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Hidden higher-order vulnerabilities in simplicial complexes revealed by fixed-index spectral robustness
Phys. Rev. E 114, 034304 – Published 8 September, 2026
DOI: https://doi.org/10.1103/n4ps-wmd7
Abstract
Robustness of simplicial complexes under triangle deletion is often characterized by the instantaneous smallest positive eigenvalue of the Hodge 1-Laplacian. We show that this observable can change spectral index when the kernel grows, and therefore it need not follow the same nonharmonic edge-space eigenmode along the deletion process. To remove this ambiguity, we define fixed-index spectral robustness by monitoring the eigenvalue whose index is selected by the first nonzero eigenvalue of the intact complex. The corresponding triangle sensitivity follows from first-order perturbation theory and identifies simplices that produce the steepest initial decrease of this monitored eigenvalue. Across synthetic and empirical clique complexes, a small fraction of triangles can drive the monitored eigenvalue to zero while the 1-skeleton and graph-level robustness measures remain unchanged. These results show that higher-order robustness is a genuinely simplicial spectral property that cannot be inferred from graph-level connectivity alone.
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