Like other social systems, in collaborative filtering a small number of "influential" users may have a large impact on the recommendations of other users, thus affecting the overall behavior of the system. Identifying influential users and studying their impact on other users is an important problem because it provides insight into how small groups can inadvertently or intentionally affect the behavior of the system as a whole. Modeling these influences can also shed light on patterns and relationships that would otherwise be difficult to discern, hopefully leading to more transparency in how the system generates personalized content. In this work we first formalize the notion of "influence" in collaborative filtering using an Influence Discrimination Model. We then empirically identify and characterize influential users and analyze their impact on the system under different underlying recommendation algorithms and across three different recommendation domains: job, movie and book recommendations. Insights from these experiments can help in designing systems that are not only optimized for accuracy, but are also tuned to mitigate the impact of influential users when it might lead to potential imbalance or unfairness in the system's outcomes.