# American Institute of Mathematical Sciences

• Previous Article
Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model
• NHM Home
• This Issue
• Next Article
From a systems theory of sociology to modeling the onset and evolution of criminality
September  2015, 10(3): 443-475. doi: 10.3934/nhm.2015.10.443

## A model of riots dynamics: Shocks, diffusion and thresholds

 1 Ecole des Hautes Etudes en Sciences Sociales and CNRS, Centre d'Analyse et de Mathématique Sociales (CAMS, UMR8557), 190-198, avenue de France - 75013 Paris, France 2 Ecole des Hautes Etudes en Sciences Sociales and CNRS, Centre d'Analyse et de Mathématique Sociales (CAMS, UMR8557), 190-198 avenue de France - 75013 Paris, France 3 UNC Chapel Hill, Department of Mathematics, Phillips Hall, CB # 3250, Chapel Hill, NC 27599-3250, United States

Received  November 2014 Revised  February 2015 Published  July 2015

We introduce and analyze several variants of a system of differential equations which model the dynamics of social outbursts, such as riots. The systems involve the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. Our models include the effects of exogenous and endogenous factors as well as various propagation mechanisms. From numerical and mathematical analysis of these models we show that the assumptions made on how different locations influence one another and how the tension in the system disperses play a major role on the qualitative behavior of bursts of social unrest. Furthermore, we analyze here various properties of these systems, such as the existence of traveling wave solutions, and formulate some new open mathematical problems which arise from our work.
Citation: Henri Berestycki, Jean-Pierre Nadal, Nancy Rodíguez. A model of riots dynamics: Shocks, diffusion and thresholds. Networks and Heterogeneous Media, 2015, 10 (3) : 443-475. doi: 10.3934/nhm.2015.10.443
##### References:
 [1] S. Alcaide, Movimiento 15-M: Los Ciudadanos Exigen Reconstruir La Política,, 2011., (). [2] H. Arendt, Crises of the Republic: Lying in Politics; Civil Disobedience; on Violence; Thoughts on Politics and Revolution, Houghton Mifflin Harcourt, 1972. [3] P. Baudains, A. Braithwaite and S. D. Johnson, Spatial Patterns in the 2011 London Riots, Policing, 7 (2012), 21-31. doi: 10.1093/police/pas049. [4] J.-Ph. Bouchaud, C. Borghesi and P. Jensen, On the emergence of an "intention field'' for socially cohesive agents, Journal of Statistical Mechanics: Theory and Experiment, (2014), P03010, 15 pp. [5] P. C. Bressloff and Z. P. Kilpatrick, Two-dimesional bumps in piecewise smooth neural fields with synaptic depression, Physica D, 239 (2010), 1048-1060. doi: 10.1016/j.physd.2010.02.016. [6] H. Berestycki and J.-P. Nadal, Self-organised critical hot spots of criminal activity, European Journal of Applied Mathematics, 21 (2010), 371-399. doi: 10.1017/S0956792510000185. [7] H. Berestycki and N. Rodríguez, Analysis of a heterogeneous model for riot dynamics : the effect of censorship of information, to appear in the European Journal of Applied Mathematics, (2015), 28 pp. [8] G. Le Bon, Psychologie des Foules, The Crowd: A Study of the Popular Mind, Editions Felix Alcan, 1895 (9th ed. 1905), Viking Press, New York, 1960. [9] J.-Ph. Bouchaud, Crises and collective socio-economic phenomena: Simple models and challenges, Journal of Statistical Physics, 151 (2013), 567-606. doi: 10.1007/s10955-012-0687-3. [10] D. Braha, Global civil unrest: Contagion, self-organization, and prediction, PloS one, 7 (2012), e48596, 1-9. doi: 10.1371/journal.pone.0048596. [11] R. Clarke and C. Lett, What happened when Michael Brown met Officer Darren Wilson,, 2014., (). [12] R. Crane and D. Sornette, Robust dynamic classes revealed by measuring the response function of a social system, Proceedings of the National Academy of Sciences of the United States of America, 105 (2008), 15649-15653. doi: 10.1073/pnas.0803685105. [13] J. D. Delk, Fires & Furies: The LA Riots, What Really Happened, ETC Publications, 1995. [14] T. P. Davies, H. M. Fry, A. G. Wilson and S. R Bishop, A mathematical model of the London riots and their policing, Scientific Reports, 3 (2013), 1-9. doi: 10.1038/srep01303. [15] C. Fizgerald, The Final Report: The L.A. Riots,, 2006., (). [16] M. W. Flamm, Law and Order: Street Crime, Civil Unrest, and the Crisis of Liberalism in the 1960s, Columbia University Press, 2005. [17] S. González-Bailón, J. Borge-Holthoefer, A. Rivero and Y. Moreno, The dynamics of protest recruitment through an online network, Scientific reports, 1 (2011), 1-7. [18] M. B. Gordon, J-P. Nadal, D. Phan and V. Semeshenko, Discrete choices under social influence: Generic properties, Mathematical Models and Methods in Applied Sciences (M3AS), 19 (2009), 1441-1481. doi: 10.1142/S0218202509003887. [19] M. Granovetter, Threshold models of collective behavior, American Journal of Sociology, 83 (1978), 1420-1443. doi: 10.1086/226707. [20] A. G. Hawkes, Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58 (1971), 83-90. doi: 10.1093/biomet/58.1.83. [21] J. C. Lang and H. De Sterck, The Arab Spring: A simple compartmental model for the dynamics of a revolution, Mathematical Social Sciences, 69 (2014), 12-21. doi: 10.1016/j.mathsocsci.2014.01.004. [22] L. Li, H. Deng, A. Dong, Y. Chang and H. Zha, Identifying and Labeling Search Tasks via Query-based Hawkes Processes, Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, (2014), 731-740. doi: 10.1145/2623330.2623679. [23] W. Lowery, C. D. Leonnig and M. Berman, Even before Michael Brown's slaying in Ferguson, racial questions hung over police,, 2014., (). [24] M. Lynch, The Arab Uprising The Unfinished Revolutions Of The New Middleeast, Public Affairs, New York, first edition, 2005. [25] B. Moore, Injustice: The Social Bases of Obedience and Revolt, White Plains, New York, 1978. [26] G. O. Mohler, M. B. Short, P. J. Brantingham, F. P. 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. [27] L. Mucchielli, Autumn 2005: A review of the most important riot in the history of french contemporary society, Journal of Ethnic and Migration Studies, 35 (2009), 731-751. doi: 10.1080/13691830902826137. [28] D. J. Myers, The diffusion of collective violence: Infectiousness, susceptibility, and mass media networks, The American Journal of Sociology, 106 (2000), 173-208. doi: 10.1086/303110. [29] T. Newburn, The Ferguson riots may seem similar to those in UK in 2011 - but there are stark contrasts,, 2014., (). [30] Y. Ogata, Space-time point-process models for earthquake occurrences, Annals of the Institute of Statistical Mathematics, 50 (1998), 379-402. doi: 10.1023/A:1003403601725. [31] P. Peralva, Emeutes urbaines en france. les émeutes françaises racontées aux brésiliens, HAL archives ouvertes, https://hal.archives-ouvertes.fr/halshs-0048422, 2010. [32] J. Salter, Police shooting of black teenager in St. Louis reignites anger,, 2014., (). [33] T. C. Schelling, Hockey helmets, concealed weapons, and daylight saving: A study of binary choices with externalities, Journal of Conflict Resolution, 17 (1973), 381-428. doi: 10.1177/002200277301700302. [34] M. B. Short, M. R. D'Orsogna, P. J. Brantingham and G. E. Tita, Measuring and modeling repeat and near-repeat burglary effects, Journal of Quantitative Criminology, 25 (2009), 325-339. doi: 10.1007/s10940-009-9068-8. [35] M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029. [36] D. A. Snow, R. Vliegenthart and C. Corrigall-Brown, Framing the French riots: A comparative study of frame variation, Social Forces, 86 (2007), 385-415. doi: 10.1093/sf/86.2.385. [37] M. Taylor, P. Lewis and H. Clifton, Why the riots stopped: Fear, rain and a moving call for peace, The Guardian, December 2011. [38] M. Tsodyks, K. Pawelzik and H. Markram, Neural networks with dynamics synapses, Neural computation, 10 (1998), 821-835. doi: 10.1162/089976698300017502. [39] A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, translatio edition, 1994. [40] H. R. Wilson and J. D. Cowan, Excitatory and inhibit interneurons, Biophysics, 12 (1972), 1-24. [41] J. K. Walton and D. Seddon, Free Markets and Food Riots: The Politics of Global Adjustment, Wiley-Blackwell, 2008. doi: 10.1002/9780470712962.

show all references

##### References:
 [1] S. Alcaide, Movimiento 15-M: Los Ciudadanos Exigen Reconstruir La Política,, 2011., (). [2] H. Arendt, Crises of the Republic: Lying in Politics; Civil Disobedience; on Violence; Thoughts on Politics and Revolution, Houghton Mifflin Harcourt, 1972. [3] P. Baudains, A. Braithwaite and S. D. Johnson, Spatial Patterns in the 2011 London Riots, Policing, 7 (2012), 21-31. doi: 10.1093/police/pas049. [4] J.-Ph. Bouchaud, C. Borghesi and P. Jensen, On the emergence of an "intention field'' for socially cohesive agents, Journal of Statistical Mechanics: Theory and Experiment, (2014), P03010, 15 pp. [5] P. C. Bressloff and Z. P. Kilpatrick, Two-dimesional bumps in piecewise smooth neural fields with synaptic depression, Physica D, 239 (2010), 1048-1060. doi: 10.1016/j.physd.2010.02.016. [6] H. Berestycki and J.-P. Nadal, Self-organised critical hot spots of criminal activity, European Journal of Applied Mathematics, 21 (2010), 371-399. doi: 10.1017/S0956792510000185. [7] H. Berestycki and N. Rodríguez, Analysis of a heterogeneous model for riot dynamics : the effect of censorship of information, to appear in the European Journal of Applied Mathematics, (2015), 28 pp. [8] G. Le Bon, Psychologie des Foules, The Crowd: A Study of the Popular Mind, Editions Felix Alcan, 1895 (9th ed. 1905), Viking Press, New York, 1960. [9] J.-Ph. Bouchaud, Crises and collective socio-economic phenomena: Simple models and challenges, Journal of Statistical Physics, 151 (2013), 567-606. doi: 10.1007/s10955-012-0687-3. [10] D. Braha, Global civil unrest: Contagion, self-organization, and prediction, PloS one, 7 (2012), e48596, 1-9. doi: 10.1371/journal.pone.0048596. [11] R. Clarke and C. Lett, What happened when Michael Brown met Officer Darren Wilson,, 2014., (). [12] R. Crane and D. Sornette, Robust dynamic classes revealed by measuring the response function of a social system, Proceedings of the National Academy of Sciences of the United States of America, 105 (2008), 15649-15653. doi: 10.1073/pnas.0803685105. [13] J. D. Delk, Fires & Furies: The LA Riots, What Really Happened, ETC Publications, 1995. [14] T. P. Davies, H. M. Fry, A. G. Wilson and S. R Bishop, A mathematical model of the London riots and their policing, Scientific Reports, 3 (2013), 1-9. doi: 10.1038/srep01303. [15] C. Fizgerald, The Final Report: The L.A. Riots,, 2006., (). [16] M. W. Flamm, Law and Order: Street Crime, Civil Unrest, and the Crisis of Liberalism in the 1960s, Columbia University Press, 2005. [17] S. González-Bailón, J. Borge-Holthoefer, A. Rivero and Y. Moreno, The dynamics of protest recruitment through an online network, Scientific reports, 1 (2011), 1-7. [18] M. B. Gordon, J-P. Nadal, D. Phan and V. Semeshenko, Discrete choices under social influence: Generic properties, Mathematical Models and Methods in Applied Sciences (M3AS), 19 (2009), 1441-1481. doi: 10.1142/S0218202509003887. [19] M. Granovetter, Threshold models of collective behavior, American Journal of Sociology, 83 (1978), 1420-1443. doi: 10.1086/226707. [20] A. G. Hawkes, Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58 (1971), 83-90. doi: 10.1093/biomet/58.1.83. [21] J. C. Lang and H. De Sterck, The Arab Spring: A simple compartmental model for the dynamics of a revolution, Mathematical Social Sciences, 69 (2014), 12-21. doi: 10.1016/j.mathsocsci.2014.01.004. [22] L. Li, H. Deng, A. Dong, Y. Chang and H. Zha, Identifying and Labeling Search Tasks via Query-based Hawkes Processes, Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, (2014), 731-740. doi: 10.1145/2623330.2623679. [23] W. Lowery, C. D. Leonnig and M. Berman, Even before Michael Brown's slaying in Ferguson, racial questions hung over police,, 2014., (). [24] M. Lynch, The Arab Uprising The Unfinished Revolutions Of The New Middleeast, Public Affairs, New York, first edition, 2005. [25] B. Moore, Injustice: The Social Bases of Obedience and Revolt, White Plains, New York, 1978. [26] G. O. Mohler, M. B. Short, P. J. Brantingham, F. P. 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. [27] L. Mucchielli, Autumn 2005: A review of the most important riot in the history of french contemporary society, Journal of Ethnic and Migration Studies, 35 (2009), 731-751. doi: 10.1080/13691830902826137. [28] D. J. Myers, The diffusion of collective violence: Infectiousness, susceptibility, and mass media networks, The American Journal of Sociology, 106 (2000), 173-208. doi: 10.1086/303110. [29] T. Newburn, The Ferguson riots may seem similar to those in UK in 2011 - but there are stark contrasts,, 2014., (). [30] Y. Ogata, Space-time point-process models for earthquake occurrences, Annals of the Institute of Statistical Mathematics, 50 (1998), 379-402. doi: 10.1023/A:1003403601725. [31] P. Peralva, Emeutes urbaines en france. les émeutes françaises racontées aux brésiliens, HAL archives ouvertes, https://hal.archives-ouvertes.fr/halshs-0048422, 2010. [32] J. Salter, Police shooting of black teenager in St. Louis reignites anger,, 2014., (). [33] T. C. Schelling, Hockey helmets, concealed weapons, and daylight saving: A study of binary choices with externalities, Journal of Conflict Resolution, 17 (1973), 381-428. doi: 10.1177/002200277301700302. [34] M. B. Short, M. R. D'Orsogna, P. J. Brantingham and G. E. Tita, Measuring and modeling repeat and near-repeat burglary effects, Journal of Quantitative Criminology, 25 (2009), 325-339. doi: 10.1007/s10940-009-9068-8. [35] M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029. [36] D. A. Snow, R. Vliegenthart and C. Corrigall-Brown, Framing the French riots: A comparative study of frame variation, Social Forces, 86 (2007), 385-415. doi: 10.1093/sf/86.2.385. [37] M. Taylor, P. Lewis and H. Clifton, Why the riots stopped: Fear, rain and a moving call for peace, The Guardian, December 2011. [38] M. Tsodyks, K. Pawelzik and H. Markram, Neural networks with dynamics synapses, Neural computation, 10 (1998), 821-835. doi: 10.1162/089976698300017502. [39] A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, translatio edition, 1994. [40] H. R. Wilson and J. D. Cowan, Excitatory and inhibit interneurons, Biophysics, 12 (1972), 1-24. [41] J. K. Walton and D. Seddon, Free Markets and Food Riots: The Politics of Global Adjustment, Wiley-Blackwell, 2008. doi: 10.1002/9780470712962.
 [1] Joelma Azevedo, Juan Carlos Pozo, Arlúcio Viana. Global solutions to the non-local Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2515-2535. doi: 10.3934/dcdsb.2021146 [2] Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347 [3] Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681 [4] E. S. Van Vleck, Aijun Zhang. Competing interactions and traveling wave solutions in lattice differential equations. Communications on Pure and Applied Analysis, 2016, 15 (2) : 457-475. doi: 10.3934/cpaa.2016.15.457 [5] Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001 [6] Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 [7] Imran H. Biswas, Indranil Chowdhury. On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian. Communications on Pure and Applied Analysis, 2016, 15 (3) : 907-927. doi: 10.3934/cpaa.2016.15.907 [8] Huxiao Luo, Xianhua Tang, Zu Gao. Sign-changing solutions for non-local elliptic equations with asymptotically linear term. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1147-1159. doi: 10.3934/cpaa.2018055 [9] Florent Berthelin, Paola Goatin. Regularity results for the solutions of a non-local model of traffic flow. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3197-3213. doi: 10.3934/dcds.2019132 [10] Stig-Olof Londen, Hana Petzeltová. Convergence of solutions of a non-local phase-field system. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 653-670. doi: 10.3934/dcdss.2011.4.653 [11] Juan C. Pozo, Vicente Vergara. Fundamental solutions and decay of fully non-local problems. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 639-666. doi: 10.3934/dcds.2019026 [12] Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126 [13] Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255 [14] Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053 [15] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [16] Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032 [17] Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 [18] Aijun Zhang. Traveling wave solutions of periodic nonlocal Fisher-KPP equations with non-compact asymmetric kernel. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022061 [19] Kyudong Choi. Persistence of Hölder continuity for non-local integro-differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1741-1771. doi: 10.3934/dcds.2013.33.1741 [20] Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417

2020 Impact Factor: 1.213