Across all the realizations, SSM/gSSM consistently outperform classical spectral and mean-field methods in modeling critical transitions at both microscopic and macroscopic scales, establishing SSM-based reduction as a robust, interpretable tool for nonlinear networked systems with broad applicability to epidemiology, ecology, and engineered networks.
Abstract
Complex networked systems are prevalent in biology, engineering, and the social sciences, yet their high-dimensional, nonlinear dynamics pose major challenges for analysis and prediction. A mathematically rigorous route to simplification is to represent system behavior on a low-dimensional, smooth invariant manifold known as a spectral submanifold (SSM). Here we present a comprehensive SSM reduction framework and its globalized extension (gSSM) for dimensionality reduction in large-scale nonlinear networks. Our approach yields accurate global and node-level predictions across synthetic and real networks, including highly heterogeneous topologies and systems with higher-order interactions. Crucially, SSM is a robust tipping-point predictor: even at low truncation order (e.g., $O(2)$) it reliably identifies the onset of sustained activity, while higher orders and gSSM capture post-onset amplitudes and saturation. Consistently, the reduction collapses the full network dynamics to a one-dimensional system, offering clarity and efficiency. Across all the realizations, SSM/gSSM consistently outperform classical spectral and mean-field methods in modeling critical transitions at both microscopic and macroscopic scales, establishing SSM-based reduction as a robust, interpretable tool for nonlinear networked systems with broad applicability to epidemiology, ecology, and engineered networks.
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
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.
A novel large-scale delayed neural network with dual-star architecture, incorporating both second-order and third-order interactions, and the accurate function fitting achieved by the model provides new insights for the design and control of neural networks.
Yue Qiu, Min Xiao, Yonghui Sun et al.· Chaos· 0 citations
This work introduces an alternative criterion based on the persistent topological cycles in which each node participates---a measure of mesoscale integration that captures features beyond local connectivity---and demonstrates that persistent topology captures information about brain network control that scalar energy summaries miss.
Carter Sale, Marco Coraggio, Mengsen Zhang et al.· 0 citations
Complex systems are difficult to study not only because they are nonlinear, multiscale, and nonstationary, but because their scientifically relevant organization is often distributed across components, relations, and interaction orders. Topology provides a mathematical language for describing that organization through connectedness, recurrence, branching, closure, cavities, and persistence across scale. This review synthesizes persistent homology, Mapper, simplicial complexes, hypergraphs, and relation-level operator methods through a unified workflow from empirical data to representation, topological construction, output, and scientific interpretation. Across nonlinear dynamics, finance, neuroscience, biology, ecology, materials, and engineered systems, topological and topology-inspired methods make state-space organization, collective constraints, and structural reorganization available as observables that can be integrated with statistics, dynamics, mechanistic models, and machine learning. The review distinguishes the claim that a representation makes structure visible from the stronger claim that it improves detection or prediction, and it summarizes comparative evidence where such benchmarks exist. Prospective early-warning evidence remains uneven, but several studies demonstrate useful structural diagnostics, data-efficient classification, anomaly detection, and reductions in false alarms. The central conclusion is that topology is most valuable when representation is treated as a scientific hypothesis and topological descriptions are connected to domain-matched inference and mechanism.
Comparisons are fundamental to science: experiment against model, one organism against another, a system against itself across time. Because many systems, from brains to climate, are characterized by how they evolve in time, it is a natural goal to compare their dynamics. Dynamical systems comparison is well defined, but has been intractable for nonlinear, high-dimensional, noisy, and partially observed data. As a result, standard comparison methods have focused on the geometry or topology of data. Here we present Dynamical Similarity Analysis (DSA), a class of metrics to compare systems by their temporal evolution. Its foundation is Koopman Operator theory, which recasts nonlinear systems as linear operators. We estimate these operators from data, then compare the operators across systems. The computation is fast, scalable, and robust to noise and partial observation. It is also differentiable. DSA identifies dynamical structure that geometric and topological methods miss. It matches recordings from the head direction circuit to ring attractor models. It shows that macaque motor cortex dynamics for two reaching tasks drift apart across years despite preserved behavior, and that primary motor cortex breaks from premotor cortex as movement begins. As an optimization objective, it induces neural networks to learn never-before hypothesized solutions that run counter to their inductive biases. Thus, DSA transforms the dynamics of a system into an object that can be measured, compared, and optimized.
Mitchell Ostrow, Adam J. Eisen, L. Kozachkov et al.· bioRxiv· 0 citations
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