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.