Periodic Reporting for period 1 - HICODY (Mathematical Challenges of Higher-Order Interactions in Collective Dynamics, and Applications)
Reporting period: 2022-09-01 to 2024-08-31
The mathematical analysis of collective dynamics has experienced a prominent growth in the last years, leading to new frontiers with cutting-edge fields in physics, biology and social sciences (e.g. complex networks, active matter or crowd dynamics). Collective dynamics is ubiquitous in science and appears in areas like social sciences, biology and neuroscience. Examples encompass the alignment of a flock, the emergence of opinion consensus, or neural synchronization. Most biological systems consist of a large amount of agents (often ranging from a billion to a million units). Hence, uncovering the mechanisms responsible for the cooperation and the spontaneous formation of large-scale self-organized patterns in the full population is a fundamental question with strong scientific implications.
It is remarkable that a handful of breakthrough papers appeared in the transition of the millennium, triggering thousands of contributions over the last decade. From a mathematical perspective, the main breakthrough is that unveiling self-organization in a large group of agents can be tackled using strong mathematical methods from nonlinear and nonlocal PDEs. Typically, nonlinear ODEs/PDEs are used to describe the evolution of these complex systems at the various scales of description: microscopic (ODEs), mesoscopic (kinetic PDEs) and macroscopic (fluid-type PDEs). Classical methods have transcended applications and gave rise to strong advances in nonlinear PDE, optimal transport, statistical mechanics or fluid mechanics with seminal works by a large community of international researchers.
The main research goal of HICODY was to investigate an emerging class of collective dynamics models describing the evolution of large populations of self-organized agents. More specifically, the cornerstone in HICODY was to develop innovative techniques in the analysis of nonlinear PDEs to go beyond the classical setting with pairwise interactions (PWI) and address more realistic scenarios governed by higher-order interactions (HOI). The common ground in the previous literature was the imposing premise that cooperation is ruled by PWI. However, the up-to-date experimental evidence suggests that PWI are often not sufficient to properly shape collective behavior in real-world systems, and it conjectures that communication rather is a collective nonlinear action at the level of groups. To date, these HOI had remained mostly unexplored due to their mathematical complexity, leaving the experimental community without appropriate models to test hypotheses. The ultimate aim of HICODY was to contribute to filling this gap and providing novel mathematical analysis and computational tools that scientists need to explain complex patterns in nature.
This project was divided into three main blocks:
1) The first one aimed to deriving the rigorous kinetic and fluid-type PDEs arising in statistical mechanics from the underlying microscopic description, thus characterizing the evolution of the probability distribution of agents and representing HOI at larger scales. Besides, the formation of patterns from multiple interactions was analyzed in several examples of velocity alignment and synchronization dynamics among others.
2) The second block focused on the study of long-time behavior in a particular integro-differential equation arising in quantitative genetics which describes the dynamics of a population structured by phenotypical traits and governed by Darwinian selection and sexual reproduction. Understanding how patterns emerge from the corresponding recombination of traits during individual interactions is of crucial importance in genetics.
3) Finally, the last block integrated an interdisciplinary approach which combined analytical and computational tools to face the demanding technical level in an innovative application to neuroscience. Specifically, novel activity patterns were explored in large ensembles of neurons with PWI but also HOI.
It is remarkable that a handful of breakthrough papers appeared in the transition of the millennium, triggering thousands of contributions over the last decade. From a mathematical perspective, the main breakthrough is that unveiling self-organization in a large group of agents can be tackled using strong mathematical methods from nonlinear and nonlocal PDEs. Typically, nonlinear ODEs/PDEs are used to describe the evolution of these complex systems at the various scales of description: microscopic (ODEs), mesoscopic (kinetic PDEs) and macroscopic (fluid-type PDEs). Classical methods have transcended applications and gave rise to strong advances in nonlinear PDE, optimal transport, statistical mechanics or fluid mechanics with seminal works by a large community of international researchers.
The main research goal of HICODY was to investigate an emerging class of collective dynamics models describing the evolution of large populations of self-organized agents. More specifically, the cornerstone in HICODY was to develop innovative techniques in the analysis of nonlinear PDEs to go beyond the classical setting with pairwise interactions (PWI) and address more realistic scenarios governed by higher-order interactions (HOI). The common ground in the previous literature was the imposing premise that cooperation is ruled by PWI. However, the up-to-date experimental evidence suggests that PWI are often not sufficient to properly shape collective behavior in real-world systems, and it conjectures that communication rather is a collective nonlinear action at the level of groups. To date, these HOI had remained mostly unexplored due to their mathematical complexity, leaving the experimental community without appropriate models to test hypotheses. The ultimate aim of HICODY was to contribute to filling this gap and providing novel mathematical analysis and computational tools that scientists need to explain complex patterns in nature.
This project was divided into three main blocks:
1) The first one aimed to deriving the rigorous kinetic and fluid-type PDEs arising in statistical mechanics from the underlying microscopic description, thus characterizing the evolution of the probability distribution of agents and representing HOI at larger scales. Besides, the formation of patterns from multiple interactions was analyzed in several examples of velocity alignment and synchronization dynamics among others.
2) The second block focused on the study of long-time behavior in a particular integro-differential equation arising in quantitative genetics which describes the dynamics of a population structured by phenotypical traits and governed by Darwinian selection and sexual reproduction. Understanding how patterns emerge from the corresponding recombination of traits during individual interactions is of crucial importance in genetics.
3) Finally, the last block integrated an interdisciplinary approach which combined analytical and computational tools to face the demanding technical level in an innovative application to neuroscience. Specifically, novel activity patterns were explored in large ensembles of neurons with PWI but also HOI.
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.
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.
The outcomes of HICODI are at the forefront of research in mathematical collective dynamics and their applications to physical and biological systems. Despite their recent publication, the works have already been cited by several international researchers. The researcher has been invited to prestigious conferences to present these findings, including BIRS, Equadiff, CIRM, NYU Abu Dhabi, and Sorbonne Université.
This is a project in theoretical mathematics. This means that all of the works proved new results which beat the state-of-the-art, and that in the long-term, the corresponding findings may have a big impact, but that there are no direct societal consequences. Particularly for this project, besides expanding our knowledge it also transfers ideas between mathematics, physics and biology.
The concept of fibered Wasserstein gradient flows has found unexpected applications in opinion dynamics models with heterogeneities and in machine learning, where infinitely deep and wide residual neural networks have been modeled as fibered gradient flows. It has also influenced optimal control problems in Wasserstein spaces and has been extended to general metric bundles, opening new avenues for studying multi-agent systems on manifolds and more complex spaces.
The mean-field limit of collective dynamics models with higher-order interactions (HOI) is another key outcome, providing a correct kinetic model even for non-uniform hypergraphs. This is expected to impact the study of phase transitions and numerical experiments in real-world networks.
The results on quantitative genetic models have influenced both mathematical biology and the field of functional inequalities, inspiring new anisotropic transport-information inequalities. While L^2-based versions are well-established, the new L^\infty version derived in this action is expected to significantly impact the analysis of complex non-linear models.
These achievements will also benefit the training of new researchers, with a new PhD student joining the team to build on the project's objectives.
This is a project in theoretical mathematics. This means that all of the works proved new results which beat the state-of-the-art, and that in the long-term, the corresponding findings may have a big impact, but that there are no direct societal consequences. Particularly for this project, besides expanding our knowledge it also transfers ideas between mathematics, physics and biology.
The concept of fibered Wasserstein gradient flows has found unexpected applications in opinion dynamics models with heterogeneities and in machine learning, where infinitely deep and wide residual neural networks have been modeled as fibered gradient flows. It has also influenced optimal control problems in Wasserstein spaces and has been extended to general metric bundles, opening new avenues for studying multi-agent systems on manifolds and more complex spaces.
The mean-field limit of collective dynamics models with higher-order interactions (HOI) is another key outcome, providing a correct kinetic model even for non-uniform hypergraphs. This is expected to impact the study of phase transitions and numerical experiments in real-world networks.
The results on quantitative genetic models have influenced both mathematical biology and the field of functional inequalities, inspiring new anisotropic transport-information inequalities. While L^2-based versions are well-established, the new L^\infty version derived in this action is expected to significantly impact the analysis of complex non-linear models.
These achievements will also benefit the training of new researchers, with a new PhD student joining the team to build on the project's objectives.