In this paper, we introduce a social network perspective for characterizing polarized information ecosystems. We apply our framework to a large-scale dataset of climate change conversations on Twitter. Leveraging a stance detection algorithm, we quantify the linksharing behaviors of Believers and Disbelievers of anthropogenic climate change. We generate networks of web domains based on co-sharing by Believers and Disbelievers, and we characterize these networks in terms of both structure and content. While Believers outnumber Disbelievers in our dataset, our results showed that Disbelievers are responsible for sharing over three times as many unique domains. However, for every 10% increment in a domain's proportion of Disbeliever sharers, the same domain is shared 58% less. Structurally, we observed that domain clusters associated with Believers were significantly smaller, denser, and less connected to other domains. Content-wise, we additionally found that Disbelievers were the near-exclusive sharers of majority right-wing and known fake news sources, whereas both Believers and Disbelievers shared left-wing and centrist domains. Collectively, our findings indicate that climate change Believers rely on a popular, well-consolidated, and exclusive set of mainstream web domains. In contrast, Disbelievers draw on a diverse and fragmented collection of fringe information sources. Although these results suggest the marginal status of climate change skepticism, they also troublingly point to its expansive ecosystem of online information featuring multiple entry points. We conclude with directions for future research and potential implications for science communication and climate change policymakers.