Ergodic and Metastable Behavior of Stochastic Popularity Dynamics in Social Networks

Abstract

This paper studies popularity dynamics in social networks, where users' engagement with competing channels is driven by intrinsic content quality, social influence, and popularity feedback. We introduce a stochastic model in which users interact through a directed influence network and channel exposure depends jointly on these three mechanisms. We first characterize the resulting dynamics in the absence of intrinsic quality, showing that popularity concentrates asymptotically on a single channel, leading to a winner-takes-all regime. When all channels have positive intrinsic quality, we establish ergodicity of the popularity process and characterize its long-run behavior. We further show that, when intrinsic quality is small, the system exhibits metastable behavior: it can remain for long periods near configurations dominated by a single channel before transitioning to another. These results highlight how weak quality differences can coexist with strong path dependence induced by popularity feedback. We then investigate platform-level interventions and propose controls that compensate for network-induced exposure biases and mitigate excessive concentration. Analytical results are illustrated through numerical simulations and an empirical analysis of YouTube data, whose observed popularity dynamics are consistent with the proposed stochastic framework. The results provide a mathematical framework for understanding the interplay between quality, social influence, and reinforcement in online popularity dynamics.