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Mathematical Modeling of Social Relationships pp 1–15Cite as

Introduction to the Mathematical Modeling of Social Relationships

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Abstract

Mathematics has been used to describe the assumptions and determine their consequences in the physical sciences over the last three centuries. More recently, mathematics has been used to develop a deeper understanding of complex biological phenomena and data. We are now witnessing the expansion of mathematical methods into the social sciences. In this chapter, we trace that trail of mathematics from the physical sciences, to the biological sciences, to the social sciences and review mathematical methods that provide new tools to understand social relationships.

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References

  • Alon, U. (2006). An introduction to systems biology: Design principles of biological circuits. Boca Raton: Chapman & Hall/CRC Mathematical and Computational Biology.

    Google Scholar 

  • Amit, D. J. (1989). Modeling brain function: The world of attractor neural networks. New York: Cambridge University Press.

    Book  Google Scholar 

  • Arrow, K. J., Karlin, S., & Suppes, P. (Eds.). (1960). Mathematical methods in the social sciences, 1959. Stanford: Stanford University Press.

    Google Scholar 

  • Babcock, J. C., Gottman, J. M., Ryan, K. D., & Gottman, J. S. (2013). A component analysis of a brief psycho-educational couples’ workshop: One-year follow-up results. Journal of Family Therapy, 35(3), 252–280 https://doi.org/10.1111/1467-6427.12017

    Article  Google Scholar 

  • Bak, P. (1997). How nature works: The science of self-organized criticality. New York: Oxford University Press.

    Google Scholar 

  • Bastian, M., Heymann, S., & Jacomy, M. (2009). Gephi: An open source software for exploring and manipulating networks. International AAAI Conference on Weblogs and Social Media. https://gephi.org/users/publications/

  • Batty, M., & Longley, P. (1994). Fractal cities. New York: Academic Press.

    Google Scholar 

  • Bonabeau, E. (2002). Agent-based modeling: Methods and techniques for simulating human systems. Proceedings of the National Academy of Sciences (USA), 99(Suppl. 3), 7280–7287 https://doi.org/10.1073/pnas.082080899

    Article  Google Scholar 

  • Boyd, J. P. (1969). The algebra of group kinship. Journal of Mathematical Psychology, 6(1), 139–167 https://doi.org/10.1016/0022-2496(69)90032-7

    Article  Google Scholar 

  • Brown, C., & Liebovitch, L. S. (2010). Fractal analysis: Quantitative Applications in the social sciences (Vol. 165). Los Angeles: Sage.

    Book  Google Scholar 

  • Cioffi-Revilla, C. (Ed.). (2017). Computation and social science. Springer International Publishing.

    Google Scholar 

  • Coleman, J. S. (1964). Introduction to mathematical sociology. New York: The Free Press of Glencoe.

    Google Scholar 

  • Coleman, P. T. (2011). The five percent: Finding solutions to seemingly impossible conflicts. New York: Public Affairs.

    Google Scholar 

  • Coleman, P. T., Vallacher, R. R., Nowak, A., & Bui-Wrzosinska, L. (2007). Intractable conflict as an attractor: A dynamical systems approach to conflict escalation and intractability. American Behavioral Scientist, 50(11), 1454–1475 https://doi.org/10.1177/0002764207302463

    Article  Google Scholar 

  • Fernandez-Rosales, I. Y., Liebovitch, L. S., & Guzman-Vargas, L. (2015). The dynamic consequences of cooperation and competition in smallworld networks. PLoS One, 10(4), e0126234 https://doi.org/10.1371/journal.pone.0126234

    Article  Google Scholar 

  • Forrester, J. W. (1973). World dynamics. Cambridge: Wright-Allen Press.

    Google Scholar 

  • Freeman, L. C. (2004). The development of social network analysis: A study in the sociology of science. Vancouver: Empirical Press.

    Google Scholar 

  • Garabedian, P. R. (1964). Partial differential equations. New York: Wiley.

    Google Scholar 

  • Gardner, M. (1970). Mathematical games – the fantastic combinations of John Conway’s new solitaire game “life”. Scientific American, 223, 120–123 https://doi.org/10.1038/scientificamerican1070-120

    Article  Google Scholar 

  • Geller, A. (2011). Growing social structure: An empirical multiagent excursion into kinship in rural north-west frontier province. Structure and Dynamics, 5(1). http://escholarship.org/uc/item/4ww6x6gm

  • Gottman, J., Swanson, C., & Swanson, K. (2002). A general systems theory of marriage: Nonlinear difference equation modeling of marital interaction. Personality and Social Psychology Review, 6(4), 326–340 https://doi.org/10.1207/S15327957PSPR0604_07

    Article  Google Scholar 

  • Gottman, J. M., Murray, J. D., Swanson, C. C., Tyson, R., & Swanson, K. R. (2005). The mathematics of marriage: Dynamic nonlinear models. Cambridge, MA: MIT Press.

    Google Scholar 

  • Granovetter, M. (1983). Threshold models of diffusion and collective behavior. Journal of Mathematical Sociology, 9, 165–179 https://doi.org/10.1080/0022250X.1983.9989941

    Article  Google Scholar 

  • Haken, H. (1983). Synergetics, an introduction: Nonequilibrium phase transitions and self-organization in physics, chemistry, and biology (3rd ed.). New York: Springer.

    Book  Google Scholar 

  • Holland, J. H. (1998). Emergence: From chaos to order. Cambridge, MA: Perseus Books.

    Google Scholar 

  • Horowitz, I. L. (2006). Big five and little five: Measuring revolutions in social science. Society, 43(3), 9–12 https://doi.org/10.1007/BF02687589

    Article  Google Scholar 

  • IBM. (2016). Watson for Oncology. http://www.ibm.com/smarterplanet/us/en/ibmwatson/watson-oncology.html. Accessed 14 June 2016.

  • Katchalsky, A., & Curran, P. F. (1965). Nonequilibrium thermodynamics in biophysics. Cambridge, MA: Harvard University Press.

    Book  Google Scholar 

  • Kelso, J. A. S. (1995). Dynamic patterns: The self-organization of brain and behavior. Cambridge, MA: MIT Press.

    Google Scholar 

  • Kulesza, W. M., Cisłak, A., Vallacher, R. R., Nowak, A., Czekiel, M., & Bedynska, S. (2015). The face of the chameleon: The experience of facial mimicry for the mimicker and the mimickee. The Journal of Social Psychology, 155(6), 590–604 https://doi.org/10.1080/00224545.2015.1032195

    Article  Google Scholar 

  • Levenson, R. W., & Gottman, J. M. (1985). Physiological and affective predictors of change in relationship satisfaction. Journal of Personality and Social Psychology, 49(1), 85–94 https://doi.org/10.1037/0022-3514.49.1.85

    Article  Google Scholar 

  • Liebovitch, L. S. (1998). Fractals and chaos: Simplified for the life sciences. New York: Oxford University Press.

    Google Scholar 

  • Liebovitch, L. S. (2006). Why the eye is round. In J. Fischbarg (Ed.), The biology of the eye, Advances in organ biology (Vol. 10). New York: Elsevier. http://people.qc.cuny.edu/Faculty/Larry.Liebovitch/documents/roundeyes.pdf

    Google Scholar 

  • Liebovitch, L. S., Tsinoremas, N., & Pandya, A. (2007). Developing combinatorial multi-component therapies (CMCT) of drugs that are more specific and have fewer side effects than traditional one drug therapies. Nonlinear Biomed Physics, 1, 11. https://doi.org/10.1186/1753-4631-1-11

    Article  Google Scholar 

  • Liebovitch, L. S., Naudot, V., Vallacher, R. R., Nowak, A., Bui-Wrzosinska, L., & Coleman, P. T. (2008). Dynamics of two-actor cooperation-competition conflict models. Physica A, 387, 6360–6378. https://doi.org/10.1016/j.physa.2008.07.020

    Article  Google Scholar 

  • Liebovitch, L. S., Vallacher, R. R., & Michaels, J. (2010). Dynamics of cooperation–competition interaction models. Peace and Conflict, 16(2), 175–188 https://doi.org/10.1080/10781911003691625

    Article  Google Scholar 

  • Liebovitch, L. S., Peluso, P. R., Norman, M. D., Su, J., & Gottman, J. M. (2011). Mathematical model of the dynamics of psychotherapy. Cognitive Neurodynamics, 5(3), 265–275. https://doi.org/10.1007/s11571-011-9157-x

    Article  Google Scholar 

  • Luce, R. D., Bush, R. R., & Eugene, G. E. (Eds.). (1963). Handbook of mathematical psychology. New York: Wiley.

    Google Scholar 

  • Marr, C., & Huett, M.-T. (2009). Outer-totalistic cellular automata on graphs. Physics Letters A, 373, 546–549 https://doi.org/10.1016/j.physleta.2008.12.013

    Article  Google Scholar 

  • McCammon, J. A., & Harvey, S. C. (1987). Dynamics of Proteins and Nucleic Acids. New York: Cambridge University Press.

    Book  Google Scholar 

  • Meinhardt, H. (1995). The algorithmic beauty of sea shells. New York: Springer.

    Book  Google Scholar 

  • Murray, J. D. (1989). Mathematical biology. New York: Springer.

    Book  Google Scholar 

  • Newman, J. R. (1956). The world of mathematics (Vol. 4). New York: Simon and Schuster.

    Google Scholar 

  • Newtson, D. (1994). The perception and coupling of behavior waves. In R. R.Vallacher & A. Nowak (Eds.), Dynamical systems in social psychology (pp. 139–167). San Diego: Academic Press.

    Google Scholar 

  • Nowak, A. (2004). Dynamical minimalism: Why less is more in psychology. Personality and Social Psychology Review, 8(2), 183–192. https://doi.org/10.1207/s15327957pspr0802_12

    Article  Google Scholar 

  • Nowak, A., & Vallacher, R. R. (1998). Dynamical social psychology. New York: The Guilford Press.

    Google Scholar 

  • Nowak, A., Szamrej, J., & Latané, B. (1990). From private attitude to public opinion: A dynamic theory of social impact. Psychological Review, 97(3), 362 https://doi.org/10.1037/0033-295X.97.3.362

    Article  Google Scholar 

  • O’Neil, R., & Schutt, R. (2013). Doing data science: Straight talk from the frontline. Sebastopol: O’Reilly Media.

    Google Scholar 

  • Peluso, P. R., Liebovitch, L. S., Gottman, J. M., Norman, M. D., & Su, J. (2011). A mathematical model of psychotherapy: An investigation using dynamic non-linear equations to model the therapeutic relationship. Psychotherapy Research, 22(1), 40–55 https://doi.org/10.1080/10503307.2011.622314

    Article  Google Scholar 

  • Prigogine, I. (1980). From being to becoming: Time and complexity in the physical sciences. San Francisco: W. H. Freeman.

    Google Scholar 

  • Richardson, L. F. (1960). Arms and insecurity. Pittsburgh: The Boxwood Press.

    Google Scholar 

  • Shannon, C. E., Weaver, W., & Burks, A. W. (1949). The mathematical theory of communication. Urbana: University of Illinois Press.

    Google Scholar 

  • Stacey, R. D. (1992). Managing the unknowable. San Francisco: Jossey-Bass.

    Google Scholar 

  • Strogatz, S. H. (1994). Nonlinear dynamics and chaos. Cambridge, MA: Perseus Books Group.

    Google Scholar 

  • Tesser, A. (1980). When individual dispositions and social pressure conflict: A catastrophe. Human Relations, 33(6), 393–407 https://doi.org/10.1177/001872678003300604

    Article  Google Scholar 

  • Troy, D. (2015). Mapping urban social networks as a means for understanding cities. Bloomberg Data for Good Exchange 2015.

    Google Scholar 

  • Turcotte, D. L., & Rundle, J. B. (2002). Self-organizing complexity in the physical, biological, and social sciences. Proceedings of the National Academy of Sciences (USA), 99(Suppl. 1), 2463–2465 https://doi.org/10.1073/pnas.012579399

    Article  Google Scholar 

  • Vallacher, R. R., & Nowak, A. E. (1994). Dynamical systems in social psychology. San Diego: Academic Press.

    Google Scholar 

  • Vallacher, R. R., Coleman, P. T., Nowak, A., Bui-Wrzosinska, L., Liebovitch, L. S., Kugler, K. G., & Bartoli, A. (2013). Attracted to conflict: Dynamic foundations of destructive social relations. New York: Springer.

    Book  Google Scholar 

  • Vallacher, R. R., Read, S. J., & Nowak, A. (2017). Computational social psychology. New York: Routledge.

    Google Scholar 

  • von Bertalanffy, L. (1969). General system theory: Foundations, development, applications. New York: G. Braziller.

    Google Scholar 

  • von Neumann, J., & Morgenstern, O. (1945). The theory of games and economic behavior. Princeton: Princeton University Press.

    Google Scholar 

  • Wiener, N. (1948). Cybernetics or control and communication in the animal and the machine. New York: Wiley.

    Google Scholar 

  • Wigner, E. P. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics, 13(1), 1–14. https://doi.org/10.1002/cpa.3160130102

    Article  Google Scholar 

  • Wilson, T. P. (2010). On the role of mathematics in the social sciences. The Journal of Mathematical Sociology, 10(3–4), 221–239 https://doi.org/10.1080/0022250X.1984.9989970

    Google Scholar 

  • Wolfram, S. (1994). Cellular automata and complexity: Collected papers. Reading: Addison-Wesley Pub. Co.

    Google Scholar 

  • Zimmerman, B., Lindberg, C., & Plsek, P. E. (1998). Edgeware: Insights from complexity science for health care leaders. Irving: VHA Inc..

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Conflict Resolution Studies, College of Arts, Humanities, and Social Sciences, Nova Southeastern University, Fort Lauderdale, FL, USA

    Urszula Strawinska-Zanko

  2. Departments of Physics and Psychology, Queens College, City University of New York, Flushing, NY, USA

    Larry S. Liebovitch

  3. Physics Program, The Graduate Center, City University of New York, New York, NY, USA

    Larry S. Liebovitch

  4. Adjunct Senior Research Scholar, Advanced Consortium for Cooperation, Conflict, and Complexity (AC4), The Earth Institute at Columbia University, New York, NY, USA

    Larry S. Liebovitch

Authors
  1. Urszula Strawinska-Zanko
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  2. Larry S. Liebovitch
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Correspondence to Urszula Strawinska-Zanko .

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Editors and Affiliations

  1. Department of Conflict Resolution Studies, College of Arts, Humanities, and Social Sciences, Nova Southeastern University, Fort Lauderdale, Florida, USA

    Urszula Strawinska-Zanko

  2. Departments of Physics and Psychology, Queens College, City University of New York, Flushing, New York, USA

    Larry S. Liebovitch

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Strawinska-Zanko, U., Liebovitch, L.S. (2018). Introduction to the Mathematical Modeling of Social Relationships. In: Strawinska-Zanko, U., Liebovitch, L. (eds) Mathematical Modeling of Social Relationships. Computational Social Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-76765-9_1

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  • DOI: https://doi.org/10.1007/978-3-319-76765-9_1

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