# American Institute of Mathematical Sciences

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

## Gang rivalry dynamics via coupled point process networks

 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

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.
Citation: M. B. Short, G. O. Mohler, P. J. Brantingham, G. E. Tita. Gang rivalry dynamics via coupled point process networks. Discrete & 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, (1999). Google Scholar [2] C. Boehm, Blood Revenge: The Anthropology of Feuding in Montenegro and Other Tribal Societies,, University of Pennsylvania Press, (1987). Google Scholar [3] M. Cooney, Warriors and Peacemakers: How Third Parties Shape Violence,, New York University Press, (1998). Google Scholar [4] D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes,, $2^{nd}$ edition, (2008). Google Scholar [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. Google Scholar [6] M. Egesdal, C. Fathauer, K. Louie and J. Neuman, Statistical and stochastic modeling of gang rivalries in Los Angeles,, SIURO, 3 (2010), 72. doi: 10.1137/09S010459. Google Scholar [7] J. Fagan, The social organization of drug use and drug dealing among urbang gangs,, Criminology, 27 (1989), 633. Google Scholar [8] G. Farrell and K. Pease (eds.), Repeat Victimization,, Criminal Justice Press, (2001). Google Scholar [9] M. R. Gottfredson and T. Hirshi, A General Theory of Crime,, Stanford University Press, (1990). Google Scholar [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. doi: 10.1016/j.physa.2011.05.040. Google Scholar [11] J. Hespanha, An efficient MATLAB Algorithm for Graph Partitioning,, Technical report, (2004). Google Scholar [12] B. A. Jacobs and R. Wright, Street Justice: Retaliation in the Criminal Underworld,, Cambridge University Press, (2006). doi: 10.1017/CBO9780511816055. Google Scholar [13] S. D. Johnson, Repeat burglary victimisation: A tale of two theories,, Journal of Experimental Criminology, 4 (2008), 215. doi: 10.1007/s11292-008-9055-3. Google Scholar [14] P. Jones, P. J. Brantingham and L. Chayes, Statistical models of criminal behavior: The effects of law enforcement actions,, M3AS, 20 (2010), 1397. doi: 10.1142/S0218202510004647. Google Scholar [15] M. W. Klein and C. L. Maxson, Street Gang Patterns and Policies,, Oxford University Press, (2006). doi: 10.1093/acprof:oso/9780195163445.001.0001. Google Scholar [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. doi: 10.1057/sj.2011.21. Google Scholar [17] C. Maxson, Street gangs,, in Crime and Public Policy (eds. J. Q. Wilson and J. Petersilia), (2011), 158. Google Scholar [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. doi: 10.1198/jasa.2011.ap09546. Google Scholar [19] Y. Ogata, Space-time point process models for earthquake occurrences,, Ann. Inst. Statist. Math., 50 (1998), 379. doi: 10.1023/A:1003403601725. Google Scholar [20] Y. Ogata, On lewis' simulation method for point processes,, IEEE, 27 (1981), 23. doi: 10.1109/TIT.1981.1056305. Google Scholar [21] Y. Ogata, Statistical models for earthquake occurrences and residual analysis for point processes,, Journal of American Statistical Association, 83 (1988), 9. doi: 10.2307/2288914. Google Scholar [22] A. V. Papachristos, Murder by Structure: A Network Theory of Gang Homicide,, Ph.D thesis, (2007). Google Scholar [23] A. V. Papachristos, Murder by structure: Dominance relations and the social structure of gang homicide,, American Journal of Sociology, 115 (2009), 74. Google Scholar [24] S. Phillips and M. Cooney, Aiding peace, abetting violence: Third parties and the management of conflict,, American Sociological Review, 70 (2005), 334. doi: 10.1177/000312240507000207. Google Scholar [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. doi: 10.1093/aler/2.1.58. Google Scholar [26] A. B. Pitcher, Adding police to a mathematical model of burglary,, European Journal of Applied Mathematics, 21 (2010), 401. doi: 10.1017/S0956792510000112. Google Scholar [27] S. Ross, Simulation,, Second edition. Statistical Modeling and Decision Science. Academic Press, (1997). Google Scholar [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. doi: 10.1080/00045600903550428. Google Scholar [29] N. Rodriguez and A. L. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior,, M3AS, 20 (2010), 1425. doi: 10.1142/S0218202510004696. Google Scholar [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. Google Scholar [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. doi: 10.1177/0022427806298356. Google Scholar [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, (2003). Google Scholar [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. doi: 10.1142/S0218202508003029. Google Scholar [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. doi: 10.1073/pnas.0910921107. Google Scholar [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. doi: 10.1007/s10940-009-9068-8. Google Scholar

show all references

##### References:
 [1] E. Anderson, Code of the Street: Decency, Violence, and the Moral Life of the Inner City,, Norton, (1999). Google Scholar [2] C. Boehm, Blood Revenge: The Anthropology of Feuding in Montenegro and Other Tribal Societies,, University of Pennsylvania Press, (1987). Google Scholar [3] M. Cooney, Warriors and Peacemakers: How Third Parties Shape Violence,, New York University Press, (1998). Google Scholar [4] D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes,, $2^{nd}$ edition, (2008). Google Scholar [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. Google Scholar [6] M. Egesdal, C. Fathauer, K. Louie and J. Neuman, Statistical and stochastic modeling of gang rivalries in Los Angeles,, SIURO, 3 (2010), 72. doi: 10.1137/09S010459. Google Scholar [7] J. Fagan, The social organization of drug use and drug dealing among urbang gangs,, Criminology, 27 (1989), 633. Google Scholar [8] G. Farrell and K. Pease (eds.), Repeat Victimization,, Criminal Justice Press, (2001). Google Scholar [9] M. R. Gottfredson and T. Hirshi, A General Theory of Crime,, Stanford University Press, (1990). Google Scholar [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. doi: 10.1016/j.physa.2011.05.040. Google Scholar [11] J. Hespanha, An efficient MATLAB Algorithm for Graph Partitioning,, Technical report, (2004). Google Scholar [12] B. A. Jacobs and R. Wright, Street Justice: Retaliation in the Criminal Underworld,, Cambridge University Press, (2006). doi: 10.1017/CBO9780511816055. Google Scholar [13] S. D. Johnson, Repeat burglary victimisation: A tale of two theories,, Journal of Experimental Criminology, 4 (2008), 215. doi: 10.1007/s11292-008-9055-3. Google Scholar [14] P. Jones, P. J. Brantingham and L. Chayes, Statistical models of criminal behavior: The effects of law enforcement actions,, M3AS, 20 (2010), 1397. doi: 10.1142/S0218202510004647. Google Scholar [15] M. W. Klein and C. L. Maxson, Street Gang Patterns and Policies,, Oxford University Press, (2006). doi: 10.1093/acprof:oso/9780195163445.001.0001. Google Scholar [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. doi: 10.1057/sj.2011.21. Google Scholar [17] C. Maxson, Street gangs,, in Crime and Public Policy (eds. J. Q. Wilson and J. Petersilia), (2011), 158. Google Scholar [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. doi: 10.1198/jasa.2011.ap09546. Google Scholar [19] Y. Ogata, Space-time point process models for earthquake occurrences,, Ann. Inst. Statist. Math., 50 (1998), 379. doi: 10.1023/A:1003403601725. Google Scholar [20] Y. Ogata, On lewis' simulation method for point processes,, IEEE, 27 (1981), 23. doi: 10.1109/TIT.1981.1056305. Google Scholar [21] Y. Ogata, Statistical models for earthquake occurrences and residual analysis for point processes,, Journal of American Statistical Association, 83 (1988), 9. doi: 10.2307/2288914. Google Scholar [22] A. V. Papachristos, Murder by Structure: A Network Theory of Gang Homicide,, Ph.D thesis, (2007). Google Scholar [23] A. V. Papachristos, Murder by structure: Dominance relations and the social structure of gang homicide,, American Journal of Sociology, 115 (2009), 74. Google Scholar [24] S. Phillips and M. Cooney, Aiding peace, abetting violence: Third parties and the management of conflict,, American Sociological Review, 70 (2005), 334. doi: 10.1177/000312240507000207. Google Scholar [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. doi: 10.1093/aler/2.1.58. Google Scholar [26] A. B. Pitcher, Adding police to a mathematical model of burglary,, European Journal of Applied Mathematics, 21 (2010), 401. doi: 10.1017/S0956792510000112. Google Scholar [27] S. Ross, Simulation,, Second edition. Statistical Modeling and Decision Science. Academic Press, (1997). Google Scholar [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. doi: 10.1080/00045600903550428. Google Scholar [29] N. Rodriguez and A. L. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior,, M3AS, 20 (2010), 1425. doi: 10.1142/S0218202510004696. Google Scholar [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. Google Scholar [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. doi: 10.1177/0022427806298356. Google Scholar [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, (2003). Google Scholar [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. doi: 10.1142/S0218202508003029. Google Scholar [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. doi: 10.1073/pnas.0910921107. Google Scholar [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. doi: 10.1007/s10940-009-9068-8. Google Scholar
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