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Preface to special issue on mathematics of social systems

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  • 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.

    For more information please click the “Full Text” above.
    Mathematics Subject Classification: Primary: 91C29, 35B35, 35B36, 60K30, 94C15; Secondary: 70F10, 70F45.


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    A. B. T. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents, Discrete and Continuous Dynamical Systems, 34 (2014), 1249-1278.


    D. Balagué, J. A. Carrillo and Y. Yao, Confinement for repulsive-attractive kernels, Discrete and Continuous Dynamical Systems, 34 (2014), 1227-1248.


    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), 1279-1309.


    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), 1683-1714.doi: 10.1088/0951-7715/24/6/001.


    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), 1311-1333.


    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),1335-1354.


    F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.doi: 10.1109/TAC.2007.895842.


    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), 104302.doi: 10.1103/PhysRevLett.96.104302.


    R. Ghosh and K. Lerman, Rethinking Centrality: The role of dynamical processes in social network analysis, Discrete and Continuous Dynamical Systems, 34 (2014), 1355-1372.


    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, 1-13. doi: 10.1186/2190-8532-2-1.


    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), 015203.doi: 10.1103/PhysRevE.84.015203.


    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), 1373-1410.


    M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.doi: 10.1007/s002850200144.


    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), 1411-1436.


    S. G. McCalla, Paladins as predators: Invasive waves in a spatial evolutionary adversarial game, Discrete and Continuous Dynamical Systems, 34 (2014), 1437-1457.


    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), 1249-1267.doi: 10.1142/S0218202508003029.


    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), 462-483.doi: 10.1137/090759069.


    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), 3961-3965.


    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), 066114.doi: 10.1103/PhysRevE.82.066114.


    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), 1459-1477.


    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), p. 115013.doi: 10.1088/0266-5611/27/11/115013.


    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), 1226-1229.


    X.-S. Wang, H. Wang and J. Wu., Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303-3324.doi: 10.3934/dcds.2012.32.3303.


    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), 1479-1506.

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