When we search online for content, we are constantly exposed to rankings. For example, web search results are presented as a ranking, and online bookstores often show us lists of best-selling books. While popularity-based ranking algorithms (like Google's PageRank) have been extensively studied in previous works, we still lack a clear understanding of their potential systemic consequences. In this work, we fill this gap by introducing a new model of network growth that allows us to compare the properties of the networks generated under the influence of different ranking algorithms. We show that by correcting for the omnipresent age bias of popularity-based ranking algorithms, the resulting networks exhibit a significantly larger agreement between the nodes' inherent quality and their long-term popularity, and a less concentrated popularity distribution. To further promote popularity diversity, we introduce and validate a perturbation of the original rankings where a small number of randomly-selected nodes are promoted to the top of the ranking. Our findings move the first steps toward a model-based understanding of the long-term impact of popularity-based ranking algorithms, and could be used as an informative tool for the design of improved information filtering tools.