Hybrid Entropy-Based Metrics for k-Hop Environment Analysis in Complex Networks

Two hybrid, entropy-guided node metrics are proposed for the k-hop environment: Entropy-Weighted Redundancy (EWR) and Normalized Entropy Density (NED). The central idea is to couple local Shannon entropy with neighborhood density/redundancy so that structural heterogeneity around a vertex is captured even when classical indices (e.g., degree or clustering) are similar. The metrics are formally defined and shown to be bounded, isomorphism-invariant, and stable under small edge edits. Their behavior is assessed on representative topologies (Erdős–Rényi, Barabási–Albert, Watts–Strogatz, random geometric graphs, and the Zephyr quantum architecture). Across these settings, EWR and NED display predominantly negative correlation with degree and provide information largely orthogonal to standard centralities; vertices with identical degree can differ by factors of two to three in the proposed scores, revealing bridges and heterogeneous regions. These properties indicate utility for vulnerability assessment, topology-aware optimization, and layout heuristics in engineered and quantum networks.