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In this paper we introduce a mathematical model that captures some of thesalient features of recommender systems that are based on popularity and thattry to exploit social ties among the users. We show that, under very generalconditions, the market always converges to a steady state, for which we areable to give an explicit form. Thanks to this we can tell rather precisely howmuch a market is altered by a recommendation system, and determine the power ofusers to influence others. Our theoretical results are complemented byexperiments with real world social networks showing that social graphs preventlarge market distortions in spite of the presence of highly influential users.

The Limits of Popularity-Based Recommendations, and the Role of Social Ties