Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Gang rivalry dynamics via coupled point process networks

Pages: 1459 - 1477, Volume 19, Issue 5, July 2014      doi:10.3934/dcdsb.2014.19.1459

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M. B. Short - School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States (email)
G. O. Mohler - Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, CA 95053, United States (email)
P. J. Brantingham - Department of Anthropology, UCLA, Los Angeles, CA 90095, United States (email)
G. E. Tita - Department of Criminology, Law and Society, UC Irvine, Irvine, CA 92697, United States (email)

Abstract: We introduce a point process model for inter-gang violence driven by retaliation -- a core feature of gang behavior -- and multi-party inhibition. Here, a coupled system of state-dependent jump stochastic differential equations is used to model the conditional intensities of the directed network of gang rivalries. The system admits an exact simulation strategy based upon Poisson thinning. The model produces a wide variety of transient or stationary weighted network configurations and we investigate under what conditions each type of network forms in the continuum limit. We then fit the model to gang violence data provided by the Hollenbeck district of the Los Angeles Police Department to measure the levels of excitation and inhibition present in gang violence dynamics, as well as the stability of gang rivalries in Hollenbeck.

Keywords:  Stochastic dynamical systems, point processes, network dynamics, crime modeling.
Mathematics Subject Classification:  60G55, 70K50, 91D99.

Received: March 2011;      Revised: June 2012;      Available Online: April 2014.