By Lina Sorg
Collective dynamics refers to organisms or groups that interact in social ways to create a macroscopically-observable phenomenon. Phase transitions are observable in many models representing collective dynamics; the dynamics often change with the model parameters. As part of a minisymposium presentation entitled “A Coupled Reaction-Diffusion Model for Gang Territorial Development” during the SIAM Conference on the Life Sciences, Alethea Barbaro presents an agent-based discrete model that simulates gang territorial development. The model is based on graffiti markings on a two-dimensional lattice.
Much like territorial animals, street gangs typically establish a home base region and physically mark their territories to identify claimed areas and/or maintain status or economic interests. “In a city, you know a gang is present based on graffiti,” Barbaro said. Gangs are responsible for many of the violent crimes in the United States and worldwide. They often compete for open territory, especially around border areas, but rarely venture into territory that is marked with another gang’s symbols. Gang graffiti may also signal the presence of illicit activities, such as extortion or drug trafficking, in the area.
Barbaro begins by thinking microscopically with a lattice model. To keep things simple, the model assumes that there are only two competing gangs, and randomly distributes red and blue agents representative of those gangs. Barbaro’s research group also bases the model on the assumption that gang agents avoid territory marked by other gangs, are repelled by the presence of opposing agents, and move via a biased random walk. In the model, gang members interact only via graffiti field, not directly with other gang members.
Barbaro's lattice model assumes that there are only two competing gangs, and randomly distributes red and blue agents that represent those gangs. Image credit: Alethea Barbaro, LS16 presentation.
The model allows Barbaro to study and evaluate the dynamics and steady-state solutions of gang members and their corresponding graffiti markings. Her group employs numerical simulations to demonstrate that territory formation and gang segregation—which act as the model’s parameters—are varied. They then obtain a continuum system of equations representing territorial development, and use it to perform a linear stability analysis. Additionally, use of discrete dynamics exposes the importance of graffiti’s density, as it controls the probability of agents moving to a new site. The changing size of the model’s beta term is also significant. As beta grows, agents of the same gang clump together; this is the segregated phase. As beta shrinks, the agents are equally likely to move in any direction, regardless of the amount of graffiti. This is called the well-mixed phase.
Ultimately, the model’s numerical results show that graffiti rate, inverse temperature, and rate of graffiti decay all affect phase transition in the collective dynamics of gang activity. Understanding gang graffiti offers a clearer indication of agent interactions both within a single gang and within rival gangs.
||Lina Sorg is the associate editor of SIAM News.