The project has resulted in two peer-reviewed publications, two preprints submitted for review, and two more works expected to be completed and submitted within six to nine months.
We started with preparatory work before addressing objectives O1 and O2. Most real-world multi-agent systems display a high level of heterogeneity, with individual characteristics differing significantly among neighboring agents, which impacts global dynamics. At large scales, this heterogeneity manifests as an infinite system of coupled PDEs representing the evolution of the probability distribution of agents with certain features, each taking the form of a fibered continuity equation. In this action, we introduced the fibered Wasserstein space as the correct functional setting to analyze these equations. This space is defined through an optimal transport problem that penalizes transport across different fibers. We identified a weakly Riemannian structure for this space, allowing for subdifferential and differential calculus (similar to Otto calculus). This framework helped us identifying a gradient flow structure for a wide range of heterogeneous dynamics models like aggregation, flocking, and synchronization for the first time in the literature.
For objective O1, we focused on multi-agent systems with heterogeneous interaction networks, such as social networks, political affiliations, or neuronal connections. These systems are complex, and describing them individually becomes infeasible, making large population limits crucial. This is addressed by the mean-field limit, which approximates the system through its population density. In this action, we studied general multi-agent systems involving higher-order interactions (HOI) using hypergraphs. We showed that, under regular conditions, the mean-field limit can be described by a kinetic equation where the hypergraph limit is encoded by an unbounded-rank hypergraphon, capturing infinitely many interaction levels.
We then explored synchronization models involving HOI in collaboration with Seoul National University, the secondment institution. This work, part of objective O2, built on numerical experiments physicist showing that binary and ternary interactions lead to stable equilibria transitioning from monopolar to bipolar configurations based on the relative coupling strengths. The fibered Wasserstein space was again fundamental, revealing a hidden gradient flow structure. However, due to the non-convex nature of the energy functional, local convexity regions vary with coupling strengths, explaining the observed phase transitions. Further work was required to rigorously prove convergence rates to equilibrium in relevant regimes, and these outcomes are being prepared for publication.
We also addressed objective O2 by studying Fisher’s infinitesimal model in quantitative genetics, which describes the evolution of continuous traits under sexual reproduction and selection. This model involves three individuals (two parents and one offspring), representing ternary interactions and non-conservative dynamics. In this action, we obtained quantitative convergence rates to a unique pseudo-equilibrium under strong selection for the first time. We started with quadratic selection and extended the results to general convex selection cases.
Lastly, for objective O3, we investigated activity patterns in new variants of the Kuramoto model, relevant for understanding abnormal brain dynamics such as those in epilepsy. We focused on models that capture both phase and amplitude dynamics, like the Stuart-Landau model. In this action, we found that amplitude heterogeneity introduces new phenomena not observed in previous studies, such as transitions from active incoherence to active phase-locking and then to death phase-locking, before returning to active phase-locking. This leader-driven dynamics in intermediate coupling strengths challenges previous findings and is being finalized for publication.
