With new review and roadmap papers, VU mathematicians Raffaella Mulas, Magnus Botnan, and Christian Bick are at the edge of research on higher-order network interactions.
Collaboration networks are an example where traditional models based on pairwise interactions fall short: Three researchers may co-author a research article without them without any pair of them having written a paper jointly before. Thus, such genuine three-way interactions cannot be adequately captured by graph models, which capture pairwise links. Further examples arise across disciplines, from neuroscience and social systems to biological and technological networks, where interactions are of “higher order” in the sense that they naturally involve more than two entities at a time. Hypergraphs provide a flexible mathematical framework to describe and analyze higher-order interaction networks.
The Department of Mathematics of the Vrije Universiteit Amsterdam has developed into one of the leading research centers for research on higher-order networks, hypergraphs, and their applications. For example, the VU coordinates the EU-funded Marie Curie Doctoral Network BeyondTheEdge, which is led by VU mathematician Christian Bick and coordinated by the VU. Bringing together key researchers in this rapidly advancing field, VU mathematician Raffaella Mulas co-organized a four-week special program at the Alan Turing Institute in London in the summer of 2024.
A key outcome of this program was the development of the comprehensive perspective Hypergraphs and simplicial complexes in focus: A roadmap for future research in higher-order interactions, which has appeared in Journal of Physics: Complexity. This roadmap paper was written collaboratively by participants of the program—including VU mathematicians Raffaella Mulas, Magnus Botnan, and Christian Bick—and coordinated by Mulas and her workshop co-organizers. This forward-looking roadmap identifies major theoretical questions, computational challenges, and emerging application areas to serve as a guide for future research directions.
At the same time, the review Collective dynamics on higher-order networks, which was co-authored by Bick and just appeared in Nature Reviews Physics takes stock of recent developments on dynamical properties of higher-order networks. Despite the specific focus on network dynamical systems, it also links recent results to other strands of research within the Amsterdam Center for Dynamics and Computation.
Further work in this area is also forthcoming, including the book Spectra of Discrete Structures by Mulas and collaborators, which is related to the spectral foundations of such networks.
Hypergraphs and simplicial complexes in focus: A roadmap for future research in higher-order interactions, Journal of Physics: Complexity; doi:10.1088/2632-072X/ae3c4e (Open Access)
Collective dynamics on higher-order networks, Nature Reviews Physics; doi:10.1038/s42254-025-00916-3
Spectra of Discrete Structures, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Forthcoming (Open Access)