Higher Order Renormalization of NETworks

Description du projet

Les réseaux d’ordre supérieur et leurs comportements collectifs émergents

De nombreux systèmes du monde réel, du cerveau aux réseaux sociaux en passant par les économies, impliquent des interactions collectives entre de nombreux éléments. Comprendre comment les comportements collectifs émergent à différentes échelles constitue un défi fondamental qui reste à relever. La renormalisation, qui consiste à redéfinir les paramètres afin d’éliminer les phénomènes «aberrants» découlant de la théorie mais non observés expérimentalement, a été appliquée avec succès aux interactions structurelles par paires. Elle reste encore largement inexplorée en ce qui concerne les structures d’ordre supérieur et leur dynamique. Soutenu par le programme Actions Marie Skłodowska-Curie, le projet HORNET vise à combler cette lacune en développant un cadre de renormalisation rigoureux pour les réseaux d’ordre supérieur. Il entend ainsi étudier comment les comportements collectifs évoluent à différentes échelles, en validant ses résultats à partir d’ensembles de données réels, notamment des connectomes cérébraux, des interactions sociales et des systèmes économiques.

Objectif

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.