  Recent studies suggest that social media usage -- while linked to an
increased diversity of information and perspectives for users -- has
exacerbated user polarization on many issues. A popular theory for this
phenomenon centers on the concept of "filter bubbles": by automatically
recommending content that a user is likely to agree with, social network
algorithms create echo chambers of similarly-minded users that would not have
arisen otherwise. However, while echo chambers have been observed in real-world
networks, the evidence for filter bubbles is largely post-hoc.
  In this work, we develop a mathematical framework to study the filter bubble
theory. We modify the classic Friedkin-Johnsen opinion dynamics model by
introducing another actor, the network administrator, who filters content for
users by making small changes to the edge weights of a social network (for
example, adjusting a news feed algorithm to change the level of interaction
between users).
  On real-world networks from Reddit and Twitter, we show that when the network
administrator is incentivized to reduce disagreement among users, even
relatively small edge changes can result in the formation of echo chambers in
the network and increase user polarization. We theoretically support this
observed sensitivity of social networks to outside intervention by analyzing
synthetic graphs generated from the stochastic block model. Finally, we show
that a slight modification to the incentives of the network administrator can
mitigate the filter bubble effect while minimally affecting the administrator's
target objective, user disagreement.
