In a network, pairs of individual elements, or nodes, connect to each other; those connections can represent a sprawling system with myriad individual links. A hypergraph goes deeper: It gives researchers a way to model complex, dynamical systems where interactions among three or more individuals—or even among groups of individuals—may play an important part. Instead of edges that connect pairs of nodes, it is based on hyperedges that connect groups of nodes. Hypergraphs can represent higher-order interactions that represent collective behaviors like swarming in fish, birds, or bees, or processes in the brain.
Scientists usually use a hypergraph model to predict dynamic behaviors. But the opposite problem is interesting, too. What if researchers can observe the dynamics but don’t have access to a reliable model? Yuanzhao Zhang, an SFI Complexity Postdoctoral Fellow, has an answer.
In a paper published in Nature Communications, Zhang and his collaborators describe a novel algorithm that can infer the structure of a hypergraph using only the observed dynamics.
Their algorithm uses time-series data — observations collected at even intervals over a period — to construct hypergraphs (and other representations of higher-order interactions) that produce the observed patterns. It can be applied to any dataset assumed to have some underlying mathematical structure, Zhang says. Time-series data are useful for studying the spread of disease or the behavior of financial markets, biological systems, and many other situations.
Notably, the approach only requires the data; it doesn’t require prior knowledge about the system or how individual nodes behave. “That’s the main advantage,” Zhang says. “It opens up a lot more possibilities, and you can apply it to systems for which you don’t know the underlying dynamics.”
He points to brain function as an example. Researchers can collect observational time-series data, but they don’t have a good model for how everything fits together. “Obviously we cannot cut open our brains and see what’s actually going on,” he says. “But we can learn something by looking at data from brain recordings.”
In the new paper, Zhang and his collaborators verified their approach by testing it on time-series data, ensuring that it produced a known underlying structure. Then, they applied it to electroencephalogram (EEG) data collected from more than 100 human subjects. An EEG measures electrical activity in various areas of the brain over time, collected through sensors stuck to a person’s scalp. The resulting report looks like a series of waves.
Most known connections in the brain are pairwise, connecting one brain region to another. However, using their new algorithm, Zhang and his collaborators unearthed a hypergraph model that accurately captured connections in the EEG data among three or more regions. That suggests higher-order interactions play an important and underappreciated role in shaping macroscopic patterns of brain activity.
The researchers used their model to identify the most frequent types of interactions among brain regions. “What’s really interesting is that the top six prominent hyperedges all pointed towards the prefrontal cortex, which is known to be one of the information processing hubs in the brain,” Zhang says. The current work can infer a model of a few hundred nodes; in the future, he hopes to scale up to larger networks.
Read the paper "Hypergraph reconstruction from dynamics" in Nature Communications (March 19, 2025) at doi: 10.1038/s41467-025-57664-2