The widespread emergence of opinion polarization is often attributed to the rise of social media and the internet. These platforms can promote selective exposure, leading to the formation of echo chambers where individuals are only exposed to viewpoints that reinforce their existing beliefs. However, experimental evidence shows that exposure to opposing views through cross-cutting ties is common in both online and offline social contexts, which frequently involve long-standing personal relationships. To account for these facts, we have developed an opinion model that applies to static contact structures. In this model, an agent’s influence over their neighbors depends on the similarity of their opinions. Our findings suggest that polarization can indeed emerge in such static structures and, driven by an increased narrow-mindedness, even in presence of non-negligible cross-cutting ties. Interestingly, the polarized opinion distributions generated by our model closely resemble those obtained in surveys about highly polarized issues. This has allowed us to categorize various issues based on their controversial nature, shedding light on the factors that contribute to opinion polarization.