arXiv subjects

Vincent Thibeault

Publications and source records attributed to Vincent Thibeault.

10 recordsLinked to original sources

arXiv2026

Operator-theoretic approach to the partial integration of randomly coupled phase oscillators

In our previous work [arXiv:2504.06248], we adopted Koopman theory to link the existence of different constants of motion to the presence of specific network motifs of Kuramoto oscillators. Yet, it remains to be shown how the partial integration can be carried out using the Koopman generator and its eigenfunctions. In this paper, we construct a random graph from network motifs that admit Koopman eigenfunctions and conserved quantities, and use it to define a partially integrable Kuramoto model. We perform the partial integration of the introduced model when there are monomial eigenfunctions and conserved cross-ratios, while providing an operator-theoretic derivation of the Watanabe-Strogatz transformation based on Magnus expansion and a recent result on closed forms of the Baker-Campbell-Hausdorff formula [arXiv:1502.06589].

math.DS↗

arXiv2025

Kuramoto meets Koopman: Constants of motion, symmetries, and network motifs

Conserved quantities in phase-oscillator dynamics are well established for identically coupled oscillators, or groups thereof, but the explicit connectivity conditions under which more complex networks admit constants of motion remain difficult to identify. Using Koopman theory, we derive general conditions for the existence of distinct conserved quantities in the Kuramoto model with heterogeneous phase lags on any weighted, directed, and signed graph. To this end, we find Koopman eigenfunctions and continuous Lie symmetries that generate different families of constants of motion. The derived conditions reveal a broad class of network motifs that support conserved quantities and we detect these motifs in hundreds of complex empirical networks. The results thus point to connectivity patterns that can preserve phase relationships over time and motivate further investigations of Koopman spectral properties for dynamics on complex networks.

nlin.AO↗

arXiv2024

On the reconstruction limits of complex networks

Network reconstruction consists in retrieving the hidden interaction structure of a system from observations. Many reconstruction algorithms have been proposed, although less research has been devoted to describe their theoretical limitations. In this work, we take a first-principles approach and build on our earlier definition of reconstructability-the fraction of structural information recoverable from data. We relate this quantity to the true data-generating (TDG) process and delineate an information-theoretic reconstruction limit, i.e., the upper bound of the mutual information between the true underlying graph and any graph reconstructed from observations. These concepts lead us to a principled numerical method to assess the validity of empirically reconstructed networks, based on model selection and a quantity we introduce: the reconstruction index. This index approximates the reconstructability from data, quantifies the variability of the reconstructed network ensemble, and is shown to predict reconstruction error without requiring knowledge of the true underlying network. We characterize this method and test it on empirical time series and networks.

stat.AP↗

arXiv2024

The low-rank hypothesis of complex systems

Complex systems are high-dimensional nonlinear dynamical systems with intricate interactions among their constituents. To make interpretable predictions about their large-scale behavior, it is typically assumed, without a clear statement, that these dynamics can be reduced to a few number of equations involving a low-rank matrix describing the network of interactions -- what we call the low-rank hypothesis. Our paper sheds light on this assumption and questions its validity. By leveraging fundamental theorems on singular value decomposition, we expose the hypothesis for various random graphs, either by making explicit their low-rank formulation or by demonstrating the exponential decrease of their singular values. Notably, we verify the hypothesis experimentally for real networks by revealing the rapid decrease of their singular values, which has major consequences on their effective ranks. We then evaluate the impact of the low-rank hypothesis for general dynamical systems on networks through an optimal dimension reduction. This allows us to prove that recurrent neural networks can be exactly reduced, and to connect the rapidly decreasing singular values of real networks to the dimension reduction error of the nonlinear dynamics they support, be it microbial, neuronal or epidemiological. Finally, we prove that higher-order interactions naturally emerge from the dimension reduction, thus providing theoretical insights into the origin of higher-order interactions in complex systems.

nlin.AO↗

arXiv2022

Dimension reduction of dynamics on modular and heterogeneous directed networks

Dimension reduction is a common strategy to study non-linear dynamical systems composed by a large number of variables. The goal is to find a smaller version of the system whose time evolution is easier to predict while preserving some of the key dynamical features of the original system. Finding such a reduced representation for complex systems is, however, a difficult task. We address this problem for dynamics on weighted directed networks, with special emphasis on modular and heterogeneous networks. We propose a two-step dimension-reduction method that takes into account the properties of the adjacency matrix. First, units are partitioned into groups of similar connectivity profiles. Each group is associated to an observable that is a weighted average of the nodes' activities within the group. Second, we derive a set of conditions that must be fulfilled for these observables to properly represent the original system's behavior, together with a method for approximately solving them. The result is a reduced adjacency matrix and an approximate system of ODEs for the observables' evolution. We show that the reduced system can be used to predict some characteristic features of the complete dynamics for different types of connectivity structures, both synthetic and derived from real data, including neuronal, ecological, and social networks. Our formalism opens a way to a systematic comparison of the effect of various structural properties on the overall network dynamics. It can thus help to identify the main structural driving forces guiding the evolution of dynamical processes on networks.

math.DS↗

arXiv2022

Duality between predictability and reconstructability in complex systems

Predicting the evolution of a large system of units using its structure of interaction is a fundamental problem in complex system theory. And so is the problem of reconstructing the structure of interaction from temporal observations. Here, we find an intricate relationship between predictability and reconstructability using an information-theoretical point of view. We use the mutual information between a random graph and a stochastic process evolving on this random graph to quantify their codependence. Then, we show how the uncertainty coefficients, which are intimately related to that mutual information, quantify our ability to reconstruct a graph from an observed time series, and our ability to predict the evolution of a process from the structure of its interactions. Interestingly, we find that predictability and reconstructability, even though closely connected by the mutual information, can behave differently, even in a dual manner. We prove how such duality universally emerges when changing the number of steps in the process, and provide numerical evidence of other dualities occurring near the criticality of multiple different processes evolving on different types of structures.

cond-mat.stat-mech↗

arXiv2021

Social confinement and mesoscopic localization of epidemics on networks

Recommendations around epidemics tend to focus on individual behaviors, with much less efforts attempting to guide event cancellations and other collective behaviors since most models lack the higher-order structure necessary to describe large gatherings. Through a higher-order description of contagions on networks, we model the impact of a blanket cancellation of events larger than a critical size and find that epidemics can suddenly collapse when interventions operate over groups of individuals rather than at the level of individuals. We relate this phenomenon to the onset of mesoscopic localization, where contagions concentrate around dominant groups.

physics.soc-ph↗

arXiv2020

Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks

Simple models of infectious diseases tend to assume random mixing of individuals, but real interactions are not random pairwise encounters: they occur within various types of gatherings such as workplaces, households, schools, and concerts, best described by a higher-order network structure. We model contagions on higher-order networks using group-based approximate master equations, in which we track all states and interactions within a group of nodes and assume a mean-field coupling between them. Using the Susceptible-Infected-Susceptible dynamics, our approach reveals the existence of a mesoscopic localization regime, where a disease can concentrate and self-sustain only around large groups in the network overall organization. In this regime, the phase transition is smeared, characterized by an inhomogeneous activation of the groups. At the mesoscopic level, we observe that the distribution of infected nodes within groups of a same size can be very dispersed, even bimodal. When considering heterogeneous networks, both at the level of nodes and groups, we characterize analytically the region associated with mesoscopic localization in the structural parameter space. We put in perspective this phenomenon with eigenvector localization and discuss how a focus on higher-order structures is needed to discern the more subtle localization at the mesoscopic level. Finally, we discuss how mesoscopic localization affects the response to structural interventions and how this framework could provide important insights for a broad range of dynamics.

physics.soc-ph↗

arXiv2020

Threefold way to the dimension reduction of dynamics on networks: an application to synchronization

Several complex systems can be modeled as large networks in which the state of the nodes continuously evolves through interactions among neighboring nodes, forming a high-dimensional nonlinear dynamical system. One of the main challenges of Network Science consists in predicting the impact of network topology and dynamics on the evolution of the states and, especially, on the emergence of collective phenomena, such as synchronization. We address this problem by proposing a Dynamics Approximate Reduction Technique (DART) that maps high-dimensional (complete) dynamics unto low-dimensional (reduced) dynamics while preserving the most salient features, both topological and dynamical, of the original system. DART generalizes recent approaches for dimension reduction by allowing the treatment of complex-valued dynamical variables, heterogeneities in the intrinsic properties of the nodes as well as modular networks with strongly interacting communities. Most importantly, we identify three major reduction procedures whose relative accuracy depends on whether the evolution of the states is mainly determined by the intrinsic dynamics, the degree sequence, or the adjacency matrix. We use phase synchronization of oscillator networks as a benchmark for our threefold method. We successfully predict the synchronization curves for three phase dynamics (Winfree, Kuramoto, theta) on the stochastic block model. Moreover, we obtain the bifurcations of the Kuramoto-Sakaguchi model on the mean stochastic block model with asymmetric blocks and we show numerically the existence of periphery chimera state on the two-star graph. This allows us to highlight the critical role played by the asymmetry of community sizes on the existence of chimera states. Finally, we systematically recover well-known analytical results on explosive synchronization by using DART for the Kuramoto-Sakaguchi model on the star graph.

nlin.AO↗

arXiv2020

Detecting structural perturbations from time series with deep learning

Small disturbances can trigger functional breakdowns in complex systems. A challenging task is to infer the structural cause of a disturbance in a networked system, soon enough to prevent a catastrophe. We present a graph neural network approach, borrowed from the deep learning paradigm, to infer structural perturbations from functional time series. We show our data-driven approach outperforms typical reconstruction methods while meeting the accuracy of Bayesian inference. We validate the versatility and performance of our approach with epidemic spreading, population dynamics, and neural dynamics, on various network structures: random networks, scale-free networks, 25 real food-web systems, and the C. Elegans connectome. Moreover, we report that our approach is robust to data corruption. This work uncovers a practical avenue to study the resilience of real-world complex systems.

physics.soc-ph↗