The Vibrational Coherence Index (ICV): A Normalized Local-to-Global Metric for Alignment in DeGroot Opinion Dynamics

Description This work introduces the Vibrational Coherence Index (ICV), a novel normalized metric designed to quantify local and global agreement in opinion dynamics governed by the classical DeGroot consensus model. Unlike traditional disagreement or Laplacian-based smoothness measures, the ICV provides an interpretable bounded score in the interval 0,1, expressing the degree of alignment between each node and the weighted average of its neighborhood. The framework enables coherence analysis at multiple scales: Node-level coherence, capturing local deviations, Cluster-level coherence, revealing internal community alignment, Network-level coherence, measuring global stability of the system. The ICV complements existing consensus metrics by offering a unified and scalable tool for detecting emergent stability patterns, local misalignments, and structural coherence in networked systems. This contribution bridges classical consensus theory with modern network coherence perspectives and provides a robust foundation for future studies on dynamic topologies, perturbations, and adversarial influence in distributed multi-agent environments.