AI 中文总结
研究针对图向量场的局部交互组织量化空白，提出SheafIQ层论框架，在多类系统中揭示了传统方法未捕捉的互补组织信息，拓展了网络分析维度。
AI 中文摘要
图结构上的向量场自然出现在各类生物与工程系统中，其中节点上定义了向量值状态并通过网络交互演化。现有方法主要刻画图拓扑或单个信号，但通常未量化节点关联向量间的局部交互如何在图上组织。为解决这一局限，本文提出名为SheafIQ的层论框架，其将相邻向量表示在公共的边关联坐标系中，将局部不兼容性映射为残差能量分布，并通过熵量化其全局组织。在蛋白质、功能脑网络、城市交通系统及电网中，SheafIQ均揭示了超出传统图与信号描述符的互补组织信息。更广泛而言，该方法建立了一个统一的信息论框架，用于量化几何图上向量值状态的组织，将网络分析从单纯的图拓扑扩展出去。
英文摘要
Vector fields on graph structures naturally arise in diverse biological and engineered systems, where vector-valued states are defined on the nodes and evolve through the network interactions. Existing methods primarily characterize either the graph topology or individual signals, but generally do not quantify how local interactions among node-associated vectors are organized across the graph. To address this limitation, a sheaf-theoretic framework, termed SheafIQ, is proposed to represent neighboring vectors in a common edge-associated coordinate system, map local incompatibilities to a residual energy distribution, and quantify its global organization through entropy. Across proteins, functional brain networks, urban traffic systems, and power grids, SheafIQ consistently reveals complementary organizational information beyond conventional graph- and signal-based descriptors. More broadly, it establishes a unified information-theoretic framework for quantifying the organization of vector-valued states on geometric graphs, extending network analysis beyond graph topology alone.
