Graphs are a highly expressive abstraction for modeling entities and their relations, such as molecular structures, social networks, and traffic networks. Deep Graph Networks (DGNs) have emerged as a family of deep learning models that can effectively process and learn such structured information. However, learning effective information propagation patterns within DGNs remains a critical challenge that heavily influences the model capabilities, both in the static domain and in the temporal domain (where features and/or topology evolve). Given this challenge, this thesis investigates the dynamics of information propagation within DGNs for static and dynamic graphs, focusing on their design as dynamical systems. Throughout this work, we provide theoretical and empirical evidence to demonstrate the effectiveness of our proposed architectures in propagating and preserving long-term dependencies between nodes, and in learning complex spatio-temporal patterns from irregular and sparsely sampled dynamic graphs. In summary, this thesis provides a comprehensive exploration of the intersection between graphs, deep learning, and dynamical systems, offering insights and advancements for the field of graph representation learning and paving the way for more effective and versatile graph-based learning models.

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