It is proved that networks of n nodes with local dimension d can be exactly reduced from n d dimensions to a fixed macroscopic system of dimension m d, where m is the number of fundamental solutions required by the nodal dynamics.
Abstract
We establish a unified framework for exact dimensional reductions in network dynamical systems using Lie-Scheffers theory. For network dynamical systems with \emph{mean-field Lie-Scheffers structure}, we prove that networks of $n$ nodes with local dimension $d$ can be exactly reduced from $ n d $ dimensions to a fixed macroscopic system of dimension $ m d $, where $m$ is the number of fundamental solutions required by the nodal dynamics. Crucially, the superposition principle resulting from the Lie-algebraic structure allows the mean-field coupling to be expressed explicitly in terms of the macroscopic variables, yielding a \emph{closed} self-consistent system independent of network size. This reduction collapses the high-dimensional network flow onto invariant manifolds parameterized by $ \gamma = d(n-m) $ independent constants of motion. Our framework rigorously explains known reductions and provides a \emph{systematic method to discover new ones}. We illustrate the theory with ensembles of Riccati equations (encompassing the Kuramoto model and Theta neuron model), quasi-linear ODEs, and generalized Bernoulli equations, explicitly deriving the macroscopic flows and conserved quantities for each case.
A one-dimensional reduction for dynamical systems on networks with purely higher-order interactions is developed, formulated through an effective higher-order interaction strength, associated with the triangular interactions of the underlying network and the dynamical system's effective state.
We derive a one-dimensional reduction for nonlinear dynamics on simplicial complexes containing both pairwise and triangular (higher-order) interactions. The effective state is defined using a mixed weight determined by the pairwise and triangular degrees of each node. The resulting reduced equation retains two structural coefficients, associated separately with the pairwise and higher-order coupling channels. A fluctuation expansion identifies the closure assumptions underlying the reduction and shows how deviations of individual node states from the effective state contribute to the approximation error. We numerically validate the proposed framework on Gene-regulatory dynamics, the double-well system, and SIS spreading. The states of the reduced model are compared with full-network simulations through coupling-parameter sweeps, steady-state branch calculations, and progressive node-removal experiments on synthetic and real-world networks. The reduced model successfully reproduces the principal transitions and steady-state branches in all three dynamical systems considered. Agreement is strongest for relatively homogeneous networks and deteriorates when structural heterogeneity produces a broader distribution of node states. The closure diagnostics account for this loss of accuracy and indicate when a single effective state is no longer sufficient. The reduction therefore provides a tractable description of resilience in systems with coexisting pairwise and higher-order interactions.
Amit Tiwari, C. Hens, Prosenjit Kundu· 0 citations
Within the scope of the qualitative study of dynamical systems via graph-based methods, we extend the notion of structural graph to dynamical systems generated by actions of monoids, as abstract algebraic structures. The setting considers the action of an ordered monoid (not necessarily \(\mathbb{N},\ \mathbb{Z}\) or \(\mathbb{R}\)) on a metric space in a general sense, and therefore includes a wider class of dynamical systems than those described solely by ordinary differential or difference equations. We then investigate several fundamental properties of the structural graph, such as connectedness and degeneracy. In particular, it is shown that in the finite case, the structural graph associated to a \(C^{1}\)-vector field on a closed differential manifold is necessarily connected. As illustrative examples, we consider the classical van der Pol oscillator on the Poincaré sphere, the mathematical pendulum on the phase plane and the nonanalytic case of the center-focus. And as a perspective, we suggest work about what could be considered as a new form of linearization of dynamical systems via their structural graph, and address the question about a possible use for investigating global qualitative properties such as structural stability.
Bouchra Tarhzout, A. Ouadfel, F. Zinoun· International Journal of Ana...· 0 citations
We focus on a stochastic system that models a collection of $N$ identical oscillators with unidirectional coupling, perturbed by white noise. The unidirectional coupling breaks the symmetry of mutual dependence among the oscillators. It is shown that the associated random dynamical system $\phi$ admits a dynamical order, meaning that $\phi$ admits a simple asymptotic one-dimensional structure that is totally ordered with respect to the standard order in $\mathbb{R}^N$. Our approach takes a novel geometric perspective: we introduce a random dynamical system $\Phi$ on a smooth Riemannian manifold $M$, diffeomorphic to $\mathbb{S}^1 \times \mathbb{R}^{N-1}$, by ``wrapping''the random system $\phi$ from $\mathbb{R}^N$ onto $M$. By choosing an appropriate random cone field $\mathcal{C}_M$ on $M$, we show that $\Phi$ is a differentially positive random system on $M$, which is a random counterpart of the differentially positive systems introduced by Forni and Sepulchre. We further demonstrate that the traditional well-known horizontal curves can be identified as the conal curves on $M$, thereby providing a crucial tool for establishing the dynamical order of the random system $\phi$.
Classical fractons are Hamiltonian systems that can develop attractors after projection onto configuration or shape variables, although the full phase space admits none. We study a scale-invariant, dipole-conserving two-parameter family of fracton Hamiltonians $H_{\alpha,\beta}$. By separating coordinates into scale and shape, we obtain autonomous shape dynamics that admit fixed points which leave a purely scale evolution of the form $R(t)\propto |t|^{\alpha/(\alpha-\beta)}$. The shape fixed points, which determine the distribution of the expanding particles, are central configurations of power-law Riesz potentials. The distinguished model $(\alpha,\beta)=(-2,1)$ is unique: its scale evolution takes the Einstein-de Sitter form $R(t)\propto |t|^{2/3}$, its fixed-point equation is the equal-mass Newtonian central-configuration, its large-$N$ distribution is a homogeneous ball, and its homothetic trajectories admit a zero-energy Newtonian gravitational dual. The fixed points are locally stable, and simulations at moderate $N$ approach them from random initial data. Large $N$ simulations reveal a richer class of fixed-points: bound clusters of approximately fixed physical size retain internal motion, while their centers approach unequal-mass Newtonian central configurations and preserve large-scale homogeneity. A scale-separation conjecture yields an effective unequal-mass fracton dynamics for the centers and a corresponding zero-energy Newtonian gravitational dual. Trajectories generically exhibit a bidirectional arrow of time: scale and shape complexity grow away from a Janus point, while Boltzmann entropy grows logarithmically. Together, these features reproduce the salient structure of a flat matter-dominated cosmology. In the distinguished fracton model, all these cosmological analogues emerge as attractor properties, making it a toy model for cosmological dynamics without fine-tuning.
A. Singh, Dileep P. Jatkar, S. L. Sondhi et al.· 0 citations
We establish marginal and functional convex order comparisons for graphon mean-field systems on $\mathbb{R}^d$, including the infinite-horizon setting. A key difficulty is that convex order cannot in general be transferred through the usual particle approximation, which requires us to work directly with Euler schemes for the graphon mean-field system. Under a suitable dissipativity condition, we establish Euler approximation estimates that are uniform in both time and the agent label, and combine them with forward-backward induction arguments to obtain the convex order comparisons. For the infinite-horizon problem, dissipativity provides the required uniform-in-time stability, while an exponentially weighted $L^2$-space enables trajectory-level convergence and leads to the functional convex order on $[0,\infty)$. As an application, we derive value-function comparisons for a class of one-dimensional linear-quadratic graphon mean-field games.
