TopoNets 2026

Graph Neural Networks (GNNs) have achieved notable success in learning on graph-structured data in areas such as biochemistry, drug discovery, and materials science. However, GNNs are not without challenges. In this talk we discuss two challenges that arise in when the learning task depends on higher-order information: First, deeper GNNs exhibit instability due to the convergence of node representations, known as oversmoothing, which can reduce their effectiveness in learning long-range interactions and dependencies. We propose and study unitary graph convolutions, which allow for deeper networks that provably avoid oversmoothing during training. Our experimental results confirm that Unitary GNNs achieve competitive performance on benchmark datasets. Second, GNNs have limited expressivity in that there are fundamental function classes that they cannot learn. An effective remedy for limited expressivity is the use of encodings, which augment the input graph with additional structural information. We propose novel encodings based on discrete Ricci curvature, which lead to significant gains in empirical performance and expressivity by capturing higher-order relational information. We then consider the more general question of how higher-order information can be leveraged most effectively in graph learning. We propose a set of encodings that are computed on a hypergraph parametrization of the input graph and provide theoretical and empirical evidence for their effectiveness.

In highly productive agricultural landscapes, human impacts combine with naturally variable climate and geology to fundamentally alter processes in the critical zone (CZ), the layer of the Earth’s surface that includes root-soil-vegetation-atmosphere interfaces. For example, artificial drainage and fertilizer applications impact riverine transport of water and nutrients, and abrupt transitions in vegetation states impact land-atmosphere fluxes of water and energy. Subsystem components may switch from being nearly independent to tightly synchronized, and drivers may elicit non-linear or threshold responses. These shifts have implications for predictive understanding in Earth and environmental science, but are challenging to identify and quantify. This presentation will explore information theory and other data-driven methods to identify process connectivity, predictability, and temporal regimes in CZ behaviors, based on multivariate high frequency datasets, spatially gridded data, and ecohydrological model results.  In a modeling context, these metrics can be considered as measures of “functional performance”, or the extent to which our models “get the right answers for the right reasons”.

Percolation establishes the connectivity of complex networks and is one of the most fundamental critical phenomena for studying complex systems. On simple networks, percolation displays a second-order phase transition; on multiplex networks, the percolation transition can become discontinuous. However, little is known about percolation in networks with higher-order interactions. Here, I will show that percolation can be turned into a fully fledged dynamical process when higher-order interactions are taken into account. By introducing signed triadic interactions, where a node can regulate the interactions between two other nodes, we define triadic percolation. We uncover that in this paradigmatic model the connectivity of the network changes in time, and the order parameter undergoes period doubling and a route to chaos.

I will present a general theory for triadic percolation that accurately predicts the full phase diagram on random graphs, confirmed by extensive numerical simulations, and show that real network topologies reveal similar phenomenology. I will then discuss recent extensions showing that triadic interactions can induce dynamical topological patterns in spatial higher-order networks, generate rich spatio-temporal activity regimes in an in silico neural medium, and produce even richer bifurcation diagrams in multilayer networks. These results radically change our understanding of percolation and may be used to study complex systems where functional connectivity changes dynamically and non-trivially, such as neural, climate and ecological systems.

The study of higher-order networks has flourished in recent years to explore how different multibody interactions can affect the structure and dynamics of complex systems. Yet, most higher-order descriptions remain rooted on the concept of nodes, representing graphs through the hyperdegree distributions of nodes, or the sizes and overlaps of hyperedges in terms of numbers of nodes. We introduce k-star covers as a novel structural decomposition for higher-order networks. A k-star cover is defined as a set of structures of order k overlapping on shared structures of order k-1, generalizing the notion of star graphs to higher orders: 1-stars correspond to regular star graphs, 2-stars to triangles overlapping on edges, and so on. We show that k-star covers provide a principled and computationally tractable framework for characterizing multi-body correlations in hypergraphs at multiple scales. We compute k-star covers for real-world hypergraph datasets as well as clique complexes of regular graphs, demonstrating that this decomposition captures structural information beyond standard pairwise adjacency representations. Our results suggest that k-star covers offer a promising building block for the development of more expressive models of structure and dynamics on higher-order networks.

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