The hyperparameter tuning of message-passing mechanisms in graph networks [1], as a cornerstone of their geometric learning, has proven to be a critical factor in their performance during inference and extrapolation [2]. This study presents a systematic characterization and analysis of this algorithm (highly system-dependent), focusing on its role in modeling partial differential equations (PDEs), beginning with hyperbolic systems and extending to parabolic and elliptic cases. Initial results provide insights into optimizing the number of message-passing iterations based on the differential system’s properties, enhancing the interpretability of graph networks by framing their learning process as a nonlinear spatial operator [1] deeply connected to the underlying physics. Furthermore, we establish a relationship between physical propagation speed and message-passing range, elucidating the under-reaching phenomenon and linking its behavior to the CFL (Courant-Friedrichs- Lewy) condition. For elliptic and parabolic problems, we identify a dependence on mesh geometry and connectivity, revealing how message-passing sensitivity influences the learning of these systems’ physics. Additionally, we analyze the latent space dynamics, demonstrating that avoiding under-reaching is crucial and that message aggregation acts as an error regularization mechanism in recurrent prediction schemes. For both elliptic and parabolic problems, we observe a dependence on mesh geometry and connectivity, highlighting how the sensitivity of message passing influences the learning of the underlying physics. Additionally, we analyze the regularization effect of increased message-passing iterations, once a sufficient lower bound is met. A higher number of message-passing steps enhances robustness to noise, which is crucial for achieving stable solvers over long rollouts. The findings of this work establish physics-informed principles for the design and operation of graph networks in PDE modeling, advancing the state-of-the-art in computational physics and providing new interpretability tools for these complex architectures.