Behind the Paper

Stability analysis of  high-dimensional complex systems is a central challenge across a wide range of disciplines, including neuroscience, physics, ecology, epidemiology, climate science, and engineering.  Traditionally, stability studies focused on low-dimensional models and pairwise interactions between components. Over time, the representation of complex systems as networks--where subsystems are nodes connected by edges--enabled us to identify structural patterns related to stability. 

News & Opinion

With the surge in data availability from sensors, experiments, and high-resolution simulations, there has been a shift toward data-driven methods that reconstruct system dynamics directly from observations. Importantly, modern approaches extend beyond pairwise links to quantify higher-order interactions, introducing nonlinear feedback loops and coupling effects that enrich the system’s dynamical landscape. These effects give rise not only to new fixed points but also enable robust analysis by capturing equilibrium dynamics and extending to more complex regimes, including limit cycles and chaos.

Our work follows this evolution by developing a data-driven framework that captures higher-order interactions through effective weighted adjacency matrices near empirically identified fixed points. Validated on both low- and high-dimensional nonlinear systems exhibiting tipping elements, limit cycles, and chaos, and applied to real-world data such as the Nordic Power Grid, this method highlights the growing capability of data-driven stability analysis to address complex, real-world dynamical challenges.

Our approach distinguishes itself from existing state-of-the-art methods by reconstructing both deterministic and stochastic components directly from data, leveraging Kramers-Moyal coefficients. Its moment-based formulation offers inherent robustness to noise and is particularly well-suited for high-dimensional stochastic systems, efficiently capturing pair-wise and higher-order interactions through compact tensor representations. 

The computational complexity of the estimation of pair-wise and higher-order interactions scales as N^{3(Z+1)} , where N is the dimension of the time series and Z represents the order of interactions in reconstructed dynamical  equation. We demonstrate that the CPU time of the method scales with the number of data points n approximately as n^{0.8}, for fixed values of Z and N.