Coherence Performance and Weighted Tuning in a Class of High-Clustering Networks

This study investigates network consensus performance, including first-order network coherence, delay robustness, and spectral connectivity-cost efficiency, all quantified via Laplacian spectra. To evaluate the impacts of network coupling and weight assignment on coherence performance, two types of high-clustering generalized windmill graphs are selected as models, with weights assigned based on node heterogeneity. By integrating Sylvester’s Determinant Theorem, exact solutions for the three performance indices are derived, and a weighted tuning analysis of consensus behavior is conducted based on these solutions. The proposed methodology holds substantial application potential for high-clustering networks with deterministic topologies and significant node heterogeneity.