Decomposing Force Fields as Flows on Graphs Reconstructed From Stochastic Trajectories

Ramón Dineth Nartallo-Kaluarachchi, Paul Expert, David Beers, Alexander Strang, Morten L Kringelbach, Renaud Lambiotte, Alain Goriely
Proceedings of the Third Learning on Graphs Conference, PMLR 269:4:1-4:26, 2025.

Abstract

Disentangling irreversible and reversible forces from random fluctuations is a challenging problem in the analysis of stochastic trajectories measured from real-world dynamical systems. We present an approach to approximate the dynamics of a stationary Langevin process as a discrete-state Markov process evolving over a graph-representation of phase-space, reconstructed from stochastic trajectories. Next, we utilise the analogy of the Helmholtz-Hodge decomposition of an edge-flow on a contractible simplicial complex with the associated decomposition of a stochastic process into its irreversible and reversible parts. This allows us to decompose our reconstructed flow and to differentiate between the irreversible currents and reversible gradient flows underlying the stochastic trajectories. We validate our approach on a range of solvable and nonlinear systems and apply it to derive insight into the dynamics of flickering red-blood cells and healthy and arrhythmic heartbeats. In particular, we capture the difference in irreversible circulating currents between healthy and passive cells and healthy and arrhythmic heartbeats. Our method breaks new ground at the interface of data-driven approaches to stochastic dynamics and graph signal processing, with the potential for further applications in the analysis of biological experiments and physiological recordings. Finally, it prompts future analysis of the convergence of the Helmholtz-Hodge decomposition in discrete and continuous spaces.

Cite this Paper

BibTeX
@InProceedings{pmlr-v269-nartallo-kaluarachchi25a, title = {Decomposing Force Fields as Flows on Graphs Reconstructed From Stochastic Trajectories}, author = {Nartallo-Kaluarachchi, Ram{\'o}n Dineth and Expert, Paul and Beers, David and Strang, Alexander and Kringelbach, Morten L and Lambiotte, Renaud and Goriely, Alain}, booktitle = {Proceedings of the Third Learning on Graphs Conference}, pages = {4:1--4:26}, year = {2025}, editor = {Wolf, Guy and Krishnaswamy, Smita}, volume = {269}, series = {Proceedings of Machine Learning Research}, month = {26--29 Nov}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v269/main/assets/nartallo-kaluarachchi25a/nartallo-kaluarachchi25a.pdf}, url = {https://proceedings.mlr.press/v269/nartallo-kaluarachchi25a.html}, abstract = {Disentangling irreversible and reversible forces from random fluctuations is a challenging problem in the analysis of stochastic trajectories measured from real-world dynamical systems. We present an approach to approximate the dynamics of a stationary Langevin process as a discrete-state Markov process evolving over a graph-representation of phase-space, reconstructed from stochastic trajectories. Next, we utilise the analogy of the Helmholtz-Hodge decomposition of an edge-flow on a contractible simplicial complex with the associated decomposition of a stochastic process into its irreversible and reversible parts. This allows us to decompose our reconstructed flow and to differentiate between the irreversible currents and reversible gradient flows underlying the stochastic trajectories. We validate our approach on a range of solvable and nonlinear systems and apply it to derive insight into the dynamics of flickering red-blood cells and healthy and arrhythmic heartbeats. In particular, we capture the difference in irreversible circulating currents between healthy and passive cells and healthy and arrhythmic heartbeats. Our method breaks new ground at the interface of data-driven approaches to stochastic dynamics and graph signal processing, with the potential for further applications in the analysis of biological experiments and physiological recordings. Finally, it prompts future analysis of the convergence of the Helmholtz-Hodge decomposition in discrete and continuous spaces.} }
Endnote
%0 Conference Paper %T Decomposing Force Fields as Flows on Graphs Reconstructed From Stochastic Trajectories %A Ramón Dineth Nartallo-Kaluarachchi %A Paul Expert %A David Beers %A Alexander Strang %A Morten L Kringelbach %A Renaud Lambiotte %A Alain Goriely %B Proceedings of the Third Learning on Graphs Conference %C Proceedings of Machine Learning Research %D 2025 %E Guy Wolf %E Smita Krishnaswamy %F pmlr-v269-nartallo-kaluarachchi25a %I PMLR %P 4:1--4:26 %U https://proceedings.mlr.press/v269/nartallo-kaluarachchi25a.html %V 269 %X Disentangling irreversible and reversible forces from random fluctuations is a challenging problem in the analysis of stochastic trajectories measured from real-world dynamical systems. We present an approach to approximate the dynamics of a stationary Langevin process as a discrete-state Markov process evolving over a graph-representation of phase-space, reconstructed from stochastic trajectories. Next, we utilise the analogy of the Helmholtz-Hodge decomposition of an edge-flow on a contractible simplicial complex with the associated decomposition of a stochastic process into its irreversible and reversible parts. This allows us to decompose our reconstructed flow and to differentiate between the irreversible currents and reversible gradient flows underlying the stochastic trajectories. We validate our approach on a range of solvable and nonlinear systems and apply it to derive insight into the dynamics of flickering red-blood cells and healthy and arrhythmic heartbeats. In particular, we capture the difference in irreversible circulating currents between healthy and passive cells and healthy and arrhythmic heartbeats. Our method breaks new ground at the interface of data-driven approaches to stochastic dynamics and graph signal processing, with the potential for further applications in the analysis of biological experiments and physiological recordings. Finally, it prompts future analysis of the convergence of the Helmholtz-Hodge decomposition in discrete and continuous spaces.
APA
Nartallo-Kaluarachchi, R.D., Expert, P., Beers, D., Strang, A., Kringelbach, M.L., Lambiotte, R. & Goriely, A.. (2025). Decomposing Force Fields as Flows on Graphs Reconstructed From Stochastic Trajectories. Proceedings of the Third Learning on Graphs Conference, in Proceedings of Machine Learning Research 269:4:1-4:26 Available from https://proceedings.mlr.press/v269/nartallo-kaluarachchi25a.html.

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