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.

翻译：大多数针对有向交互的统计和机器学习方法聚焦于变量间的两两效应。即便现有循环模型主要通过节点级依赖关系表示反馈机制，使得大规模递归结构难以估计与比较。这一局限在生物与神经系统中尤为突出，这些系统中交互高度递归且包含大量重叠循环。我们提出一种变分框架用于循环交互的统计推断。有向交互被表示为单纯复形上的边流，并在能量最小化动力系统下演化。由此产生的动力学将瞬态交互分量与持久谐波流分离，生成捕捉稳定递归组织的低维循环空间。该框架不枚举单个循环，而是将循环交互表示为希尔伯特空间中的元素，从而支持投影、平均、比较及群体级统计推断。我们建立了谐波投影的理论性质，包括循环空间特征化、方差缩减及群体推断。模拟实验表明，与现有有向交互方法相比，该方法在密集递归系统中对循环结构的恢复显著提升。应用于400名人类受试者的静息态fMRI数据，该框架揭示了通过边级平均无法检测的可重复大规模循环组织。这些结果为高维动力系统中递归交互的研究提供了可扩展的统计框架。