Selim Haj Ali

• Understanding how collective patterns emerge on graphs is a fundamental challenge across disciplines, from biological and ecological networks to computational and physical systems. This thesis explores the interplay between network topology and emergent dynamics using minimal models and spectral graph techniques. A first investigation focuses on network inference, showing that Turing patterns encode structural information about the underlying graph, which we use to infer missing links. The second study investigates multistability in reaction-diffusion networks, showing how local spectral gaps influence the attractor landscape of Turing patterns using a heuristic binary classification algorithm. Finally, the third study applies the sandpile model to soil erosion processes, bridging concepts from self-organised criticality and connectivity-based geomorphology to investigate the role of minimal models in empirical research. This thesis combines theoretical analysis, computational modelling and empirical validation to highlight how structure shapes dynamics across different contexts and illustrate the potential of minimal models as predictive, explanatory and exploratory tools for complex systems.