# Dynamical systems and complex networks: A Koopman operator perspective

arXiv: https://arxiv.org/abs/2405.08940 · HTML: https://mlsanity.com/paper/2405.08940/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 95 references that are in the map.
- First posted on arXiv 14 May 2024.

## Builds on (most cited first, 23 of 23)

- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Koopman Operator Dynamical Models: Learning, Analysis and Control](https://mlsanity.com/paper/2102.02522/) (2021)
- [Koopman operator-based model reduction for switched-system control of PDEs](https://mlsanity.com/paper/1710.06759/) (2017)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Sparsity-promoting dynamic mode decomposition](https://mlsanity.com/paper/1309.4165/) (2013)
- [The Multiverse of Dynamic Mode Decomposition Algorithms](https://mlsanity.com/paper/2312.00137/) (2023)
- [Linear identification of nonlinear systems: A lifting technique based on the Koopman operator](https://mlsanity.com/paper/1605.04457/) (2016)
- [Data-Driven Model Predictive Control using Interpolated Koopman Generators](https://mlsanity.com/paper/2003.07094/) (2020)
- [Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1807.09331/) (2018)
- [Kernel methods for detecting coherent structures in dynamical data](https://mlsanity.com/paper/1904.07752/) (2019)
- [Robust Tube-based Model Predictive Control with Koopman Operators--Extended Version](https://mlsanity.com/paper/2108.13011/) (2021)
- [A kernel-based approach to molecular conformation analysis](https://mlsanity.com/paper/1809.11092/) (2018)
- [Koopman-based spectral clustering of directed and time-evolving graphs](https://mlsanity.com/paper/2204.02951/) (2022)
- [Tensor-based computation of metastable and coherent sets](https://mlsanity.com/paper/1908.04741/) (2019)
- [Transfer operators on graphs: Spectral clustering and beyond](https://mlsanity.com/paper/2305.11766/) (2023)
- [Broken detailed balance and entropy production in directed networks](https://mlsanity.com/paper/2402.19157/) (2024)
- [Data-driven model reduction of agent-based systems using the Koopman generator](https://mlsanity.com/paper/2012.07718/) (2020)
- [Modularity of Directed Networks: Cycle Decomposition Approach](https://mlsanity.com/paper/1407.8039/) (2014)
- [Data-driven Influence Based Clustering of Dynamical Systems](https://mlsanity.com/paper/2204.02373/) (2022)

## Built on by (most cited first, 9 of 9)

- [Learning graphons from data: Random walks, transfer operators, and spectral clustering](https://mlsanity.com/paper/2507.18147/) (2025)
- [How deep is your network? Deep vs. shallow learning of transfer operators](https://mlsanity.com/paper/2509.19930/) (2025)
- [Clustering Time-Evolving Networks Using the Spatio-Temporal Graph Laplacian](https://mlsanity.com/paper/2407.12864/) (2024)
- [Data-driven system identification using quadratic embeddings of nonlinear dynamics](https://mlsanity.com/paper/2501.08202/) (2025)
- [Data-driven approximation of transfer operators for mean-field stochastic differential equations](https://mlsanity.com/paper/2509.09891/) (2025)
- [Taming Epilepsy: Mean Field Control of Whole-Brain Dynamics](https://mlsanity.com/paper/2603.18035/) (2026)
- [Optimization of randomized neural networks for transfer operator approximation](https://mlsanity.com/paper/2605.23689/) (2026)
- [Integral Formulation of QENDy for Robust Nonlinear System Identification](https://mlsanity.com/paper/2606.11629/) (2026)
- [Functional dynamic mode decomposition: Learning infinite-dimensional systems from data](https://mlsanity.com/paper/2609.29159/) (2026)

## Abstract (arXiv)

The Koopman operator has entered and transformed many research areas over the last years. Although the underlying conceptunicodex2013representing highly nonlinear dynamical systems by infinite-dimensional linear operatorsunicodex2013has been known for a long time, the availability of large data sets and efficient machine learning algorithms for estimating the Koopman operator from data make this framework extremely powerful and popular. Koopman operator theory allows us to gain insights into the characteristic global properties of a system without requiring detailed mathematical models. We will show how these methods can also be used to analyze complex networks and highlight relationships between Koopman operators and graph Laplacians.