# American Institute of Mathematical Sciences

July  2014, 19(5): i-v. doi: 10.3934/dcdsb.2014.19.5i

## Preface to special issue on mathematics of social systems

 1 University of California Los Angeles, Department of Mathematics, 520 Portola Plaza Box 951555, Los Angeles, CA 90095-1555

Published  April 2014

This special issue is an outgrowth of a minisyposium titled Mathematics of Social Systems" held at the 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications, held in Orlando, FL in July 2012. Presenters from that session were invited to submit papers that were reviewed using the usual procedures of the DCDS journals, along with additional authors from the field. Mathematics has already had a significant impact on basic research involving fundamental problems in physical sciences, biological sciences, computer science and engineering. Examples include understanding of the equations of incompressible fluid dynamics, shock wave theory and compressible gas dynamics, ocean modeling, algorithms for image processing and compressive sensing, and biological problems such as models for invasive species, spread of disease, and more recently systems biology for modeling of complex organisms and complex patterns of disease. This impact has yet to come to fruition in a comprehensive way for complex social behavior. While computational models such as agent-based systems and well-known statistical methods are widely used in the social sciences, applied mathematics has not to date had a core impact in the social sciences at the level that it achieves in the physical and life sciences. However in recent years we have seen a growth of work in this direction and ensuing new mathematics problems that must be tackled to understand such problems. Technical approaches include ideas from statistical physics, nonlinear partial differential equations of all types, statistics and inverse problems, and stochastic processes and social network models. The collection of papers presented in this issue provides a backdrop of the current state of the art results in this developing new research area in applied mathematics. The body of work encompasses many of the challenges in understanding these discrete complex systems and their related continuum approximations.

Citation: Andrea L. Bertozzi. Preface to special issue on mathematics of social systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : i-v. doi: 10.3934/dcdsb.2014.19.5i
##### References:
 [1] A. B. T. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents,, Discrete and Continuous Dynamical Systems, 34 (2014), 1249.   Google Scholar [2] D. Balagué, J. A. Carrillo and Y. Yao, Confinement for repulsive-attractive kernels,, Discrete and Continuous Dynamical Systems, 34 (2014), 1227.   Google Scholar [3] J. Bedrossian and N. Rodríguez, Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $R^d$,, Discrete and Continuous Dynamical Systems, 34 (2014), 1279.   Google Scholar [4] J. Bedrossian, N. Rodríguez and A. L. Bertozzi, Local and global well-posedness for aggregation equations and Keller-Segel models with degenerate diffusion,, Nonlinearity, 24 (2011), 1683.  doi: 10.1088/0951-7715/24/6/001.  Google Scholar [5] M. Burger, M. Di Francesco, P. A. Markowich and M. T. Wolfram, Mean field games with nonlinear mobilities in pedestrian dynamics,, Discrete and Continuous Dynamical Systems, 34 (2014), 1311.   Google Scholar [6] Y. S. Cho, A. Galstyan, P. J. Brantingham and G. Tita, Latent self-exciting point process model for spatial-temporal networks,, Discrete and Continuous Dynamical Systems, 34 (2014), 1335.   Google Scholar [7] F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Transactions on Automatic Control, 52 (2007), 852.  doi: 10.1109/TAC.2007.895842.  Google Scholar [8] M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006).  doi: 10.1103/PhysRevLett.96.104302.  Google Scholar [9] R. Ghosh and K. Lerman, Rethinking Centrality: The role of dynamical processes in social network analysis,, Discrete and Continuous Dynamical Systems, 34 (2014), 1355.   Google Scholar [10] R. A. Hegemann, E. A. Lewis and A. L. Bertozzi, An estimate & Score Algorithm for simultaneous parameter estimation and reconstruction of incomplete data on social networks,, Security Informatics, 2 (): 1.  doi: 10.1186/2190-8532-2-1.  Google Scholar [11] T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions,, Phys. Rev. E, 84 (2011).  doi: 10.1103/PhysRevE.84.015203.  Google Scholar [12] T. Kolokolnikov, M. J. Ward and J. Wei, The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime,, Discrete and Continuous Dynamical Systems, 34 (2014), 1373.   Google Scholar [13] M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models,, J. Math. Biol., 45 (2002), 219.  doi: 10.1007/s002850200144.  Google Scholar [14] A. Mackey, T. Kolokolnikov and A. L. Bertozzi, Two-species particle aggregation and stability of co-dimension one solutions,, Discrete and Continuous Dynamical Systems, 34 (2014), 1411.   Google Scholar [15] S. G. McCalla, Paladins as predators: Invasive waves in a spatial evolutionary adversarial game,, Discrete and Continuous Dynamical Systems, 34 (2014), 1437.   Google Scholar [16] 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. Meth. Appl. Sci., 18 (2008), 1249.  doi: 10.1142/S0218202508003029.  Google Scholar [17] M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime - hotpsots, bifurcations, and suppression,, SIAM J. Appl. Dyn. Sys., 9 (2010), 462.  doi: 10.1137/090759069.  Google Scholar [18] M. B. Short, P. J. Brantingham, A. L. Bertozzi and G. E. Tita, Dissipation and displacement of hotpsots in reaction-diffusion models of crime,, Proc. Nat. Acad. Sci., 107 (2010), 3961.   Google Scholar [19] M. B. Short, P. J. Brantingham and M. R. D'Orsogna, Cooperation and punishment in an adversarial game: How defectors pave the way to a peaceful society,, Phys. Rev. E, 82 (2010).  doi: 10.1103/PhysRevE.82.066114.  Google Scholar [20] M. B. Short, G. O. Mohler, P. J. Brantingham and G. E. Tita, Gang rivalry dynamics via coupled point process networks,, Discrete and Continuous Dynamical Systems, 34 (2014), 1459.   Google Scholar [21] A. Stomakhin, M. B. Short and A. L. Bertozzi, Reconstruction of missing data in social networks based on temporal patterns of interactions,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/11/115013.  Google Scholar [22] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase in a system of self-driven particles,, Phys. Rev. Lett., 75 (1995), 1226.   Google Scholar [23] X.-S. Wang, H. Wang and J. Wu., Traveling waves of diffusive predator-prey systems: Disease outbreak propagation,, Discrete Contin. Dyn. Syst., 32 (2012), 3303.  doi: 10.3934/dcds.2012.32.3303.  Google Scholar [24] J. R. Zipkin, M. B. Short and A. L. Bertozzi, Cops on the dots in a mathematical model of urban crime and police response,, Discrete and Continuous Dynamical Systems, 34 (2014), 1479.   Google Scholar

show all references

##### References:
 [1] A. B. T. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents,, Discrete and Continuous Dynamical Systems, 34 (2014), 1249.   Google Scholar [2] D. Balagué, J. A. Carrillo and Y. Yao, Confinement for repulsive-attractive kernels,, Discrete and Continuous Dynamical Systems, 34 (2014), 1227.   Google Scholar [3] J. Bedrossian and N. Rodríguez, Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $R^d$,, Discrete and Continuous Dynamical Systems, 34 (2014), 1279.   Google Scholar [4] J. Bedrossian, N. Rodríguez and A. L. Bertozzi, Local and global well-posedness for aggregation equations and Keller-Segel models with degenerate diffusion,, Nonlinearity, 24 (2011), 1683.  doi: 10.1088/0951-7715/24/6/001.  Google Scholar [5] M. Burger, M. Di Francesco, P. A. Markowich and M. T. Wolfram, Mean field games with nonlinear mobilities in pedestrian dynamics,, Discrete and Continuous Dynamical Systems, 34 (2014), 1311.   Google Scholar [6] Y. S. Cho, A. Galstyan, P. J. Brantingham and G. Tita, Latent self-exciting point process model for spatial-temporal networks,, Discrete and Continuous Dynamical Systems, 34 (2014), 1335.   Google Scholar [7] F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Transactions on Automatic Control, 52 (2007), 852.  doi: 10.1109/TAC.2007.895842.  Google Scholar [8] M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006).  doi: 10.1103/PhysRevLett.96.104302.  Google Scholar [9] R. Ghosh and K. Lerman, Rethinking Centrality: The role of dynamical processes in social network analysis,, Discrete and Continuous Dynamical Systems, 34 (2014), 1355.   Google Scholar [10] R. A. Hegemann, E. A. Lewis and A. L. Bertozzi, An estimate & Score Algorithm for simultaneous parameter estimation and reconstruction of incomplete data on social networks,, Security Informatics, 2 (): 1.  doi: 10.1186/2190-8532-2-1.  Google Scholar [11] T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions,, Phys. Rev. E, 84 (2011).  doi: 10.1103/PhysRevE.84.015203.  Google Scholar [12] T. Kolokolnikov, M. J. Ward and J. Wei, The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime,, Discrete and Continuous Dynamical Systems, 34 (2014), 1373.   Google Scholar [13] M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models,, J. Math. Biol., 45 (2002), 219.  doi: 10.1007/s002850200144.  Google Scholar [14] A. Mackey, T. Kolokolnikov and A. L. Bertozzi, Two-species particle aggregation and stability of co-dimension one solutions,, Discrete and Continuous Dynamical Systems, 34 (2014), 1411.   Google Scholar [15] S. G. McCalla, Paladins as predators: Invasive waves in a spatial evolutionary adversarial game,, Discrete and Continuous Dynamical Systems, 34 (2014), 1437.   Google Scholar [16] 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. Meth. Appl. Sci., 18 (2008), 1249.  doi: 10.1142/S0218202508003029.  Google Scholar [17] M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime - hotpsots, bifurcations, and suppression,, SIAM J. Appl. Dyn. Sys., 9 (2010), 462.  doi: 10.1137/090759069.  Google Scholar [18] M. B. Short, P. J. Brantingham, A. L. Bertozzi and G. E. Tita, Dissipation and displacement of hotpsots in reaction-diffusion models of crime,, Proc. Nat. Acad. Sci., 107 (2010), 3961.   Google Scholar [19] M. B. Short, P. J. Brantingham and M. R. D'Orsogna, Cooperation and punishment in an adversarial game: How defectors pave the way to a peaceful society,, Phys. Rev. E, 82 (2010).  doi: 10.1103/PhysRevE.82.066114.  Google Scholar [20] M. B. Short, G. O. Mohler, P. J. Brantingham and G. E. Tita, Gang rivalry dynamics via coupled point process networks,, Discrete and Continuous Dynamical Systems, 34 (2014), 1459.   Google Scholar [21] A. Stomakhin, M. B. Short and A. L. Bertozzi, Reconstruction of missing data in social networks based on temporal patterns of interactions,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/11/115013.  Google Scholar [22] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase in a system of self-driven particles,, Phys. Rev. Lett., 75 (1995), 1226.   Google Scholar [23] X.-S. Wang, H. Wang and J. Wu., Traveling waves of diffusive predator-prey systems: Disease outbreak propagation,, Discrete Contin. Dyn. Syst., 32 (2012), 3303.  doi: 10.3934/dcds.2012.32.3303.  Google Scholar [24] J. R. Zipkin, M. B. Short and A. L. Bertozzi, Cops on the dots in a mathematical model of urban crime and police response,, Discrete and Continuous Dynamical Systems, 34 (2014), 1479.   Google Scholar
 [1] H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433 [2] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457 [3] Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219 [4] A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441 [5] Qiang Fu, Yanlong Zhang, Yushu Zhu, Ting Li. Network centralities, demographic disparities, and voluntary participation. Mathematical Foundations of Computing, 2020, 3 (4) : 249-262. doi: 10.3934/mfc.2020011 [6] Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251 [7] Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346 [8] Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446 [9] Shuyang Dai, Fengru Wang, Jerry Zhijian Yang, Cheng Yuan. A comparative study of atomistic-based stress evaluation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020322 [10] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [11] Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026 [12] Håkon Hoel, Gaukhar Shaimerdenova, Raúl Tempone. Multilevel Ensemble Kalman Filtering based on a sample average of independent EnKF estimators. Foundations of Data Science, 2020  doi: 10.3934/fods.2020017 [13] Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074 [14] Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071 [15] Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464 [16] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341 [17] S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435 [18] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 [19] Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013 [20] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

2019 Impact Factor: 1.27