The proliferation of social media platforms, recommender systems, and their joint societal impacts have prompted significant interest in opinion formation and evolution within social networks. In this work, we study how local dynamics in a network can drive opinion polarization. In particular, we study time evolving networks under the classic Friedkin-Johnsen opinion model. Edges are iteratively added or deleted according to simple local rules, modeling decisions based on individual preferences and network recommendations. 1 We give theoretical bounds showing how individual edge updates affect polarization, and a related measure of disagreement across edges. Via simulations on synthetic and real-world graphs, we find that the presence of two simple dynamics gives rise to high polarization: 1) confirmation bias -i.e., the preference for nodes to connect to other nodes with similar expressed opinions and 2) friend-of-friend link recommendations, which encourage new connections between closely connected nodes. We also investigate the role of fixed connections which are not subject to these dynamics. We find that even a small number of fixed edges can significantly limit polarization, but still lead to multimodal opinion distributions, which may be considered polarized in a different sense.