Mathematical Social Dynamics: Traditional and New Areas of Research

We present a review on the application of the mathematical models for research on social processes, social structures, and actors in social systems. The scope of the review is not restricted to the classical applications of mathematics such as theory of probability, statistics, stochastic processes, differential equations, and game theory. We also discuss applications of the theory of networks for social network analysis and the numerical research on dynamics of social systems. The number of these applications has increased very fast in recent years. Special attention is given to the results from the area of sociophysics, where mathematical methodology is used to analyze social systems in cooperation with the models and concepts of physics. Another special topic in his review is connected to the results from econophysics, where the mathematical methodology and theories and methods of physics are used in the studies on the dynamics of economic systems. In addition, we give several examples for the application of mathematical methods to social systems: (a) application of difference equations to model the flow of substances in channels of networks; (b) analytical solution of nonlinear equations connected to the model of waves of popularity; (c) numerical results of the waves of popularity in a model that accounts for the change in the opinion of the supporters of the ideas for positive or negative popularity of a person, material item, or a piece of information (idea, theory, ideology, etc.) In the last case, we illustrate the effectiveness of the numerical analysis to discover new effects on the studied social system. The review ends with a large list of references. These references can be used as a guide of the way of new researchers to the large field of mathematical social dynamics.