Modeling cross-platform narrative diffusion: a stable Hawkes–ODE framework for exposure and adoption dynamics

Abstract Understanding how narratives spread across social media platforms requires a framework that captures both the stochastic generation of posts and the gradual evolution of user engagement. This study proposes a cross-platform diffusion model that integrates a marked multivariate Hawkes process with a bounded exposure–adoption ordinary differential equation (ODE). The Hawkes component models event arrivals and cross-platform excitation, while the ODE captures how exposure translates into active engagement over time. Applied to the 2025 U. S. tariff-war discourse spanning 30, 493 Instagram, 11, 218 TikTok, 12, 252 X, and 30, 493 YouTube posts collected between January and May 2025, the model reveals interpretable and stable cross-platform influence dynamics. The estimated branching matrix ( (B) = 0. 523) lies well below the critical threshold ( (B) = 1), indicating a stable, self-damped diffusion regime in which cross-platform cascades persist but do not explode, with dominant pathways from short-form and real-time platforms toward long-form commentary channels. The exposure–adoption ODE was validated through a forward-prediction experiment, training on a percentage of each trajectory and evaluating on the future, demonstrating accurate extrapolation of engagement trends and confirming the model’s predictive coherence. Together, these results demonstrate that the proposed framework provides a mathematically grounded and interpretable lens for analyzing how narratives emerge, propagate, and stabilize across interconnected media ecosystems.