Special Session 136: Collective Dynamics in Large Networks: From Kuramoto to Spin Models

XY-model meets graphons

Artem Alexandrov Steklov Institute of Mathematics Russia Co-Author(s):

Abstract: Initially developed as limiting objects that describe large networks, graphons laid the foundation to consider generalizations of mean-field spin models, like Ising model or XY-model. Statistical physics tells us that the thermodynamic limit is crucial to study phase transitions and other critical phenomena. For dense (and not so dense) graphs, this limit can be described with help of graphons or $L^p$-graphons. I will show how to use graphons to analyze phase transitions in XY model and its generalization with quenched disorder. Graphons provides a simple tool for derive and solve self-consistency equations by analyzing the spectrum of related functional operators. Moreover, for certain graphons, we can observe metastability or so-called transient states. To illustrate these statements, we discuss the Kuramoto model in the context of polariton arrays.

Mean-field limits for interacting particle systems on adaptive dynamical networks

Nathalie Ayi Sorbonne Universite France Co-Author(s):

Abstract: A central distinction in interacting particle systems is between indistinguishable and non-exchangeable cases. In the latter, particle identity plays a key role, and interactions are naturally modeled by systems of ODEs on weighted graphs. In this talk, we study such systems and their large-population behavior, highlighting the connection between graph limit and mean-field approaches. Our main focus is on systems evolving on adaptive dynamical networks, where agents interact through weighted graphs whose structure evolves over time in a coupled manner with the agents' states. In the dense-graph regime, we show that the large-population limit is described by a Vlasov-type equation posed on an extended phase space including agent states, identities, and evolving interaction weights. We present two complementary derivations of this limit: a mean-field approach in the spirit of Sznitman's work, under a structural assumption on the weight dynamics, and a deterministic graph limit approach that allows us to remove this restriction.

Kuramoto's Energy Landscape in random geometric graphs.

Pablo Groisman Universidad de Buenos Aires Argentina Co-Author(s):

Abstract: We'll consider the energy function of the Kuramoto model in random geometric graphs on a given manifold, with particular focus on the d-dimensional torus and the 2-sphere. As the number of nodes diverges, with high probability there are no (smooth) local minima other than the phase-locked state in the case of the sphere, while in the torus there is at least one local minimum for each homotopy class of continuous functions from the manifold to the circle. This is proved for d = 1 and conjectured for every d > 1. We'll then specialize to the cycle graph, where we can go beyond existence and describe the volume and geometry of the basins of attraction of each twisted state, shedding light on the long-time behavior of the system.

Critical threshold for synchronizability of high-dimensional Kuramoto oscillators under higher-order interactions

Dohyun Kim Sungkyunkwan University Korea Co-Author(s):    Hyungjin Huh

Abstract: Collective synchronization in the Kuramoto model has been widely studied, but the original model is defined on the unit circle and only includes pairwise interactions. In this presentation, based on [Huh and Kim, Chaos (2024), 123119], we study a high-dimensional Kuramoto model with both two-body and three-body interactions. Let $\kappa_1$ and $\kappa_2$ denote the corresponding interaction strengths. We show that complete synchronization can emerge if $\kappa_1+\kappa_2>0$, whereas it cannot occur if $\kappa_1+\kappa_2

Adaptive and Higher-Order Dynamics of Kuramoto-type Models

Christian Kuehn TUM Germany Co-Author(s):

Abstract: In my talk I shall provide a short survey of recent results I have obtained with a number of co-authors on considering Kuramoto-type models, where adaptive (or co-evolutionary) and higher-order (or polyadic) aspects play a key role. In particular, I shall discuss structural issues (e.g., how do simplicial and non-simplicical aspects form adaptively or arise via modelling), mean-field limits (on graph limits), as well as solutions to the resulting bifurcation problems. This highlights the importance of combining different mathematical approaches to understand Kuramoto-type models from first principles up to eventually understanding their complex dynamics.

Interacting dynamical systems on self-similar networks

Georgi Medvedev Drexel University USA Co-Author(s):

Abstract: We develop a continuum limit and mean-field theory for interacting particle systems (IPS) on self-similar networks, a new class of discrete models whose large-scale behavior gives rise to nonlocal evolution equations on fractal domains. This work extends the graphon-based framework for IPS, used to derive continuum and mean-field limits in the non-exchangeable setting, to situations where the spatial domain is fractal rather than Euclidean. The motivation arises from both physical models naturally formulated on fractals and real-world networks exhibiting hierarchical or quasi-self-similar structure. The main result of this work is an explicit isomorphism between self-similar IPS and graphon IPS, which allows us to justify the continuum and mean-field limits in the self-similar setting.

Twisted states of Kuramoto oscillators on self-similar sets

Matthew Mizuhara The College of New Jersey USA Co-Author(s):

Abstract: Coupled phase oscillators provide a prototypical setting for studying spontaneous synchronization and other pattern formation. The interplay between network topology and emergent phenomena remains a central theme in their study. Some real-world networks, such as coupled neurons and the Internet, exhibit self-similarity at multiple scales, so there is a need to analyze coupled oscillators on hierarchical/fractal structures. To that end, we study Kuramoto oscillators on self-similar networks approximating fractals. We show that the complex topology of fractals gives rise to a rich diversity of equilibria, generalizing well-studied twisted states found on simpler networks. We introduce an approach for constructing and classifying these states by combining tools from fractal geometry, topology, and analysis. Our method relies on harmonic extensions, which provide exact solutions to the Laplace equation on the fractal, and uses T-convergence to establish convergence of the nonlinear problems. This work is in collaboration with Georgi Medvedev and is supported by the U.S. National Science Foundation.

Mean-field approach to finite-size fluctuations in coupled oscillator systems

Oleh Omelchenko University of Potsdam Germany Co-Author(s):    Oleh Omel`chenko and Georg Gottwald

Abstract: Networks of coupled phase oscillators are one of the most studied dynamical systems with numerous applications in physics, chemistry, biology, and engineering. A variety of methods exists to explain their properties and dynamics in the thermodynamic limit, when the network size tends to infinity. However, the behavior of such systems in the more realistic case of a finite number of oscillators still remains poorly understood. In this talk, we revisit the paradigmatic Kuramoto-Sakaguchi model describing synchronization transitions in networks of all-to-all coupled heterogeneous phase oscillators, and propose an ab initio approach for characterizing analytically the statistical properties of finite-size fluctuations in this system. Our framework is applicable to any stationary partially synchronized state and does not require any prior knowledge about its structure. Moreover, it is sufficiently general such that it can be applied to a broader class of interacting particle systems.

Formation of clusters and coarsening in weakly interacting diffusions

Grigorios A Pavliotis Imperial College London England Co-Author(s):    Nicolai Gerber, Rishabh Gvalani, Martin Hairer, Andre Schlichting

Abstract: We study the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.

Stability analysis of Kuramoto model with time delay via a generalized spectral theory

Haozhe Shu Tohoku university Japan Co-Author(s):    Haozhe Shu

Abstract: Known as a relatively simple mean-field model of coupled phase oscillators, the Kuramoto model plays an important role in studying the collective synchronization phenomenon. In this talk, the linear stability of incoherent state to the continuum limit of the Kuramoto model with time delay is investigated. Here, it is known that due to the presence of continuous spectrum on the imaginary axis, stability analysis meets obstacles even if no eigenvalue exists on the right complex half-plane. To address it, a generalized spectral theory on a Gelfand triple is utilized. Under some analyticity condition, resolvent operator is extended to a generalized resolvent. It is shown that continuous singularities (continuous spectrum) disappear from the Riemann surface of the generalized resolvent due to the change of topology via the Gelfand triple. Then, the contour deformation is applied in the inverse Laplace representation to show stability in a weak topology.

Numerical Methods for Stochastic McKean-Vlasov Equations

Gideon Simpson Drexel University USA Co-Author(s):

Abstract: We study numerical schemes for integrating the stochastic McKean-Vlasov equations, with goals of both identifying metastable states and sampling from equilibrium. In particular, the metastable solutions may correspond to invariant distributions of the underlying interacting particle system. This is the case when the metastable solutions are finite temperature analogs of time independent solutions to the zero temperature McKean-Vlasov. This is joint work in collaboration with M. Ottobre.