Modeling, analysis, and manipulation of co-evolution between appraisal dynamics and opinion dynamics

The evolution of interpersonal relationships to structural balance involves various complex social psychological mechanics, including interpersonal influences, opinion dynamics, person–person homophily, person–entity homophily, etc. In this paper, we propose a discrete-time nonlinear dynamical system that characterizes the complex interplay among all the above mechanisms. Specifically, interpersonal appraisals are updated based on person–person homophily and person–opinion homophily; the social influence matrix is built according to the interpersonal appraisals; and the opinion dynamics evolve on the social influence network. Via theoretical analysis, we characterize the finite-time behavior and the equilibrium set of the model. In addition, we establish the equivalence among social balance, modulus consensus, convergence, and the non-vanishing appraisal condition. Moreover, we provide sufficient conditions such that the current faction formation in social balance is changed by introducing additional opinion dimensions, i.e., by setting new agendas. Numerical examples show the effectiveness of such agenda-setting strategy and indicate that sufficiently extreme topics are required in order to successfully conduct such manipulation.