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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 & 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., (). Google Scholar

[2]

H. Arendt, Crises of the Republic: Lying in Politics; Civil Disobedience; on Violence; Thoughts on Politics and Revolution,, Houghton Mifflin Harcourt, (1972). Google Scholar

[3]

P. Baudains, A. Braithwaite and S. D. Johnson, Spatial Patterns in the 2011 London Riots,, Policing, 7 (2012), 21. doi: 10.1093/police/pas049. Google Scholar

[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). Google Scholar

[5]

P. C. Bressloff and Z. P. Kilpatrick, Two-dimesional bumps in piecewise smooth neural fields with synaptic depression,, Physica D, 239 (2010), 1048. doi: 10.1016/j.physd.2010.02.016. Google Scholar

[6]

H. Berestycki and J.-P. Nadal, Self-organised critical hot spots of criminal activity,, European Journal of Applied Mathematics, 21 (2010), 371. doi: 10.1017/S0956792510000185. Google Scholar

[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). Google Scholar

[8]

G. Le Bon, Psychologie des Foules,, The Crowd: A Study of the Popular Mind, (1895). Google Scholar

[9]

J.-Ph. Bouchaud, Crises and collective socio-economic phenomena: Simple models and challenges,, Journal of Statistical Physics, 151 (2013), 567. doi: 10.1007/s10955-012-0687-3. Google Scholar

[10]

D. Braha, Global civil unrest: Contagion, self-organization, and prediction,, PloS one, 7 (2012), 1. doi: 10.1371/journal.pone.0048596. Google Scholar

[11]

R. Clarke and C. Lett, What happened when Michael Brown met Officer Darren Wilson,, 2014., (). Google Scholar

[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. doi: 10.1073/pnas.0803685105. Google Scholar

[13]

J. D. Delk, Fires & Furies: The LA Riots, What Really Happened,, ETC Publications, (1995). Google Scholar

[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. doi: 10.1038/srep01303. Google Scholar

[15]

C. Fizgerald, The Final Report: The L.A. Riots,, 2006., (). Google Scholar

[16]

M. W. Flamm, Law and Order: Street Crime, Civil Unrest, and the Crisis of Liberalism in the 1960s,, Columbia University Press, (2005). Google Scholar

[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. Google Scholar

[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. doi: 10.1142/S0218202509003887. Google Scholar

[19]

M. Granovetter, Threshold models of collective behavior,, American Journal of Sociology, 83 (1978), 1420. doi: 10.1086/226707. Google Scholar

[20]

A. G. Hawkes, Spectra of some self-exciting and mutually exciting point processes,, Biometrika, 58 (1971), 83. doi: 10.1093/biomet/58.1.83. Google Scholar

[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. doi: 10.1016/j.mathsocsci.2014.01.004. Google Scholar

[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. doi: 10.1145/2623330.2623679. Google Scholar

[23]

W. Lowery, C. D. Leonnig and M. Berman, Even before Michael Brown's slaying in Ferguson, racial questions hung over police,, 2014., (). Google Scholar

[24]

M. Lynch, The Arab Uprising The Unfinished Revolutions Of The New Middleeast,, Public Affairs, (2005). Google Scholar

[25]

B. Moore, Injustice: The Social Bases of Obedience and Revolt,, White Plains, (1978). Google Scholar

[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. doi: 10.1198/jasa.2011.ap09546. Google Scholar

[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. doi: 10.1080/13691830902826137. Google Scholar

[28]

D. J. Myers, The diffusion of collective violence: Infectiousness, susceptibility, and mass media networks,, The American Journal of Sociology, 106 (2000), 173. doi: 10.1086/303110. Google Scholar

[29]

T. Newburn, The Ferguson riots may seem similar to those in UK in 2011 - but there are stark contrasts,, 2014., (). Google Scholar

[30]

Y. Ogata, Space-time point-process models for earthquake occurrences,, Annals of the Institute of Statistical Mathematics, 50 (1998), 379. doi: 10.1023/A:1003403601725. Google Scholar

[31]

P. Peralva, Emeutes urbaines en france. les émeutes françaises racontées aux brésiliens,, HAL archives ouvertes, (2010). Google Scholar

[32]

J. Salter, Police shooting of black teenager in St. Louis reignites anger,, 2014., (). Google Scholar

[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. doi: 10.1177/002200277301700302. Google Scholar

[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. doi: 10.1007/s10940-009-9068-8. Google Scholar

[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. doi: 10.1142/S0218202508003029. Google Scholar

[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. doi: 10.1093/sf/86.2.385. Google Scholar

[37]

M. Taylor, P. Lewis and H. Clifton, Why the riots stopped: Fear, rain and a moving call for peace,, The Guardian, (2011). Google Scholar

[38]

M. Tsodyks, K. Pawelzik and H. Markram, Neural networks with dynamics synapses,, Neural computation, 10 (1998), 821. doi: 10.1162/089976698300017502. Google Scholar

[39]

A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems,, American Mathematical Society, (1994). Google Scholar

[40]

H. R. Wilson and J. D. Cowan, Excitatory and inhibit interneurons,, Biophysics, 12 (1972), 1. Google Scholar

[41]

J. K. Walton and D. Seddon, Free Markets and Food Riots: The Politics of Global Adjustment,, Wiley-Blackwell, (2008). doi: 10.1002/9780470712962. Google Scholar

show all references

References:
[1]

S. Alcaide, Movimiento 15-M: Los Ciudadanos Exigen Reconstruir La Política,, 2011., (). Google Scholar

[2]

H. Arendt, Crises of the Republic: Lying in Politics; Civil Disobedience; on Violence; Thoughts on Politics and Revolution,, Houghton Mifflin Harcourt, (1972). Google Scholar

[3]

P. Baudains, A. Braithwaite and S. D. Johnson, Spatial Patterns in the 2011 London Riots,, Policing, 7 (2012), 21. doi: 10.1093/police/pas049. Google Scholar

[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). Google Scholar

[5]

P. C. Bressloff and Z. P. Kilpatrick, Two-dimesional bumps in piecewise smooth neural fields with synaptic depression,, Physica D, 239 (2010), 1048. doi: 10.1016/j.physd.2010.02.016. Google Scholar

[6]

H. Berestycki and J.-P. Nadal, Self-organised critical hot spots of criminal activity,, European Journal of Applied Mathematics, 21 (2010), 371. doi: 10.1017/S0956792510000185. Google Scholar

[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). Google Scholar

[8]

G. Le Bon, Psychologie des Foules,, The Crowd: A Study of the Popular Mind, (1895). Google Scholar

[9]

J.-Ph. Bouchaud, Crises and collective socio-economic phenomena: Simple models and challenges,, Journal of Statistical Physics, 151 (2013), 567. doi: 10.1007/s10955-012-0687-3. Google Scholar

[10]

D. Braha, Global civil unrest: Contagion, self-organization, and prediction,, PloS one, 7 (2012), 1. doi: 10.1371/journal.pone.0048596. Google Scholar

[11]

R. Clarke and C. Lett, What happened when Michael Brown met Officer Darren Wilson,, 2014., (). Google Scholar

[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. doi: 10.1073/pnas.0803685105. Google Scholar

[13]

J. D. Delk, Fires & Furies: The LA Riots, What Really Happened,, ETC Publications, (1995). Google Scholar

[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. doi: 10.1038/srep01303. Google Scholar

[15]

C. Fizgerald, The Final Report: The L.A. Riots,, 2006., (). Google Scholar

[16]

M. W. Flamm, Law and Order: Street Crime, Civil Unrest, and the Crisis of Liberalism in the 1960s,, Columbia University Press, (2005). Google Scholar

[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. Google Scholar

[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. doi: 10.1142/S0218202509003887. Google Scholar

[19]

M. Granovetter, Threshold models of collective behavior,, American Journal of Sociology, 83 (1978), 1420. doi: 10.1086/226707. Google Scholar

[20]

A. G. Hawkes, Spectra of some self-exciting and mutually exciting point processes,, Biometrika, 58 (1971), 83. doi: 10.1093/biomet/58.1.83. Google Scholar

[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. doi: 10.1016/j.mathsocsci.2014.01.004. Google Scholar

[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. doi: 10.1145/2623330.2623679. Google Scholar

[23]

W. Lowery, C. D. Leonnig and M. Berman, Even before Michael Brown's slaying in Ferguson, racial questions hung over police,, 2014., (). Google Scholar

[24]

M. Lynch, The Arab Uprising The Unfinished Revolutions Of The New Middleeast,, Public Affairs, (2005). Google Scholar

[25]

B. Moore, Injustice: The Social Bases of Obedience and Revolt,, White Plains, (1978). Google Scholar

[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. doi: 10.1198/jasa.2011.ap09546. Google Scholar

[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. doi: 10.1080/13691830902826137. Google Scholar

[28]

D. J. Myers, The diffusion of collective violence: Infectiousness, susceptibility, and mass media networks,, The American Journal of Sociology, 106 (2000), 173. doi: 10.1086/303110. Google Scholar

[29]

T. Newburn, The Ferguson riots may seem similar to those in UK in 2011 - but there are stark contrasts,, 2014., (). Google Scholar

[30]

Y. Ogata, Space-time point-process models for earthquake occurrences,, Annals of the Institute of Statistical Mathematics, 50 (1998), 379. doi: 10.1023/A:1003403601725. Google Scholar

[31]

P. Peralva, Emeutes urbaines en france. les émeutes françaises racontées aux brésiliens,, HAL archives ouvertes, (2010). Google Scholar

[32]

J. Salter, Police shooting of black teenager in St. Louis reignites anger,, 2014., (). Google Scholar

[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. doi: 10.1177/002200277301700302. Google Scholar

[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. doi: 10.1007/s10940-009-9068-8. Google Scholar

[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. doi: 10.1142/S0218202508003029. Google Scholar

[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. doi: 10.1093/sf/86.2.385. Google Scholar

[37]

M. Taylor, P. Lewis and H. Clifton, Why the riots stopped: Fear, rain and a moving call for peace,, The Guardian, (2011). Google Scholar

[38]

M. Tsodyks, K. Pawelzik and H. Markram, Neural networks with dynamics synapses,, Neural computation, 10 (1998), 821. doi: 10.1162/089976698300017502. Google Scholar

[39]

A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems,, American Mathematical Society, (1994). Google Scholar

[40]

H. R. Wilson and J. D. Cowan, Excitatory and inhibit interneurons,, Biophysics, 12 (1972), 1. Google Scholar

[41]

J. K. Walton and D. Seddon, Free Markets and Food Riots: The Politics of Global Adjustment,, Wiley-Blackwell, (2008). doi: 10.1002/9780470712962. Google Scholar

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