, events occur independently at a constant average rate with no memory of past events, so clustering only appears by pure chance and dissolves quickly. In this self-exciting simulation, each event actively raises the local rate for a period afterward, producing genuine, persistent, and spatially structured clustering — real crime data consistently shows more clustering than a Poisson model predicts, which is precisely why criminologists adopted Hawkes-style models instead.
George Mohler, then at Santa Clara University and later UCLA-affiliated researchers including Martin Short, P. Jeffrey Brantingham, and George Tita, adapted Hawkes' seismology mathematics to urban crime data in a series of papers beginning around 2008–2011, formally published as "Self-Exciting Point Process Modeling of Crime" in the Journal of the American Statistical Association. This work founded the subfield of mathematical criminology that treats crime as a measurable, stochastic spatiotemporal process and directly led to commercial predictive-policing software.