Higher Order Renormalization of NETworks

Objective

Complex systems exhibit collective behaviours that arise from group interactions, represented by networks beyond pairwise interactions, also known as higher-order networks. Since emergent phenomena often appear across scales, renormalization methods have recently been introduced for simple networks. However, only limited efforts have been made to extend renormalization to higher-order structures, leaving a substantial gap. This project addresses it by developing a principled renormalization framework for higher-order networks. The approach introduces two complementary methods, both based on a hidden variables formulation. The first, Multiscale Higher-Order Network Renormalization, defines ensembles of random hidden-variable hypergraphs invariant under coarse graining. The second, Geometric Higher-Order Network Renormalization, embeds hypergraphs in metric spaces and performs renormalization by geometric proximity, where the geometric coordinates are themselves hidden variables. In addition, the project will go beyond the common practice of applying renormalization exclusively to structure, by investigating its impact on higher-order dynamics. This includes processes governed by group mechanisms, such as higher-order spreading that captures diffusion over social channels, synchronization driven by group couplings, and Ising inspired models with Hamiltonians including higher-order interaction terms. Studying how these processes transform under coarse graining will reveal how collective behaviours change across scales, which dynamical features are preserved, and when macroscopic order emerges or vanishes. Finally, the two frameworks will be tested on empirical datasets, including brain connectomes, temporal social interactions and economic systems. This comparative validation will assess the distinctive multiscale signatures of higher-order interactions and support the search for universal principles governing the organization of group mechanisms across scales.