A similarity laplacian energy perspective on critical node identification in complex networks

The identification of critical nodes is fundamental to understanding complex network dynamics. Current Laplacian-based centrality measures predominantly build upon either the standard Laplacian matrix or distance Laplacian matrices. While effective, these approaches are limited to direct connections or geodesic paths, missing important higher-order structural information. This paper introduces Similarity Laplacian Centrality (SLC), which fundamentally redefines the Laplacian foundation by constructing a similarity Laplacian matrix based on Local Path index. Our key innovation lies in replacing the conventional adjacency/distance matrices with a similarity matrix that incorporates both direct and indirect relationships through paths of length 2 and 3. This allows SLC to capture functional similarities between nodes that extend beyond immediate neighborhood or shortest-path considerations. Within this novel framework, node importance is measured by the perturbation to the spectral energy of the Similarity Laplacian Matrix upon node removal. We provide theoretical guarantees and derive a computationally efficient closed-form solution that scales to large networks. Large-scale experiments conducted across multiple real-world networks demonstrate that the SLC method achieves strong performance in influence propagation, ranking accuracy, and discrimination capability across most datasets, effectively identifying critical nodes within complex networks.