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As social recommendations such as friend suggestions and people to follow become increasingly popular and influential on the growth of social media, we find that prominent social recommendation algorithms can exacerbate the under-representation of certain demographic groups at the top of the social hierarchy. To study this imbalance in online equal opportunities, we leverage new Instagram data and offer for the first time an analysis that studies the effect of gender, homophily and growth dynamics under social recommendations. Our mathematical analysis demonstrates the existence of an algorithmic glass ceiling that exhibits all the properties of the metaphorical social barrier that hinders groups like women or people of color from attaining equal representation. What raises concern is that our proof shows that under fixed minority and homophily parameters the algorithmic effect is systematically larger than the …