Polarization in Geometric Opinion Dynamics

In light of increasing recent attention to political polarization, understanding how polarization can arise poses an important theoretical question. While more classical models of opinion dynamics seem poorly equipped to study this phenomenon, a recent novel approach by H\ka zł a, Jin, Mossel, and Ramnarayan (HJMR) proposes a simple geometric model of opinion evolution that provably exhibits strong polarization in specialized cases. Moreover, polarization arises quite organically in their model: in each time step, each agent updates opinions according to their correlation/response with an issue drawn at random. However, their techniques do not seem to extend beyond a set of special cases they identify, which benefit from fragile symmetry or contractiveness assumptions, leaving open how general this phenomenon really is. In this paper, we further the study of polarization in related geometric models. We show that the exact form of polarization in such models is quite nuanced: even when strong polarization does not hold, it is possible for weaker notions of polarization to nonetheless attain. We provide a concrete example where weak polarization holds, but strong polarization provably fails. However, we show that strong polarization provably holds in many variants of the HJMR model, which are also robust to a wider array of distributions of random issues---this suggests that the form of polarization introduced by HJMR is more universal than suggested by their special cases. We also show that the weaker notions connect more readily to the theory of Markov chains on general state spaces.

[1]  Nicole Immorlica,et al.  Social Learning Under Platform Influence: Extreme Consensus and Persistent Disagreement , 2020, SSRN Electronic Journal.

[2]  Eva Tardos,et al.  Adversarial Perturbations of Opinion Dynamics in Networks , 2020, EC.

[3]  Elchanan Mossel,et al.  A Geometric Model of Opinion Polarization , 2019, Mathematics of Operations Research.

[4]  Charalampos E. Tsourakakis,et al.  Minimizing Polarization and Disagreement in Social Networks , 2017, WWW.

[5]  R. Fisher Ronald J. Fisher: A North American Pioneer in Interactive Conflict Resolution , 2016 .

[6]  J. Leicester In-Groups and Out-Groups , 2016 .

[7]  Matt Taddy,et al.  Measuring Group Differences in High-Dimensional Choices: Method and Application to Congressional Speech , 2016, Econometrica.

[8]  Jesse M. Shapiro,et al.  Measuring Polarization in High-Dimensional Data: Method and Application to Congressional Speech , 2016 .

[9]  Daniel DellaPosta,et al.  Why Do Liberals Drink Lattes?1 , 2015, American Journal of Sociology.

[10]  O. J. Harvey,et al.  Intergroup Conflict And Cooperation: The Robbers Cave Experiment , 2013 .

[11]  Aristides Gionis,et al.  Opinion Maximization in Social Networks , 2013, SDM.

[12]  O. Butkovsky Subgeometric rates of convergence of Markov processes in the Wasserstein metric , 2012, 1211.4273.

[13]  Huy L. Nguyen,et al.  On the Convergence of the Hegselmann-Krause System , 2012, arXiv.org.

[14]  David Lee,et al.  Biased assimilation, homophily, and the dynamics of polarization , 2012, Proceedings of the National Academy of Sciences.

[15]  Matthew O. Jackson,et al.  Naïve Learning in Social Networks and the Wisdom of Crowds , 2010 .

[16]  S. Fortunato,et al.  Statistical physics of social dynamics , 2007, 0710.3256.

[17]  R. Durrett Probability: Theory and Examples , 1993 .

[18]  M. Degroot Reaching a Consensus , 1974 .

[19]  R. Fisher Towards a Social-Psychological Model of Intergroup Conflict , 2016 .

[20]  Luc Rey-Bellet,et al.  Ergodic properties of Markov processes , 2006 .

[21]  P. Wallace,et al.  INTERGROUP CONFLICT AND COOPERATION , 1999 .

[22]  Noah E. Friedkin,et al.  Social influence and opinions , 1990 .

[23]  I. I. Gikhman Convergence to Markov processes , 1969 .