Many systems in the natural sciences and beyond exhibit complex relational structure that changes over time. Social networks evolve as relationships change, traffic patterns vary throughout the day, and protein–protein interactions shift with cellular conditions. Learning these dynamics from data is a challenging problem. A recent approach in this area, Graph Neural Controlled Differential Equations, extends Neural CDEs from paths on Euclidean domains to paths on graph domains. In this talk, we discuss an extension of this framework that respects the geometry of the underlying set and is equivariant to permutations of the node ordering. We will discuss empirical advantages of this modification, as well as benefits of the formulation as a continuous-time model. 

Researchers across disciplines rely on clustering to uncover meaningful patterns in noisy similarity data. Standard two-step pipelines reduce noise before clustering, introducing arbitrary parameters that often produce misleading structure. We unite noise reduction and clustering through Bayesian community detection, using information theory to balance model complexity and fit. This one-step approach automatically determines the number of clusters, avoids detecting patterns in random data, and makes full use of limited samples. Testing on synthetic benchmarks and gene expression data shows the approach yields more reliable and interpretable results than widely used alternatives, improving data-driven discovery across scientific disciplines where samples are limited or expensive.

Robustness to perturbation is a key topic in the study of complex systems occurring across a wide variety of applications from epidemiology to biochemistry. In this talk I will examine the eigenspectrum of the Jacobian matrices associated to a general class of networked dynamical systems, which contains information on how perturbations to a stationary state develop over time. I will show that stability is always determined by a spectral outlier, but with pronounced differences to the corresponding eigenvector in different regimes. Depending on model details, instability may originate in nodes of anomalously low or high degrees, or may occur everywhere in the network at once. Our results have potentially useful applications in network monitoring to predict or prevent catastrophic failures.

Recurrent neural networks (RNNs) provide a theoretical framework for understanding computation in biological neural circuits, yet classical results, such as Hopfield's model of associative memory, rely on symmetric connectivity that restricts network dynamics to gradient-like flows. In contrast, biological networks support rich time-dependent behaviour facilitated by their asymmetry. In this talk, I will introduce a general framework, known as ‘drift-diffusion matching’, for training continuous-time RNNs to represent arbitrary stochastic dynamical systems within a low-dimensional latent subspace. Allowing asymmetric connectivity, I will show that RNNs can embed the drift and diffusion of an arbitrary stochastic differential equation, including nonlinear and nonequilibrium dynamics such as chaotic attractors. As an application, we have constructed RNN realisations of stochastic systems that transiently explore various attractors through both input-driven switching and autonomous transitions driven by nonequilibrium currents, which we interpret as models of associative and sequential (episodic) memory. To elucidate how these dynamics are encoded in the network, I will introduce decompositions of the RNN based on its asymmetric connectivity and its time-irreversibility. These results extend attractor neural network theory beyond equilibrium, showing that asymmetric neural populations can implement a broad class of dynamical computations within low-dimensional manifolds, unifying ideas from associative memory, nonequilibrium statistical mechanics, and neural computation.

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.