Andrea Bertozzi University of California Los Angeles Thanks to contributions from Laura Smith, Rachel Danson, George Tita, Jeff Brantingham.

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

Movie by Alethea Barbaro.

From Metzler and Klafter, 2000.

 PDF for Brownian walk in 1D:  Using Taylor expansions in space and time and passing to the limit one gets  Propagator:

 Fourier transform

 Fickian diffusion has certain statistics, in particular ~Kt, characteristic of Brownian motion.  Sub-diffusion has different scaling ~K a t a 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(le ft)  Levy (right)

 Length of jump as well as waiting time between jumps are drawn from PDF.  Levy has jump length PDF   >0  Propagator satisfies the nonlocal PDE  Where the nonlocal operator is easily expressed in terms of its Fourier transform:

 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  as close to zero as possible.  For non-destructive searches the optimal  is close to 1, with a margin that behaves like  Here is avg distance between targets and r v is the vision distance.  Searchin g for unknown target locations  Efficiency of search can be defined as number of targets visited compare d to typical distance travelled.  For destructi ve searches (typical in crime applicatio ns) one takes m as close to 1 as possible.  For non- destructi ve searches the optimal m is close to 2, with a margin that behaves like G.M. Viswanathan et al, Nature 401, 911 (1999).

 Nonlinear diffusion – used sometimes in population dynamics.  Assume a motion in which the speed or jump length is a function of the local density.  Can be used to model anti-crowding in which dispersal is not needed if the group is sparse.  Satisfies a PDE W t = (W 1+a ) xx where a>0.  The propagator function is the well-known Barenblatt solution it has compact support and finite speed of propagation of the support.  This model arises in the contexts of animal populations such as herds or flocks.

 Continuum modeling with Levy processes applied to crime.  Dynamics on networks linked to spatial motion of criminals.  Stochastic models vs. deterministic – both are in the developmental stage.  Analysis of real data, inference from real data vs. agent based simulations.

 M. Bradonjic, A. Hagberg, A. Percus. Giant Component and Connectivity in Geographical Threshold Graphs (2007).  M. Egesdal, C. Fathauer, K. Louie, J. Neumann, Statistical Modeling of Gang Violence in Los Angeles, SIURO,  S. R. Jammalamadaka, A. Sengupta. Topics in Circular Statistics. World Scientific. Series on Multivariate Analysis Vol. 5.  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.”  Brantingham, P.J. and G. Tita. Offender Mobility and Crime Pattern Formation from First Principles. In Artificial Crime Analysis Systems: Using Computer Simulations and Geographic Information Systems, edite by L. Liu and J. Eck. pp Hershey, PA: Idea Press,  R. Metzler and J. Klafter, Physics Reports 339, p. 1-77,  G.M. Viswanathan et al, Nature 401, 911 (1999).  J. Bouchard and A. Georges, Physics Reports 195, p ,  C.M. Topaz, ALB, and M.A. Lewis, Bull.. of Math. Bio., 68(7), 2006.