Networks are fascinating because of their ability to describe complex structures found in a broad variety of systems, from arts and humanities, via the life sciences to the physical science and mathematics. Perhaps even more startling is the variety of approaches that different disciplines have contributed to the study of networks. All of these approaches have a common goal: finding simplicity in complexity. Yet complexity science has no single overarching theory of what simplicity means and how and why it can be found. In this talk I will present some well known methods and results to highlight different approaches to finding simplicity that computer science, physics and mathematics have developed. I will then highlight some less-known connections and argue that an overarching theory of simplicity may be within reach. 

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.