Preface to special issue on mathematics of social systems

Previous Article
Confinement for repulsive-attractive kernels
DCDS-B Home
This Issue
Next Article

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

Preface to special issue on mathematics of social systems

Andrea L. Bertozzi ^1,

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

Published April 2014

Abstract
Related Papers
Under a Creative Commons license

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.

Keywords: pattern formation, agent based model, Social Sciences, network analysis, stochastics..

Mathematics Subject Classification: Primary: 91C29, 35B35, 35B36, 60K30, 94C15; Secondary: 70F10, 70F4.

Citation: Andrea L. Bertozzi. Preface to special issue on mathematics of social systems. Discrete and 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), 1249-1278.
[2]	D. Balagué, J. A. Carrillo and Y. Yao, Confinement for repulsive-attractive kernels, Discrete and Continuous Dynamical Systems, 34 (2014), 1227-1248.
[3]	J. Bedrossian and N. Rodríguez, Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $mathbb{R}^{d}$, Discrete and Continuous Dynamical Systems, 34 (2014), 1279-1309.
[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), 1683-1714. doi: 10.1088/0951-7715/24/6/001.
[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), 1311-1333.
[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),1335-1354.
[7]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.
[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), 104302. doi: 10.1103/PhysRevLett.96.104302.
[9]	R. Ghosh and K. Lerman, Rethinking Centrality: The role of dynamical processes in social network analysis, Discrete and Continuous Dynamical Systems, 34 (2014), 1355-1372.
[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, 1-13. doi: 10.1186/2190-8532-2-1.
[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), 015203. doi: 10.1103/PhysRevE.84.015203.
[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), 1373-1410.
[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), 219-233. doi: 10.1007/s002850200144.
[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), 1411-1436.
[15]	S. G. McCalla, Paladins as predators: Invasive waves in a spatial evolutionary adversarial game, Discrete and Continuous Dynamical Systems, 34 (2014), 1437-1457.
[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), 1249-1267. doi: 10.1142/S0218202508003029.
[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), 462-483. doi: 10.1137/090759069.
[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), 3961-3965.
[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), 066114. doi: 10.1103/PhysRevE.82.066114.
[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), 1459-1477.
[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), p. 115013. doi: 10.1088/0266-5611/27/11/115013.
[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), 1226-1229.
[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), 3303-3324. doi: 10.3934/dcds.2012.32.3303.
[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), 1479-1506.

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), 1249-1278.
[2]	D. Balagué, J. A. Carrillo and Y. Yao, Confinement for repulsive-attractive kernels, Discrete and Continuous Dynamical Systems, 34 (2014), 1227-1248.
[3]	J. Bedrossian and N. Rodríguez, Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $mathbb{R}^{d}$, Discrete and Continuous Dynamical Systems, 34 (2014), 1279-1309.
[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), 1683-1714. doi: 10.1088/0951-7715/24/6/001.
[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), 1311-1333.
[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),1335-1354.
[7]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.
[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), 104302. doi: 10.1103/PhysRevLett.96.104302.
[9]	R. Ghosh and K. Lerman, Rethinking Centrality: The role of dynamical processes in social network analysis, Discrete and Continuous Dynamical Systems, 34 (2014), 1355-1372.
[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, 1-13. doi: 10.1186/2190-8532-2-1.
[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), 015203. doi: 10.1103/PhysRevE.84.015203.
[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), 1373-1410.
[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), 219-233. doi: 10.1007/s002850200144.
[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), 1411-1436.
[15]	S. G. McCalla, Paladins as predators: Invasive waves in a spatial evolutionary adversarial game, Discrete and Continuous Dynamical Systems, 34 (2014), 1437-1457.
[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), 1249-1267. doi: 10.1142/S0218202508003029.
[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), 462-483. doi: 10.1137/090759069.
[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), 3961-3965.
[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), 066114. doi: 10.1103/PhysRevE.82.066114.
[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), 1459-1477.
[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), p. 115013. doi: 10.1088/0266-5611/27/11/115013.
[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), 1226-1229.
[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), 3303-3324. doi: 10.3934/dcds.2012.32.3303.
[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), 1479-1506.

[1]	Holly Gaff. Preliminary analysis of an agent-based model for a tick-borne disease. Mathematical Biosciences & Engineering, 2011, 8 (2) : 463-473. doi: 10.3934/mbe.2011.8.463
[2]	Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555
[3]	Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217
[4]	Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260
[5]	Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031
[6]	Kolade M. Owolabi. Numerical analysis and pattern formation process for space-fractional superdiffusive systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 543-566. doi: 10.3934/dcdss.2019036
[7]	Weiping Li, Haiyan Wu, Jie Yang. Intelligent recognition algorithm for social network sensitive information based on classification technology. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1385-1398. doi: 10.3934/dcdss.2019095
[8]	Dieter Armbruster, Christian Ringhofer, Andrea Thatcher. A kinetic model for an agent based market simulation. Networks and Heterogeneous Media, 2015, 10 (3) : 527-542. doi: 10.3934/nhm.2015.10.527
[9]	Rumi Ghosh, Kristina Lerman. Rethinking centrality: The role of dynamical processes in social network analysis. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1355-1372. doi: 10.3934/dcdsb.2014.19.1355
[10]	Evan C. Haskell, Jonathan Bell. Pattern formation in a predator-mediated coexistence model with prey-taxis. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2895-2921. doi: 10.3934/dcdsb.2020045
[11]	R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339
[12]	Guanqi Liu, Yuwen Wang. Pattern formation of a coupled two-cell Schnakenberg model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1051-1062. doi: 10.3934/dcdss.2017056
[13]	Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182
[14]	Hongfei Xu, Jinfeng Wang, Xuelian Xu. Dynamics and pattern formation in a cross-diffusion model with stage structure for predators. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4473-4489. doi: 10.3934/dcdsb.2021237
[15]	Domènec Ruiz-Balet, Enrique Zuazua. Control of reaction-diffusion models in biology and social sciences. Mathematical Control and Related Fields, 2022, 12 (4) : 955-1038. doi: 10.3934/mcrf.2022032
[16]	Pradeep Dubey, Rahul Garg, Bernard De Meyer. Competing for customers in a social network. Journal of Dynamics and Games, 2014, 1 (3) : 377-409. doi: 10.3934/jdg.2014.1.377
[17]	Israa Mohammed Khudher, Yahya Ismail Ibrahim, Suhaib Abduljabbar Altamir. Individual biometrics pattern based artificial image analysis techniques. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 567-578. doi: 10.3934/naco.2020056
[18]	Mark G. Burch, Karly A. Jacobsen, Joseph H. Tien, Grzegorz A. Rempała. Network-based analysis of a small Ebola outbreak. Mathematical Biosciences & Engineering, 2017, 14 (1) : 67-77. doi: 10.3934/mbe.2017005
[19]	V. Lanza, D. Ambrosi, L. Preziosi. Exogenous control of vascular network formation in vitro: a mathematical model. Networks and Heterogeneous Media, 2006, 1 (4) : 621-637. doi: 10.3934/nhm.2006.1.621
[20]	Yao-Li Chuang, Tom Chou, Maria R. D'Orsogna. A network model of immigration: Enclave formation vs. cultural integration. Networks and Heterogeneous Media, 2019, 14 (1) : 53-77. doi: 10.3934/nhm.2019004

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Andrea L. Bertozzi
on Google Scholar
Andrea L. Bertozzi

[Back to Top]