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Vector Space of Cycles

arXiv 2606.08202 ↗First posted 6 Jun 2026stat.ML cs.LG physics.data-ancited by 0

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Abstract (arXiv)

Most statistical and machine learning methods for directed interactions focus on pairwise effects among variables. Even existing cyclic models represent feedback primarily through node-level dependencies, making large-scale recurrent organization difficult to estimate and compare. This limitation is particularly acute in biological and neural systems, where interactions are highly recurrent and involve many overlapping cycles. We introduce a variational framework for statistical inference on cyclic interactions. Directed interactions are represented as edge flows on a simplicial complex and evolved under an energy-minimizing dynamical system. The resulting dynamics separate transient interaction components from persistent harmonic flows, yielding a low-dimensional cycle space that captures stable recurrent organization. Rather than enumerating individual cycles, the proposed framework represents cyclic interactions as elements of a Hilbert space, enabling projection, averaging, comparison, and population-level statistical inference. We establish theoretical properties of the harmonic projection, including characterization of the cycle space, variance reduction, and population inference. Simulations demonstrate substantially improved recovery of cyclic structure in dense recurrent systems compared with existing directed-interaction methods. Applied to resting-state fMRI from 400 human subjects, the framework reveals reproducible large-scale cyclic organization that is not detectable through edgewise averaging. These results provide a scalable statistical framework for studying recurrent interactions in high-dimensional dynamical systems.

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References · 19

  1. 2018DAGs with NO TEARS: Continuous Optimization for Structure Learningcited by 745
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  11. 2014From Ordinary Differential Equations to Structural Causal Models: the deterministic casecited by 41
  12. 2018Constraint-based Causal Discovery for Non-Linear Structural Causal Models with Cycles and Latent Confounderscited by 35
7 more
  1. 2013A Discovery Algorithm for Directed Cyclic Graphscited by 33
  2. 2018Statistical Model for Dynamically-Changing Correlation Matrices with Application to Brain Connectivitycited by 4
  3. 2023Measuring Information Transfer Between Nodes in a Brain Network through Spectral Transfer Entropycited by 3
  4. 2026Counterfactual Analysis of Brain Network Dynamicscited by 2
  5. 2020Flexible Bivariate INGARCH Process With a Broad Range of Contemporaneous Correlationcited by 1
  6. 2021Hodge-Laplacian of Brain Networkscited by 1
  7. 2024Topological Analysis of Seizure-Induced Changes in Brain Hierarchy Through Effective Connectivitycited by 1

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