🤖 AI Summary
Existing approaches struggle to capture the global organization of local interactions among nodes in vector fields defined over graphs. This work proposes SheafIQ, a novel framework that uniquely integrates sheaf theory and information theory to map vectors from adjacent nodes into a unified edge-associated coordinate system. By analyzing the residual energy distribution and its entropy, SheafIQ quantifies the overall orderliness of the vector field. The method transcends conventional limitations that focus solely on graph topology or nodal signals, offering a unified information-theoretic measure for vector-valued states on geometric graphs. Applied to diverse systems—including protein–protein interaction networks, functional brain connectomes, urban traffic flows, and power grids—SheafIQ uncovers complementary organizational information beyond what classical graph- and signal-based metrics reveal.
📝 Abstract
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.