AGENT BASED MODELS FOR GANG RIVALRIES

Lectures for Math 285JUCLA

REFERENCES R. A. Hegemann, L. M. Smith, A. Barbaro, A. L. Bertozzi, S. Reid, and

G. E. Tita, Geographical influences of an emerging network of gang rivalries, Physica A, Volume 390, Issues 21-22, 15 October 2011, Pages 3894-3914

Laura M. Smith, Andrea L. Bertozzi, P. Jeffrey Brantingham, George E. Tita, and Matthew Valasik, Adaptation of an Ecological Territorial Model to Street Gang Spatial Patterns in Los Angeles Discrete and Continuous Dynamical Systems A, 32(9), pp. 3223 - 3244, 2012.

Laura Smith, PhD Thesis, UCLA, 2012 Incorporating Spatial Information into Density Estimates and Street Gang Models.

G. Tita, K. Riley, G. Ridgeway, C. Grammich, A. Abrahamse, P. Greenwood, Reducing gun violence: Results from an intervention in East Los Angeles,Natl. Inst. Justice, RAND (2003).

G. Tita, J. Cohen, J. Engberg, An ecological study of the location of gang set space, Soc. Probl. 52 (2) (2005) 272–299.

Hollenbeck Google Earth Image

Gang networks and self-excitation

Rivalry network among 29 street gangs in Hollenbeck, Los AngelesTita et al. (2003)

Features of the Hollenbeck Gangs

Some gangs have been around since before World War II

Retaliatory violence is common Gangs tend to mark their “turf” by using graffiti Rivalries in Hollenbeck tend to be driven by

territorial issues Gang activity in Hollenbeck is generally isolated

from gang activity outside of Hollenbeck Makes for an excellent testbed for development

of models and algorithms.

Observed Rivalry Network Among Hollenbeck Gangs

29 Active Gangs in Hollenbeck

69 Rivalries Among the Gangs

A Set Space is a gang’s center of activity where of activity where gang member spend a large quantity of their time

Freeways and other geographic features influence the rivalry network S. Radil, C. Flint, and G. Tita,“Spatializing Social Networks: Using Social

Network Analysis to Investigate Geographies of Gang Rivalry, Territoriality,and Violence in Los Angeles.” 2010.

Prior work on modeling gang activity

Statistical modeling of gang violence in Los Angeles – SIURO 2010 Egesdal, Fathauer, Louie, and Neumann.

Each agent targets a gang with a probability based on the current rivalry strengths. This target is kept for the lifetime of the agent.

The agent moves on a lattice from one location to the next based on calculated probabilities.

These probabilities depend on the location of the agents target gang members and the target’s gang “anchor point” or set space.

The set space for each gang was determined from the locations of criminal events involving the gang.

Simulation Updates (Egesdal et al)

After all the agents have moved, new gang members are added with a certain rate

When agents of different gangs occupy the same cell, they fight with a probability based on the rivalry strength.

If two agents fight, they leave the simulation.

The rivalry strength is increased in proportion to the number of fights between gang members and decreased with time.

Schematic of Brownian Walk

From Metzler and Klafter, 2000.

Master equation derivation Brownian motion

PDF for Brownian walk in 1D:

Using Taylor expansions in space and time and passing to the limit one gets

Propagator:

Fourier Transform Brownian

Fourier transform

Random walk vs. Levy process Fickian diffusion has certain statistics, in

particular <x2(t)>~Kt, characteristic of Brownian motion. Sub-diffusion has different scaling <x2(t)>~Kata

and occurs in many systems. This is the case where jump lengths are the same but one can have a waiting time between jumps that has a long tail distribution.

Levy process occurs when the jump length is taken from a long-tailed distribution however the waiting times are normally distributed. In this case the variance is infinite and one has to look to other statistics to define the Levy behavior.

Brownian path vs. Levy path

Brownian(left)

Levy (right)

Levy Process Fokker Planck

Length of jump as well as waiting time between jumps are drawn from PDF.

Levy has jump length PDF m>0 Propagator satisfies the nonlocal PDE

Where the nonlocal operator is easily expressed in terms of its Fourier transform:

Levy walks as statistical searching strategies

Searching for unknown target locations Efficiency of search can be defined as number

of targets visited compared to typical distance travelled.

For destructive searches (crime applications) one takes m as close to zero as possible.

For non-destructive searches the optimal m is close to 1, with a margin that behaves like

Here l is avg distance between targets and rv is the vision distance.

Searching for unknown target locations

Efficiency of search can be defined as number of targets visited compared to typical distance travelled.

For destructive searches (typical in crime applications) one takes m as close to 1 as possible.

For non-destructive searches the optimal m is close to 2, with a margin that behaves like

G.M. Viswanathan et al, Nature 401, 911 (1999).

Motivation for Using an Agent-Based Model

Graph Generating Methods May produce a reasonable approximation to the

observed network Other phenomena of interest beyond the

structure of the network will not be obtained from this type of model

Agent-Based Methods Easily incorporates environmental and spatial

information inherent to the system Allows for exploring how changing the dynamics

of the individual agents or the environment can affect the evolution of the network

An Agent-Based Model Using Brownian Motion with

Semi-Permeable Boundaries

Movement Dynamics: Agents move according to Brownian Motion Agents have some probablity of crossing a

boundar (major roads, freeways, and the Los Angeles River)

Interactions: If two gang members from different gangs

cross paths, then an interaction has occurred At the end of the simulation, we exclude

rivalries where the number of interactions is mutually insignificant to both gangs

Motivation for a Biased Lévy Walk

There is compelling evidence in the literature that when people move in an unconstrained environment, the jump lengths between movements is distributed like a power law

In the presence of obstacles such as roads and buildings, the jump lengths more accurately follow a bounded power law distribution

People frequent certain locations (home, work, etc.) Gang members avoid rival gangs’ territories

D. Brockmann, L. Hufnagel, and T. Geisel. The scaling laws of human travel. Nature, 439:462-465, 2006.I. Rhee, M. Shin, S. Hong, K. Lee, and S. Chong. On the lévy-walk nature of human mobility: Do humans walk like monkeys? In IEEE INFOCOM 2008 - IEEE Conference on Computer Communications, pages 924-932. IEEE, April 2008.M. Gonzalez, C. A. Hildalgo, and A.-L. Barabasi. Understanding individual human mobility patterns. Nature Letters, 453:779-782, 2008.

An Agent-Based Model Using a Biased Levy Walkand Semi-Permeable Boundaries

Movement Dynamics: Agents are allowed to move in free space according

to a biased Lévy walk We choose the direction of bias based on the location

of other gangs’ set spaces and the location of the agent’s set Space

Agents have some probablity of crossing a boundary Interactions:

If two gang members from different gangs cross paths,then an interaction has occurred

At the end of the simulation, we exclude rivalries where the number of interactions is mutually insignificant to both gangs

Territorial Animal Characteristics - (coyotes, wolves, etc.)

Coyotes and wolves have distinct home ranges that are well-established

They create scent marks to establish cues for both their own pack as well as other packs

They may respond with avoidance to scent marks and move in the direction of their home range center (den site)

These animals have territorial patterns

Street Gang Characteristics

Gang members have distinct territories that are well-established

They create graffiti to establish cues for both their own gang as well as other gangs

They may respond with avoidance to graffiti and move in the direction of their territory’s center (set space)

These gang members have territorial patterns

Territorial Animal Model for Coyote Packs

Rivalry Network Metrics The SBLN model performs the best in the

accuracy metrics The PDE model does not perform as well The way we constructed the rivalry

network limits the connections across the semi-permeable boundaries

To evaluate the PDE model, we instead consider different data sets