Polarizing Opinion Dynamics with Confirmation Bias

Abstract

Social media and online networks have enabled discussions between users at a planetary scale on controversial topics. However, instead of seeing users converging to a consensus, they tend to partition into groups holding diametric opinions. In this work we propose an opinion dynamics model that starts from a given graph topology, and updates in each iteration both the opinions of the agents, and the listening structure of each agent, assuming there is confirmation bias. We analyze our model, both theoretically and empirically, and prove that it generates a listening structure that is likely to be polarized. We show a novel application of our model, specifically how it can be used to find polarized niches across different Twitter layers. Finally, we evaluate and compare our model to other polarization models on various synthetic datasets, showing that it yields equilibria with unique characteristics, including high polarization and low disagreement.

A Appendix

1.1 A.1 Proof of Lemma 2

Proof

Consider the opinion $x^\star _u$ of node u at equilibrium with a negative in-neighborhood, it has to satisfy Eq. (4), i.e.,

$$ x^\star _u = \alpha _u s_u +(1-\alpha _u) \sum _{v \rightarrow u} W^\star _{v \rightarrow u} x^\star _v.$$

When $x^\star _u>0$, it is necessary that $s_u>0$, otherwise $x^\star _u<0$ since the second term in the summation is negative. By rearranging the inequality $\alpha _u s_u +(1-\alpha _u) \sum _{v \rightarrow u} W^\star _{v \rightarrow u} x^\star _v>0$ we obtain $\frac{s_u}{ \sum _{v \rightarrow u} W_{v\rightarrow u} |x^\star _v|} > \frac{1-\alpha _u}{\alpha _u}$. Furthermore, $W_{v\rightarrow u}= \frac{1}{|N^{-*}_u|}$ for all in-neighbors $v \in N^{-*}_u$. To see why, for the sake of contradiction, assume without loss of generality² that there exists an arc $v \rightarrow u$ such that $W^\star _{v \rightarrow u}<\frac{1}{|N^{-*}_u|}$. Observe that each arc weight is updated in every iteration according to Eqs. (5) and (6). It is straight-forward to check that in that case $W^\star _{v \rightarrow u}$ will decrease in an iteration, contradicting its equilibrium property. Furthermore, in order for all the incoming arcs to u have the same weight, the update term $\eta x^\star _v x^\star _u$ must be equal for all $v \in N^{-*}_u$, and it must not zero-out the weight. These two facts imply that $x^\star _v=x$ for some value x for all $v \in N^{-*}_u$, and $\frac{1}{|N^{-*}_u|}-\eta x^\star _v x_u>0$ which implies the last condition.    $\blacksquare$

1.2 A.2 Example of Section 4.3

Fig. 5.

Histogram of user retweet polarity ground truth and the prediction by FJ model on debate dataset.