The main aim of Working Group 1 is to develop a rigorous analytical framework that bridges classical discrete models, such as graphs, with more general metric and higher-order structures, including metric graphs, hypergraphs, graphons, and related combinatorial environments. By extending tools traditionally confined to either discrete or continuous settings, WG1 seeks to uncover deep connections between spectrum, geometry, and topology in complex systems.

The group focuses on the study of Laplace-type operators and associated quadratic forms on discrete and metric structures, with particular attention to functional inequalities and spectral properties. A central theme is the investigation of limits of large or dense networks using probabilistic and analytic tools such as Benjamini–Schramm convergence and graphon theory. These approaches allow WG1 to rigorously describe how local interactions scale to global behavior in increasingly complex systems.

Beyond foundational theory, WG1 aims to develop mathematically sound notions of resilience, stability, and robustness for complex networks, especially in evolving or random environments. Its results are expected to provide core methods and concepts that are reused and adapted across all other Working Groups, making WG1 a natural hub for cross-WG collaboration and integration.