Topological persistence pinpoints higher-order network vulnerabilities.

Higher-order topological features extend conventional graph models by capturing multi-node interactions, enabling more accurate modeling of structural robustness in complex systems. However, understanding the structural influence in complex networks remains challenging, especially when connectivity involves multiple scales and higher-order dependencies. This paper introduces the persistent structural influence indicator, which integrates persistent homology with local geometric perturbation analysis to quantify the node influence by extracting latent higher-order topological features from complex networks. Our model effectively captures multi-scale topological features and localized structural sensitivities, providing orthogonal information to classical centrality measures. Evaluations on both synthetic and real-world networks demonstrate that the proposed model more accurately identifies structurally critical nodes, resulting in accelerated network disintegration, reducing the giant component size to 0.12 after 20% node removal compared to 0.23 for degree-based attacks, and more pronounced reductions in post attack connectivity, improves the correlation with ground-truth spreading dynamics by up to 25.1% compared to baseline methods. Furthermore, the prediction model achieves these results without reliance on domain-specific priors or extensive training, balancing interpretability, computational tractability, and structural fidelity. The proposed metric offers a robust, generalizable framework for influence quantification and structural analysis in complex networked systems.