Gang rivalry dynamics via coupled point process networks

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July 2014, 19(5): 1459-1477. doi: 10.3934/dcdsb.2014.19.1459

Gang rivalry dynamics via coupled point process networks

M. B. Short ^1, , G. O. Mohler ^2, , P. J. Brantingham ^3, and G. E. Tita ^4,

1.	School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States
2.	Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, CA 95053, United States
3.	Department of Anthropology, UCLA, Los Angeles, CA 90095, United States
4.	Department of Criminology, Law and Society, UC Irvine, Irvine, CA 92697, United States

Received March 2011 Revised June 2012 Published April 2014

Abstract
Related Papers

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: point processes, network dynamics, crime modeling., Stochastic dynamical systems.

Mathematics Subject Classification: 60G55, 70K50, 91D9.

Citation: M. B. Short, G. O. Mohler, P. J. Brantingham, G. E. Tita. Gang rivalry dynamics via coupled point process networks. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1459-1477. doi: 10.3934/dcdsb.2014.19.1459

References:

[1]	E. Anderson, Code of the Street: Decency, Violence, and the Moral Life of the Inner City, Norton, New York, 1999.
[2]	C. Boehm, Blood Revenge: The Anthropology of Feuding in Montenegro and Other Tribal Societies, University of Pennsylvania Press, Philadelphia, 1987.
[3]	M. Cooney, Warriors and Peacemakers: How Third Parties Shape Violence, New York University Press, New York, 1998.
[4]	D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, $2^{nd}$ edition, Springer, New York, 2008.
[5]	S. H. Decker and G. D. Curry, Gangs, gang homicides, and gang loyalty: Organized crimes or disorganized criminals, Journal of Criminal Justice, 30 (2002), 343-352.
[6]	M. Egesdal, C. Fathauer, K. Louie and J. Neuman, Statistical and stochastic modeling of gang rivalries in Los Angeles, SIURO, 3 (2010), 72-94. doi: 10.1137/09S010459.
[7]	J. Fagan, The social organization of drug use and drug dealing among urbang gangs, Criminology, 27 (1989), 633-670.
[8]	G. Farrell and K. Pease (eds.), Repeat Victimization, Criminal Justice Press, New York, 2001.
[9]	M. R. Gottfredson and T. Hirshi, A General Theory of Crime, Stanford University Press, 1990.
[10]	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, 390 (2011), 3894-3914. doi: 10.1016/j.physa.2011.05.040.
[11]	J. Hespanha, An efficient MATLAB Algorithm for Graph Partitioning, Technical report, 2004. Available from: http://www.ece.ucsb.edu/~hespanha/techrep.html.
[12]	B. A. Jacobs and R. Wright, Street Justice: Retaliation in the Criminal Underworld, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511816055.
[13]	S. D. Johnson, Repeat burglary victimisation: A tale of two theories, Journal of Experimental Criminology, 4 (2008), 215-240. doi: 10.1007/s11292-008-9055-3.
[14]	P. Jones, P. J. Brantingham and L. Chayes, Statistical models of criminal behavior: The effects of law enforcement actions, M3AS, 20 (2010), 1397-1423. doi: 10.1142/S0218202510004647.
[15]	M. W. Klein and C. L. Maxson, Street Gang Patterns and Policies, Oxford University Press, New York, 2006. doi: 10.1093/acprof:oso/9780195163445.001.0001.
[16]	E. Lewis, G. O. Mohler, P. J. Brantingham and A. Bertozzi, Self-exciting point process models of civilian deaths in Iraq, Security Journal, 25 (2011), 244-264. doi: 10.1057/sj.2011.21.
[17]	C. Maxson, Street gangs, in Crime and Public Policy (eds. J. Q. Wilson and J. Petersilia), Oxford University Press, New York, (2011), 158-182.
[18]	G. O. Mohler, M. B. Short, P. J. Brantingham, F. Schoenberg and G. E. Tita, Self-exciting point process modeling of crime, Journal of the American Statistical Association, 106 (2011), 100-108. doi: 10.1198/jasa.2011.ap09546.
[19]	Y. Ogata, Space-time point process models for earthquake occurrences, Ann. Inst. Statist. Math., 50 (1998), 379-402. doi: 10.1023/A:1003403601725.
[20]	Y. Ogata, On lewis' simulation method for point processes, IEEE, 27 (1981), 23-31. doi: 10.1109/TIT.1981.1056305.
[21]	Y. Ogata, Statistical models for earthquake occurrences and residual analysis for point processes, Journal of American Statistical Association, 83 (1988), 9-27. doi: 10.2307/2288914.
[22]	A. V. Papachristos, Murder by Structure: A Network Theory of Gang Homicide, Ph.D thesis, University of Chicago, 2007.
[23]	A. V. Papachristos, Murder by structure: Dominance relations and the social structure of gang homicide, American Journal of Sociology, 115 (2009), 74-128.
[24]	S. Phillips and M. Cooney, Aiding peace, abetting violence: Third parties and the management of conflict, American Sociological Review, 70 (2005), 334-354. doi: 10.1177/000312240507000207.
[25]	A. M. Piehl, D. M. Kennedy and A. A. Braga, Problem solving and youth violence: An evaluation of the Boston Gun Project, American Law and Economics Review, 2 (2000), 58-106. doi: 10.1093/aler/2.1.58.
[26]	A. B. Pitcher, Adding police to a mathematical model of burglary, European Journal of Applied Mathematics, 21 (2010), 401-419. doi: 10.1017/S0956792510000112.
[27]	S. Ross, Simulation, Second edition. Statistical Modeling and Decision Science. Academic Press, Inc., San Diego, CA, 1997.
[28]	S. M. Radil, C. Flint and G. E. Tita, Spatializing social networks: Using social network analysis to investigate geographies of gang rivalry, territoriality and violence in Los Angeles, Annals of the Association of American Geographers, 100 (2010), 307-326. doi: 10.1080/00045600903550428.
[29]	N. Rodriguez and A. L. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior, M3AS, 20 (2010), 1425-1457. doi: 10.1142/S0218202510004696.
[30]	T. A. Taniguchi, J. H. Ratcliffe and R. B. Taylor, Gang set space, drug markets, and drime around drug corners in Camden, Journal of Research in Crime and Delinquency, 48 (2011), 327-363.
[31]	G. E. Tita and G. Ridgeway, The impact of gang formation on local patterns of crime, Journal of Research in Crime and Delinquency, 44 (2007), 208-237. doi: 10.1177/0022427806298356.
[32]	G. E. Tita, J. K. Riley, G. Ridgeway, C. Grammich, A. F. Abrahamse and P. Greenwood, Reducing Gun Violence: Results from and Intervention in East Los Angeles, RAND Press, Santa Monica, 2003.
[33]	M. B. Short, M. R. D'Orsogna, V. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. Chayes, A statistical model of criminal behavior, M3AS, 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029.
[34]	M. B. Short, P. J. Brantingham, A. L. Bertozzi and G. E. Tita, Dissipation and displacement of hotspots in reaction-diffusion models of crime, PNAS, 107 (2010), 3961-3965. doi: 10.1073/pnas.0910921107.
[35]	M. B. Short, M. R. D'Orsogna, P. J. Brantingham and G. E. Tita, Measuring and modeling repeat and near-repeat burglary effects, J. Quant. Criminol., 25 (2009), 325-339. doi: 10.1007/s10940-009-9068-8.

show all references

References:

[1]	E. Anderson, Code of the Street: Decency, Violence, and the Moral Life of the Inner City, Norton, New York, 1999.
[2]	C. Boehm, Blood Revenge: The Anthropology of Feuding in Montenegro and Other Tribal Societies, University of Pennsylvania Press, Philadelphia, 1987.
[3]	M. Cooney, Warriors and Peacemakers: How Third Parties Shape Violence, New York University Press, New York, 1998.
[4]	D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, $2^{nd}$ edition, Springer, New York, 2008.
[5]	S. H. Decker and G. D. Curry, Gangs, gang homicides, and gang loyalty: Organized crimes or disorganized criminals, Journal of Criminal Justice, 30 (2002), 343-352.
[6]	M. Egesdal, C. Fathauer, K. Louie and J. Neuman, Statistical and stochastic modeling of gang rivalries in Los Angeles, SIURO, 3 (2010), 72-94. doi: 10.1137/09S010459.
[7]	J. Fagan, The social organization of drug use and drug dealing among urbang gangs, Criminology, 27 (1989), 633-670.
[8]	G. Farrell and K. Pease (eds.), Repeat Victimization, Criminal Justice Press, New York, 2001.
[9]	M. R. Gottfredson and T. Hirshi, A General Theory of Crime, Stanford University Press, 1990.
[10]	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, 390 (2011), 3894-3914. doi: 10.1016/j.physa.2011.05.040.
[11]	J. Hespanha, An efficient MATLAB Algorithm for Graph Partitioning, Technical report, 2004. Available from: http://www.ece.ucsb.edu/~hespanha/techrep.html.
[12]	B. A. Jacobs and R. Wright, Street Justice: Retaliation in the Criminal Underworld, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511816055.
[13]	S. D. Johnson, Repeat burglary victimisation: A tale of two theories, Journal of Experimental Criminology, 4 (2008), 215-240. doi: 10.1007/s11292-008-9055-3.
[14]	P. Jones, P. J. Brantingham and L. Chayes, Statistical models of criminal behavior: The effects of law enforcement actions, M3AS, 20 (2010), 1397-1423. doi: 10.1142/S0218202510004647.
[15]	M. W. Klein and C. L. Maxson, Street Gang Patterns and Policies, Oxford University Press, New York, 2006. doi: 10.1093/acprof:oso/9780195163445.001.0001.
[16]	E. Lewis, G. O. Mohler, P. J. Brantingham and A. Bertozzi, Self-exciting point process models of civilian deaths in Iraq, Security Journal, 25 (2011), 244-264. doi: 10.1057/sj.2011.21.
[17]	C. Maxson, Street gangs, in Crime and Public Policy (eds. J. Q. Wilson and J. Petersilia), Oxford University Press, New York, (2011), 158-182.
[18]	G. O. Mohler, M. B. Short, P. J. Brantingham, F. Schoenberg and G. E. Tita, Self-exciting point process modeling of crime, Journal of the American Statistical Association, 106 (2011), 100-108. doi: 10.1198/jasa.2011.ap09546.
[19]	Y. Ogata, Space-time point process models for earthquake occurrences, Ann. Inst. Statist. Math., 50 (1998), 379-402. doi: 10.1023/A:1003403601725.
[20]	Y. Ogata, On lewis' simulation method for point processes, IEEE, 27 (1981), 23-31. doi: 10.1109/TIT.1981.1056305.
[21]	Y. Ogata, Statistical models for earthquake occurrences and residual analysis for point processes, Journal of American Statistical Association, 83 (1988), 9-27. doi: 10.2307/2288914.
[22]	A. V. Papachristos, Murder by Structure: A Network Theory of Gang Homicide, Ph.D thesis, University of Chicago, 2007.
[23]	A. V. Papachristos, Murder by structure: Dominance relations and the social structure of gang homicide, American Journal of Sociology, 115 (2009), 74-128.
[24]	S. Phillips and M. Cooney, Aiding peace, abetting violence: Third parties and the management of conflict, American Sociological Review, 70 (2005), 334-354. doi: 10.1177/000312240507000207.
[25]	A. M. Piehl, D. M. Kennedy and A. A. Braga, Problem solving and youth violence: An evaluation of the Boston Gun Project, American Law and Economics Review, 2 (2000), 58-106. doi: 10.1093/aler/2.1.58.
[26]	A. B. Pitcher, Adding police to a mathematical model of burglary, European Journal of Applied Mathematics, 21 (2010), 401-419. doi: 10.1017/S0956792510000112.
[27]	S. Ross, Simulation, Second edition. Statistical Modeling and Decision Science. Academic Press, Inc., San Diego, CA, 1997.
[28]	S. M. Radil, C. Flint and G. E. Tita, Spatializing social networks: Using social network analysis to investigate geographies of gang rivalry, territoriality and violence in Los Angeles, Annals of the Association of American Geographers, 100 (2010), 307-326. doi: 10.1080/00045600903550428.
[29]	N. Rodriguez and A. L. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior, M3AS, 20 (2010), 1425-1457. doi: 10.1142/S0218202510004696.
[30]	T. A. Taniguchi, J. H. Ratcliffe and R. B. Taylor, Gang set space, drug markets, and drime around drug corners in Camden, Journal of Research in Crime and Delinquency, 48 (2011), 327-363.
[31]	G. E. Tita and G. Ridgeway, The impact of gang formation on local patterns of crime, Journal of Research in Crime and Delinquency, 44 (2007), 208-237. doi: 10.1177/0022427806298356.
[32]	G. E. Tita, J. K. Riley, G. Ridgeway, C. Grammich, A. F. Abrahamse and P. Greenwood, Reducing Gun Violence: Results from and Intervention in East Los Angeles, RAND Press, Santa Monica, 2003.
[33]	M. B. Short, M. R. D'Orsogna, V. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. Chayes, A statistical model of criminal behavior, M3AS, 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029.
[34]	M. B. Short, P. J. Brantingham, A. L. Bertozzi and G. E. Tita, Dissipation and displacement of hotspots in reaction-diffusion models of crime, PNAS, 107 (2010), 3961-3965. doi: 10.1073/pnas.0910921107.
[35]	M. B. Short, M. R. D'Orsogna, P. J. Brantingham and G. E. Tita, Measuring and modeling repeat and near-repeat burglary effects, J. Quant. Criminol., 25 (2009), 325-339. doi: 10.1007/s10940-009-9068-8.

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