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Maxime Lucas

Publications and source records attributed to Maxime Lucas.

At least 19 recordsLinked to original sources
arXiv2026

Higher-order interactions for controlling time-delayed Kuramoto model

We propose a framework for controlling the collective dynamics of the time-delayed Kuramoto model based on a delay-free, higher-order approximation of the delayed interactions. By applying the Ott--Antonsen ansatz and the second-order averaging method to the resulting higher-order Kuramoto model, we obtain a one-dimensional reduced equation for the order parameter dynamics. Numerical simulations demonstrate that the higher-order approximation predicts the dynamics of the original delayed system more accurately than the conventional pairwise approximation and enables the realization of bistability and intermediate synchronization states. Our results demonstrate the effectiveness of higher-order interpretations of time delays for the control of oscillator networks with time-delayed interactions.

nlin.AO↗
arXiv2026

Symmetry-based selection rules for higher-order interactions in coupled oscillators

Pairwise interactions among general nonlinear oscillators can be reduced, via phase reduction, to a Kuramoto-type phase coupling $\sin(- \theta_j+\theta_k )$. For higher-order interactions, multiple phase couplings exist -- such as $\sin(-2\theta_j+\theta_k+\theta_l )$ and $\sin(-\theta_j+2\theta_k-\theta_l)$. Since different nonpairwise coupling functions produce qualitatively different dynamics, it is important to understand which phase couplings should be included in coupled phase oscillator models. In this Letter, we establish selection rules for higher-order phase coupling functions. These selection rules, which can be applied without the need of explicit phase reduction, are solely based on the symmetry of the isolated oscillator velocity field and the $n$-body interaction functions. As phase reduction established the mechanistic basis for the Kuramoto model, our results provide a theoretical link between physical systems and higher-order phase models.

nlin.AO↗
arXiv2026

On the efficiency of pairwise Hamiltonian control to desynchronize the higher-order Kuramoto model

Synchronization of coupled oscillators is observed in many natural and engineered systems and emerges due to the interactions within the system. It can be both beneficial, e.g., in power grids, and harmful, e.g., in epileptic seizures. In the latter case, efficient control methods to desynchronize the systems are crucial. Recent studies have shown that interactions are not always pairwise, but higher-order, i.e., many-body, and this greatly affects the dynamics. For instance, higher-order interactions increase the linear stability of synchronized states but simultaneously shrink their attraction basin, with potentially opposite effects on control methods. Here, we use a minimally invasive pairwise control based on Hamiltonian control theory, and investigate its efficiency on phase oscillators with higher-order interactions. We show that, if the initial phases are close to the synchronized state, higher-order interactions make desynchronization more difficult to achieve. Otherwise, a non-monotonic effect appears: intermediate strengths of higher-order interactions impede desynchronization while larger ones facilitate it. In all cases, the control can desynchronize the system with a sufficient number of controlled nodes and intensity.

nlin.AO↗
arXiv2026

Minimal branching and fusion morphogenesis approaches biological multi-objective optimality

Many biological networks grow by elongation of filaments that can branch and fuse -- typical examples include fungal mycelium or slime mold. These networks must simultaneously perform multiple tasks such as transport, exploration, and robustness under finite resources. Yet, how such multi-task architectures emerge from local growth processes remains poorly understood. Here, we introduce a minimal model of spatial network morphogenesis based solely on stochastic branching, fusion, and stopping, during elongation. Despite the absence of global optimization or feedback, the model generates a broad morphospace from tree-like, to loopy, as well as hybrid architectures. By quantifying multiple functional objectives, we show that (i) these synthetic structures occupy similar regions of performance space than evolved empirical fungal networks, and (ii) that their Pareto front of optimal trade-offs lies close to that of these same fungal networks. Our results show that biological architectures approaching multi-objective optimality can arise from simple local growth rules, and identify branching and fusion as fundamental ingredients shaping the architecture of living transport networks.

nlin.AO↗
arXiv2025

Tetrahedral Core in a Sea of Competing Magnetic Phases in Graphene

We demonstrate the emergence of a robust tetrahedral magnetic ground state in monolayer graphene doped to the van Hove singularity (vHS). This noncoplanar, gapped spin configuration - featuring four equally inclined moments - has been previously identified as a candidate instability. Here, not only do we confirm its stability across all finite interactions using fully self-consistent, real-space-resolved calculations, but we also go beyond earlier work by charting the full surrounding phase diagram. In doing so, we unravel a cascade of symmetry-broken magnetic states - pseudo-tetrahedral, planar, collinear, and modulated textures - which we classify using spin structure factors and vector order parameters. These results stem from unrestricted Hartree-Fock simulations on large supercells with dense k-point sampling, enabling us to resolve interaction-driven magnetic and charge inhomogeneities. Our findings connect directly with recent ARPES and doping experiments near the vHS in graphene, and establish the tetrahedral state as the central correlated instability in this regime, offering predictive insight into emergent magnetism in correlated Dirac materials.

cond-mat.str-el↗
arXiv2025

Mixed higher-order coupling stabilizes new states

Understanding how higher-order interactions affect collective behavior is a central problem in nonlinear dynamics and complex systems. Most works have focused on a single higher-order coupling function, neglecting other viable choices. Here we study coupled oscillators with dyadic and three different types of higher-order couplings. By analyzing the stability of different twisted states on rings, we show that many states are stable only for certain combinations of higher-order couplings, and thus the full range of system dynamics cannot be observed unless all types of higher-order couplings are simultaneously considered.

nlin.AO↗
arXiv2025

Collective dynamics on higher-order networks

Higher-order interactions that nonlinearly couple more than two nodes are important in many networked systems, and their effects on collective dynamics are increasingly being studied. Here we provide an overview of this rapidly growing field, and of the techniques that can be used to describe and analyze them. We focus in particular on new phenomena and challenges that emerge when nonpairwise interactions are considered. We conclude by discussing open questions and promising future directions on the collective dynamics of higher-order networks.

nlin.AO↗
arXiv2024

Topological analysis of brain dynamical signals indicates signatures of seizure susceptibility

Epilepsy is known to drastically alter brain dynamics during seizures (ictal periods), but its effects on background (non-ictal) brain dynamics remain poorly understood. To investigate this, we analyzed an in-house dataset of brain activity recordings from epileptic zebrafish, focusing on two controlled genetic conditions across two fishlines. After using machine learning to segment and label recordings, we applied time-delay embedding and Persistent Homology -- a noise-robust method from Topological Data Analysis (TDA) -- to uncover topological patterns in brain activity. We find that ictal and non-ictal periods can be distinguished based on the topology of their dynamics, independent of genetic condition or fishline, which validates our approach. Remarkably, within a single wild-type fishline, we identified topological differences in non-ictal periods between seizure-prone and seizure-free individuals. These findings suggest that epilepsy leaves detectable topological signatures in brain dynamics even outside of ictal periods. Overall, this study demonstrates the utility of TDA as a quantitative framework to screen for topological markers of epileptic susceptibility, with potential applications across species.

physics.soc-ph↗
arXiv2024

Hamiltonian control to desynchronize Kuramoto oscillators with higher-order interactions

Synchronization is a ubiquitous phenomenon in nature. Although it is necessary for the functioning of many systems, too much synchronization can also be detrimental, e.g., (partially) synchronized brain patterns support high-level cognitive processes and bodily control, but hypersynchronization can lead to epileptic seizures and tremors, as in neurodegenerative conditions such as Parkinson's disease. Consequently, a critical research question is how to develop effective pinning control methods capable to reduce or modulate synchronization as needed. Although such methods exist to control pairwise-coupled oscillators, there are none for higher-order interactions, despite the increasing evidence of their relevant role in brain dynamics. In this work, we fill this gap by proposing a generalized control method designed to desynchronize Kuramoto oscillators connected through higher-order interactions. Our method embeds a higher-order Kuramoto model into a suitable Hamiltonian flow, and builds up on previous work in Hamiltonian control theory to analytically construct a feedback control mechanism. We numerically show that the proposed method effectively prevents synchronization in synthetics and empirical higher-order networks. Although our findings indicate that pairwise contributions in the feedback loop are often sufficient, the higher-order generalization becomes crucial when pairwise coupling is weak. Finally, we explore the minimum number of controlled nodes required to fully desynchronize oscillators coupled via an all-to-all hypergraphs.

math.DS↗
arXiv2024

Reducibility of higher-order networks from dynamics

Empirical complex systems can be characterized not only by pairwise interactions, but also by higher-order (group) interactions influencing collective phenomena, from metabolic reactions to epidemics. Nevertheless, higher-order networks' apparent superior descriptive power -- compared to classical pairwise networks -- comes with a much increased model complexity and computational cost, challenging their application. Consequently, it is of paramount importance to establish a quantitative method to determine when such a modeling framework is advantageous with respect to pairwise models, and to which extent it provides a valuable description of empirical systems. Here, we propose an information-theoretic framework, accounting for how structure affect diffusion behaviors, quantifying the entropic cost and distinguishability of higher-order interactions to assess their reducibility to lower-order structures while preserving relevant functional information. Empirical analyses indicate that some systems retain essential higher-order structure, whereas in some technological and biological networks it collapses to pairwise interactions. With controlled randomization procedures, we investigate the role of nestedness and degree heterogeneity in this reducibility process. Our findings contribute to ongoing efforts to minimize the dimensionality of models for complex systems.

physics.soc-ph↗
arXiv2024

Synergistic signatures of group mechanisms in higher-order systems

The interplay between causal mechanisms and emerging collective behaviors is a central aspect of understanding, controlling, and predicting complex networked systems. In our work, we investigate the relationship between higher-order mechanisms and higher-order behavioral observables in two representative models with group interactions: a simplicial Ising model and a social contagion model. In both systems, we find that group (higher-order) interactions show emergent synergistic (higher-order) behavior. The emergent synergy appears only at the group level and depends in a complex, non-linear way on the trade-off between the strengths of the low- and higher-order mechanisms and is invisible to low-order behavioral observables. Our work sets the basis for systematically investigating the relation between causal mechanisms and behavioral patterns in complex networked systems with group interactions, offering a robust methodological framework to tackle this challenging task.

physics.soc-ph↗
arXiv2024

Higher-order Laplacian Renormalization

We propose a cross-order Laplacian renormalization group (X-LRG) scheme for arbitrary higher-order networks. The renormalization group is a pillar of the theory of scaling, scale-invariance, and universality in physics. An RG scheme based on diffusion dynamics was recently introduced for complex networks with dyadic interactions. Despite mounting evidence of the importance of polyadic interactions, we still lack a general RG scheme for higher-order networks. Our approach uses a diffusion process to group nodes or simplices, where information can flow between nodes and between simplices (higher-order interactions). This approach allows us (i) to probe higher-order structures, defining scale-invariance at various orders, and (ii) to propose a coarse-graining scheme. We demonstrate our approach on controlled synthetic higher-order systems and then use it to detect the presence of order-specific scale-invariant profiles of real-world complex systems from multiple domains.

cond-mat.stat-mech↗
arXiv2023

Deeper but smaller: Higher-order interactions increase linear stability but shrink basins

A key challenge of nonlinear dynamics and network science is to understand how higher-order interactions influence collective dynamics. Although many studies have approached this question through linear stability analysis, less is known about how higher-order interactions shape the global organization of different states. Here, we shed light on this issue by analyzing the rich patterns supported by identical Kuramoto oscillators on hypergraphs. We show that higher-order interactions can have opposite effects on linear stability and basin stability: they stabilize twisted states (including full synchrony) by improving their linear stability, but also make them hard to find by dramatically reducing their basin size. Our results highlight the importance of understanding higher-order interactions from both local and global perspectives.

nlin.AO↗
arXiv2023

Evidence of social learning across symbolic cultural barriers in sperm whales

We provide quantitative evidence suggesting social learning in sperm whales across socio-cultural boundaries, using acoustic data from the Pacific and Atlantic Oceans. Traditionally, sperm whale populations are categorized into clans based on their vocal repertoire: the rhythmically patterned click sequences (codas) that they use. Among these codas, identity codas function as symbolic markers for each clan, accounting for 35-60% of codas they produce. We introduce a computational method to model whale speech, which encodes rhythmic micro-variations within codas, capturing their vocal style. We find that vocal style-clans closely align with repertoire-clans. However, contrary to vocal repertoire, we show that sympatry increases vocal style similarity between clans for non-identity codas, i.e. most codas, suggesting social learning across cultural boundaries. More broadly, this subcoda structure model offers a framework for comparing communication systems in other species, with potential implications for deeper understanding of vocal and cultural transmission within animal societies.

cs.SI↗
arXiv2023

A unified framework for Simplicial Kuramoto models

Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models, categorized into three main groups: "simple" models, "Hodge-coupled" models, and "order-coupled" (Dirac) models. Our framework is based on topology, discrete differential geometry as well as gradient flows and frustrations, and permits a systematic analysis of their properties. We establish an equivalence between the simple simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of manifoldness of the simplicial complex. Then, starting from simple models, we describe the notion of simplicial synchronization and derive bounds on the coupling strength necessary or sufficient for achieving it. For some variants, we generalize these results and provide new ones, such as the controllability of equilibrium solutions. Finally, we explore a potential application in the reconstruction of brain functional connectivity from structural connectomes and find that simple edge-based Kuramoto models perform competitively or even outperform complex extensions of node-based models.

nlin.AO↗
arXiv2022

Generalized Deep Thermalization for Free Fermions

In non-interacting isolated quantum systems out of equilibrium, local subsystems typically relax to non-thermal stationary states. In the standard framework, information on the rest of the system is discarded, and such states are described by a Generalized Gibbs Ensemble (GGE), maximizing the entropy while respecting the constraints imposed by the local conservation laws. Here we show that the latter also completely characterize a recently introduced projected ensemble (PE), constructed by performing projective measurements on the rest of the system and recording the outcomes. By focusing on the time evolution of fermionic Gaussian states in a tight-binding chain, we put forward a random ensemble constructed out of the local conservation laws, which we call deep GGE (dGGE). For infinite-temperature initial states, we show that the dGGE coincides with a universal Haar random ensemble on the manifold of Gaussian states. For both infinite and finite temperatures, we use a Monte Carlo approach to test numerically the predictions of the dGGE against the PE. We study in particular the $k$-moments of the state covariance matrix and the entanglement entropy, finding excellent agreement. Our work provides a first step towards a systematic characterization of projected ensembles beyond the case of chaotic systems and infinite temperatures.

quant-ph↗
arXiv2022

Simplicially driven simple contagion

Single contagion processes are known to display a continuous transition from an epidemic-free phase at low contagion rates to the epidemic state for rates above a critical threshold. This transition can become discontinuous when two simple contagion processes are coupled in a bi-directional symmetric way. However, in many cases, the coupling is not symmetric and the processes can be of a different nature. For example, risky social behaviors -- such as not wearing masks or engaging in large gatherings -- can affect the spread of a disease, and their adoption dynamics via social reinforcement mechanisms are better described by complex contagion models, rather than by the simple contagion paradigm, which is more appropriate for disease spreading phenomena. Motivated by this example, we consider a simplicial contagion (describing the adoption of a behavior) that uni-directionally drives a simple contagion (describing a disease propagation). We show that, above a critical driving strength, such driven simple contagion can exhibit both discontinuous transitions and bi-stability, which are instead absent in standard simple contagions. We provide a mean-field analytical description of the phase diagram of the system, and complement the results with Markov-chain simulations. Our results provide a novel route for a simple contagion process to display the phenomenology of a higher-order contagion, through a driving mechanism that may be hidden or unobservable in many practical instances.

physics.soc-ph↗