Real complex systems exhibit rich collective behavior, yet identifying which components of an interaction network drive such dynamics remains a central challenge. Here, we show that complexity itself can resolve this problem. In large random and empirical networks, structural disorder and heterogeneity induce spectral localization, causing Laplacian modes to concentrate on small subsets of nodes. This converts global modes into identifiable dynamical units tied to specific structural components. Exploiting this principle, we develop a node-resolved stability framework that predicts instability onsets, identifies the nodes responsible for collective transitions, and restores interpretability in systems where classical modal theories fail. In heterogeneous reaction networks, the same mechanism enables collective states beyond those usually associated with homogeneous assumptions. More broadly, our results show that complexity can be revealed, rather than obscure, the microscopic drivers of macroscopic dynamics.

A spatial graph is a graph whose nodes and edges carry spatial attributes. It is a smart modelling choice for capturing the skeleton of a shape, a blood vessel network, a porous tissue, and many other data objects with intrinsically complex geometry, often resulting in graphs with a high node and edge count. In this talk, we introduce a topological spatial graph coarsening approach based on a new framework that balances graph reduction against the preservation of topological characteristics, essential for faithfully representing the underlying shape. To capture the topological information required to calibrate the reduction level, we adapt the construction of classical topological descriptors made for point clouds (the so-called persistence diagrams) to spatial graphs. This relies on a new filtration called triangle-aware graph filtration. Our coarsening approach is parameter-free and we prove that it is equivariant under rotations, translations, and scaling of the initial spatial graph. We evaluate the performance of our method on synthetic and real spatial graphs and show that it significantly reduces the graph sizes while preserving the relevant topological information.