Lectures for Math 285J
UCLA
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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.
Rivalry network among 29 street gangs in Hollenbeck, Los Angeles
Tita et al. (2003)
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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.
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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.
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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.
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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.
From Metzler and Klafter, 2000.
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PDF for Brownian walk in 1D:
Using Taylor expansions in space and time and passing
to the limit one gets
Propagator:
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Fourier transform
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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.
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Brownian(le
ft)
Levy
(right)
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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:
G.M. Viswanathan et al, Nature 401, 911 (1999).
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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.
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Graph Generating Methods
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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
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Movement Dynamics:
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Agents move according to Brownian Motion
Agents have some probablity of crossing a boundar
(major roads, freeways, and the Los Angeles River)
Interactions:
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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
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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.
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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
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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
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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
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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
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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 semipermeable boundaries
To evaluate the PDE model, we instead
consider different data sets