Background information
With the development of new interdisciplinary branches and the advances in the sphere of theoretical physics as a conceptual framework that can be applied to other sciences, the approaches towards the analysis of the data concerning the processes within the social realm have undergone significant changes.
The findings of synergetics as the theory utilizing the spatial, temporal and functional parameters for the analysis of the multicomponent systems had a significant impact upon the methods used in natural and social sciences (Weidlich 431). Assuming that the human society can be defined as a multicomponent system consisting of individuals as active components interacting between each other and interdependent with their material environment, it can be stated that the development of a complex quantitative modeling procedure intended to estimate the variety of the possible scenarios of these macro processes was a real challenge for theoreticians. Sociodynamics is the branch aimed at meeting this ambitious objective.
Though sociodynamics can be regarded as an integral part of synergetics, the modeling procedures applied to the social sciences cannot be limited to the synergetics algorithms only because the latter has been developed for the natural sciences in the first place. The specific structure of the social system contrasting to that of natural systems, namely the inability to define the direction of the interaction between the individuals as the particles of the microlevel complicates the modeling procedures utilized by sociodynamics (Kuppers 162).
The recognition of the limitations of descriptive methodology which was ineffective for indicating the most significant variables in social sciences predetermined the shift of the emphasis towards predictive modeling (Taagepera 15). Previously, the social sciences implemented an approach of unambiguous prediction which did not allow estimating the variety of possible scenarios and depended greatly upon the choice of variables (Arrow 38). The falsifiability of certain statistical models still remained one of the most important aspects for discussing their appropriateness for the analysis of certain phenomena from the social realm. Taking into account the current level of development of other sciences, the 50-50 approach when the statistical models could be right or wrong for concrete studies only by chance, was obsolete. The growing recognition of the inappropriateness of statistical models for covering the whole scope of the complicated social processes taking place at both micro and macro levels demonstrated the importance of making the approaches taken by social scientists more scientific, namely developing feasible predictive models which would allow evaluating the variety of possible scenarios simultaneously disregarding the choice of the prioritized variables.
Due to the peculiarities of the social sphere, the history of the mathematization of the social sciences is different from that of natural sciences, for example. Historically, the idea of extending the scientific method to the social domain has been criticized because of the traditional focus on understanding the underlying processes instead of prediction (Rapoport 16). Currently, the early attempts of utilizing the differential equations for analyzing the social phenomena are only of historical interest. The following stage of the mathematization of the social sciences is the development of stochastic models focusing on random variables and their probability distributions. The simulations of social phenomena are also important tools valuable for advancing the mathematization of the social sciences as compared to the outdated method of experiments.
The application of mathematical models and the advent of simulations as an important tool opened up new opportunities for researchers working in the sphere of social sciences, allowing them to explore the variety of non-linear scenarios within the social domain.
Literature review
Taking into account the fact that the main difficulties for describing social systems mathematically were predetermined with their inherent complexity and extreme sensitivity to the wide range of external influences, it can be stated that the character of chance affectedness is essential for social sciences and requires the development of appropriate models and methods.
Historically, mathematical methods applied to the analysis of the social processes are divided into deterministic models in which the end outcome can be predicted through the analysis of the current state and the stochastic models in which the future states can be predicted only to certain degree of probability (Edling 203; Blume and Durlauf 3). The differential equations are used for describing the deterministic processes, and the main tool for the analysis of stochastic processes is Markov process with Poison process and Brownian motion as its variations. However, not all transition matrices can arise from Markov model (Singer and Spilerman 8). The affect control theory as one of the stochastic models was offered by Heise (1985) for explaining the underlying processes of people’s individual perception of the surrounding world and following decision making which is significant for the analysis of the microlevel of a social system (Smith-Lovin 172). In broad sense, the stochastic models can be divided into those following the hypothesis developed by Coleman and James named as freely forming groups according to which the equilibrium is preserved within the closed social systems and the class of theories originated by Horvarth and Foster assuming that individuals’ behavior is affected by the previous events (Horvath 516).
The recent advances in the spheres of synergetics and chaos theory (united under the term of non-linear dynamics) and statistical physics made mathematical description of complex social systems possible (Helbing 157; Williams 13). Finding an appropriate stochastic equation which will be further examined through the application of one of the methods of non-linear dynamics is central to the methodological transfer in social sciences (Troitzch 22). After describing the microlevel of the social processes stochastically, the variables of the macrolevel need to be taken into consideration for creating the links between the two levels and defining the probability distribution of macrovariables (multiple (m,n) (Weidlich 432). The main steps of the modeling procedure include:
- the configuration of macrovariables,
- collective material variables: m=(ml, … mk, … mm) , where m1…, mk are collective material variables.
- collective personal variables:
- n=(nl … ni … nl … ni …ni …nl) , where i, j, k, l, …, 1 are the alternative attitudes or between macroconfigurations and their following interpretation
- interpretation of actions and pα, pβ, …pp are subpopulations.
- transition
- the transition rates (Weidlich 434).
In describing the time-dependent transitions of the state distribution, the knowledge of a particular element undergoing changes is not compulsory though the data on the number of the elements involved into the transitions is significant. For considering the temporal evolution in analyzing the state distribution, the master equation describing the processes within the system is used:
where P(X,t) is the probability, X is a certain state and t is a particular future moment (Helbing 8). Justification of the accurate master equation requires consideration of the microscopic processes within the system. In general, the application of the stochastic models to social sciences requires paying proper attention to the peculiarities of the system itself and the processes on its microlevel which need to be related to the macrolevel.
The application of mathematical models to social sciences gave rise to debates concerning the appropriateness of this approach for explaining such complex issues as the causes of war or population growth. The main parties in the debates regarding the effectiveness of mathematical methods doubt the ability of human brains to frame a particular problem or select the most appropriate concept to solving a concrete problem. In that regard, the proponents of the game theory claim that the computer system as the rational choice player would can choose more effective solutions (Brown 80). There is evidence that mathematical techniques can be beneficial for the future development of the processes and systems which can be found not only in nature, but also the social dimension (De Marchi 162; Levin, Grenfell, Hastings and Peterson 4). On the other hand, regardless of all the advantages of the recent analytical advancements and the enhanced computation potential, the innovative approaches provide not only new opportunities, but also additional challenges, and consideration of the collective dynamics in the behaviors of individuals and relating various phenomena across the scales of different scientific disciplines (Chowell et al. 175).
Additional challenge for applying the computational models to social sciences is the so-called invisible hand regulating the seemingly spontaneous social processes. The concealed intentionality of the most influential agents predetermines the additional difficulties in adapting the mathematical models to the needs of the processes in social domain (Castelfranchi 6). Still, these challenges do not deny the opportunities of implementing the mathematical models in investigating the micro and macro levels of the social processes, but rather explain the importance of making the approaches to constructing the models more complex.
Theoretical framework
The theoretical framework applied to the analysis of the interpersonal and romantic relationships, distribution of the patent applications in the United States and African political stability include the stochastic models adapted to the peculiarities of particular social systems under analysis.
Taking into account the complexity of underlying processes taking place on the microlevels of interpersonal relationships and individual patent decisions and political choices, the deterministic models are inappropriate for covering the whole scope of possible transitions within these systems and estimating the probability of certain states at particular moments in the future (Simon 162). In that regard, the master equation need to be used for describing the time-dependent transitions within the social systems under analysis and giving serious consideration to the temporal evolution for the analysis of the state distribution. The theories of groups formation developed by Coleman and James, on the one hand, and Horvarth and Foster, on the other hand, can be applied for formulating promising hypotheses for these studies.
The modeling procedure as outlined by Weidlich (2003) can be regarded as one of the most important underpinnings of the theoretical framework important for interpreting the broad meaning of the discussed transitions and establishing the links between the changes in the microlevel of the system with the states of the macrolevel which would allow making certain generalizations and reaching the goals of applying the mathematical models to the social studies. For this reason, some mathematical techniques need to be reconstructed and adapted to the needs of social sciences (Stone 149).
Works Cited
Arrow, Kenneth. “Mathematical Models in the Social Sciences”. In Lerner, Daniel and Harold Lasswell (eds.). The Policy Sciences. Stanford University Press, 1984.
Blume, Lawrence and Steven Durlauf. Equilibrium Concepts for Social Interaction Models. New York: Cornell University, 2002.
Brown, Richard. Are Science and Mathematics Socially Constructed? A Mathematician Encounters Postmodern Interpretations of Science. Hackensack: World Scientific Publishing Co, 2009. Print.
Castelfranchi, Cristiano. “The Theory of Social Functions: Challenges for Computational Social Science and Multi-Agent Learning”. Cognitive Systems Research, 2001, 2, 5-38.
Chowell, Gerardo, James Hyman, Stephen Eubank, Carlos Castillo-Chavez. “Scaling laws for the movement of people between locations in a large city”. Physical Review, 2003.
De Marchi, Scott. Computational and mathematical modeling in the social sciences. New York: Cambridge University Press, 2005. Print.
Edling, Christopher. “Mathematics in Sociology”. Annual Review Sociology, 2002 28: 197 – 220.
Helbing, Dirk. Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes. New York: Springer, 2010. Print.
Horvath, William. “Stochastic Models of Behavior”. Stochastic Models of Behavior 1966, 12: 513 – 2520. Print.
Kuppers, Gunter and Johannes Lehhard. “Validation of Simulation: Patterns in the Social and Natural Sciences”. Journal of Artificial Societies and Social Simulation, 2005, 8(4).
Levin, Simon, Bryan Grenfell, Alan Hastings, Alan Peterson. “Mathematical and computational challenges in population biology and ecosystems science”. Science, 1997.
Rapoport, Anatol. Mathematical Models in the Social and Behavioral Sciences. New York: John Wiley and Sons, 1983. Print.
Simon, Herbert. “The Uses of Mathematics in the Social Sciences”. Mathematics and Computers in Simulation, 1978, 20, 159-166.
Singer, Burton and Seymour Spilerman. “Representation of social processes by Markov Models”. The American Journal of Sociology 1976, 82(1): 1-54.
Smith-Lovin, Lynn. Analyzing Social Interaction: Advances in Affect Control Theory. New York: Gordon and Breach Science Publishers, 1988. Print.
Stone, Richard. Mathematics in the social sciences and other essays. London: Chapman and Hall, 1966. Print.
Taagepera, Rein. Making Social Sciences More Scientific: The Need for Predictive Models. New York: Oxford University Press, 2008. Print.
Troitzch, Klaus. “Multilevel Process Modeling in the Social Sciences: Mathematical Analysis and Computer Simulation”. In Liebrand, Wim, Andrzej Nowak & Rainer Hegselmann (eds.). Computer Modeling of Social Processes. Thousand Oaks: SAGE publications, 1998. Print.
Weidlich, Wolfgang. “Sociodynamics – a systematic approach to mathematical modeling in the social sciences”. Chaos, Solitons and Fractals 2003 (18): 431 – 437. Print.
Williams, Malcolm. Science and Social Science: An Introduction. New York: Routledge, 2000.