Burglary, as defined by Uniform Crime Reporting (UCR), is the unlawful entry of a structure to commit a felony or theft. In 2005, there were more than 700 reported burglaries per 100,000 inhabitants (UCR). One of the defining characteristics or patterns found in these burglaries was the large percentage of repeated burglaries within a short period of time. Previous criminological studies have shown that the likelihood of another burglary is at its highest during the first month after the initial burglary [2]. Moreover, each of the neighboring houses also has its burglary risk increased for some amount of time [3]. Thereafter, the likelihood of burglary in the area exponentially returns to normal within less than a year. See Figure 1.

Our model relies on the fact that a house and its neighbors, to some degree, become more attractive to burglars once the house has been burglarized. Each house has an attractiveness value, which has two components: the base value and the dynamic value. The base value A_k is the fixed minimum attractiveness value of a house. The dynamic value B_k is the value that is increased by the constant value Δ as a house gets burglarized and decreased over time by the factor (1 - λ), where λ ranges from 0 to 1. Each B_k value is also determined by the neighboring attractiveness values, which are computed by the following equation:

where B_l is a neighboring dynamic attractiveness value and η is the spread constant ranging from 0 to 1.

A burglar is a criminal agent who, with a probability depending on the attractiveness value of the house, decides to burglarize the house or move. The burglar would burglarize if

where x is a random number between 0 and 1, and ε is an arbitrary constant. Otherwise, the burglar would move. The burglar's direction of movement is also probabilistically determined by the attractiveness values around its neighborhood. The more attractive a neighboring house is, the more likely the burglar will move there. Each burglar is free to move to any of its neighboring houses within the grid. In addition, after burglarizing, the burglar is randomly relocated to any house on the grid.

In the new model, a radius component is introduced to restrict the movement of burglars in order to create a more realistic model. Each burglar is initialized with a radius of sight, which is the maximum distance the burglar can ever be away from the burglar's home location. As a burglar successfully burglarizes a house, its radius of sight is increased until it reaches a specified maximum. Its random relocation after burglarizing is also restricted within the current radius of sight of that burglar. The reason for this adjustment is to take into account the fact that burglars tend to move and burglarize within or near its own neighborhood [1].

Figure 2 shows the repeated burglary pattern under the old model where burglars are unrestricted in movement. The model's parameters are as follows: A_k=0.1, Δ=100, ε=0.01, η=0.5,  λ=0.1, and initial B_k=0.0. There are 50 burglars in a square grid of 10,000 houses. As seen, the number of repeated burglaries totaled less than 50 repeated burglaries out of more than 2,000 total burglaries.

Figure 3 shows the repeated burglary pattern under the new model where burglars are restricted in movement. All the parameters are held the same except for the radius parameters. Each burglar starts with an initial radius of 3 and a maximum radius of 10 with a change in radius of 1 for each successful burglary. As seen, the new model produces a significant increase in the number of repeated burglaries. As compared to the Long Beach data, the new model produces a similar exponentially decaying trend in the future burglary risk after the initial burglary.

Figure 4 is a movie showing a simulation of burglary activities and the resulting attractiveness values at each house in the new model. The white dots are burglars. The gray dots are the home locations of the burglars. The attractiveness values range from blue for low attractiveness to red for high attractiveness.

In conclusion, introducing the radius component to the existing model does increase the number of repeated burglaries. However, the resulting numbers of burglaries per house still deviate from the actual data (Figure 5). We must further study the effects of introducing the radius component to the model while adjusting other parameters in order to represent the real world.

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This research was conducted at University of California, Los Angeles (UCLA) under the guidance of Professor Martin Short and Professor Virginia Pasour. I wish to thank the Rose Hills Foundation and the National Science Foundation for financial support (DMS-0601395) and Professor Andrea Bertozzi and Professor Andrew Bernoff for sponsoring this summer research opportunity.