zbMATH — the first resource for mathematics

A network model for polarization of political opinion. (English) Zbl 1437.91346
Summary: We propose and study a simple model for the evolution of political opinion through a population. The model includes a nonlinear term that causes individuals with more extreme views to be less receptive to external influence. Such a term was suggested in 1981 by Cobb in the context of a scalar-valued diffusion equation, and recent empirical studies support this modeling assumption. Here, we use the same philosophy in a network-based model. This allows us to incorporate the pattern of pairwise social interactions present in the population. We show that the model can admit two distinct stable steady states. This bi-stability property is seen to support polarization and can also make the long-term behavior of the system extremely sensitive to the initial conditions and to the precise connectivity structure. Computational results are given to illustrate these effects.
 [1] Axelrod, R., The dissemination of culture: A model with local convergence and global polarization, J. Confl. Resolut., 41, 203-226 (1997) [2] DeGroot, M. H., Reaching a consensus, J. Am. Stat. Assoc., 69, 118-121 (1974) · Zbl 0282.92011 [3] Friedkin, N. E.; Johnsen, E. C., Social influence and opinions, J. Math. Sociol., 15, 193-206 (1990) · Zbl 0712.92025 [4] Hegselmann, R.; Krause, U., Opinion dynamics and bounded confidence: Models, analysis, and simulation, J. Artif. Soc. Soc. Simul., 5, 2 (2002) [5] Albi, G., Pareschi, L., Toscani, G., and Zanella, M., “Recent advances in opinion modeling: Control and social influence,” in Active Particles (Springer, 2017), Vol. 1, pp. 49-98. [6] Anderson, B. D. O.; Ye, M., Recent advances in the modelling and analysis of opinion dynamics on influence networks, Int. J. Autom. Comput., 16, 129-149 (2019) [7] Blondel, V. D.; Hendrickx, J. M.; Tsitsiklis, J. N., On Krause’s multi-agent consensus model with state-dependent connectivity, IEEE Trans. Autom. Control, 54, 2586-2597 (2009) · Zbl 1367.93426 [8] Haibo, H., Competing opinion diffusion on social networks, R. Soc. Open Sci., 4, 171160 (2017) [9] Perra, N.; Rocha, L. E. C., Modelling opinion dynamics in the age of algorithmic personalisation, Sci. Rep., 9, 7261 (2019) [10] Cobb, L., “Stochastic differential equations for the social sciences,” in Mathematical Frontiers of the Social and Policy Sciences, edited by L. Cobb and R. M. Thrall (Westview Press, 1981). [11] Brandt, M. J.; Evans, A. M.; Crawford, J. T., The unthinking or confident extremist? Political extremists are more likely than moderates to reject experimenter-generated anchors, Psychol. Sci., 26, 189-202 (2015) [12] Zmigrod, L.; Rentfrow, P. J.; Robbins, T. W., The partisan mind: Is extreme political partisanship related to cognitive inflexibility?, J. Exp. Psychol. Gen., 149, 407 (2020) [13] In this work, we find it natural to use $$\epsilon$$ for the parameter that takes the form $$\epsilon^2$$ in Ref. 10. Also, since we will be using capitals to denote matrices, we use $$\theta$$ rather than $$G$$ for the long-time mean. [14] Higham, D. J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43, 525-546 (2001) · Zbl 0979.65007 [15] Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations (1992), Springer-Verlag: Springer-Verlag, Berlin · Zbl 0925.65261 [16] Klebaner, F. C., Introduction to Stochastic Calculus with Applications (1998), Imperial College Press: Imperial College Press, London · Zbl 0926.60002 [17] Murray, J. D., Mathematical Biology I. An Introduction, 3rd ed., Interdisciplinary Applied Mathematics Vol. 17 (Springer, New York, 2002). · Zbl 1006.92001 [18] Watts, D. J.; Strogatz, S. H., Collective dynamics of ‘small-world’ networks, Nature, 393, 440-442 (1998) · Zbl 1368.05139 [19] Barabási, A. L.; Albert, R., Emergence of scaling in random networks, Science, 286, 509-512 (1999) · Zbl 1226.05223 [20] Taylor, A.; Higham, D. J., CONTEST: A controllable test matrix toolbox for MATLAB, ACM Trans. Math. Softw., 35, 26:1-26:17 (2009)