The goal of this thesis is to analyze how social networks, social imitation, combined with recommendation systems, and content quality can play a crucial role in popularity dynamics in social media. In particular, we formulate a new model of popularity dynamics in social networks and study its mathematical properties. The proposed model is nonlinear and includes randomness components to capture the unpredictable fluctuations and stochastic dynamics typical of social media. Using data collected on YouTube videos over the period 2020-2023, we explore real-world trends in user engagement and video virality to gain insights into the mechanisms driving popularity dynamics.