Hidden Higher-Order Vulnerabilities in Simplicial Complexes Revealed by Branch-Consistent Functional Robustness

Abstract

Robustness of higher-order networks is often quantified by the instantaneous smallest positive eigenvalue of the Hodge $1$-Laplacian under simplex deletion. We show that this observable is generically ill-defined: along a deletion trajectory, eigenvalue branches can switch, so the quantity being monitored may correspond to different nonharmonic modes at different steps. The primary issue is therefore definitional rather than algorithmic. We resolve it by fixing the first nonharmonic branch of the intact complex and following that same branch throughout the damage process, which defines a branch-consistent functional robustness. Triangle sensitivities then follow directly from first-order perturbation theory, making the resulting mode-sensitive deletion protocol a consequence of the observable itself rather than an independent heuristic. Across synthetic and empirical clique complexes, removing only a small fraction of triangles is sufficient to drive the tracked mode to collapse, while graph-level observables remain unchanged because the $1$-skeleton is exactly preserved. The same framework also reveals bridge-like localization of functionally critical simplices and provides a compact predictor of dynamical timescales.

Figures (8)

• Figure 1: Branch-consistent functional robustness in weighted and directed clique complexes. High-sensitivity triangles are colored red; tracked mode edges are indicated in red arrows; key triangle outlined in yellow. The figure serves as an illustration of the logic: defining the tracked mode, freezing its spectral index, and scoring simplices by Mode Sensitivity. Insets show suppression of the same branch after triangle removal.
• Figure 2: Functional dismantling on synthetic clique complexes probed by the branch-consistent observable $R_{\mathrm{HO}}(f)=\widetilde{\mu}_1(f)/\widetilde{\mu}_1(0)$. Columns correspond to Erdős--Rényi, Watts--Strogatz, and spatial random ensembles; rows show undirected, directed, and weighted cases. Removing triangles ranked by mode sensitivity (MS) collapses the tracked higher-order mode after removing only a tiny fraction of simplices, in sharp contrast to both the non-tracked spectral baseline (NTS) and graph-based heuristics (Local, Graph), which display gradual decay. The dashed line indicates the critical fraction $f_c$ where the tracked mode vanishes.
• Figure 3: Failure of the instantaneous smallest positive eigenvalue as a robustness observable. (a) The branch-consistent eigenvalue $\widetilde{\mu}_1(f)$ (red) vanishes at the critical fraction $f_c$, signaling collapse of the tracked higher-order mode, whereas the instantaneous smallest positive eigenvalue $\mu_{+}^{\mathrm{inst}}(f)$ (blue) remains finite because it switches to a different branch after the tracked mode enters the kernel. (b) The kernel dimension increases at $f_c$, identifying the emergence of an additional harmonic mode. The coincidence of tracked-mode collapse and kernel growth shows that $\mu_{+}^{\mathrm{inst}}(f)$ does not monitor a fixed physical degree of freedom and therefore cannot serve as a consistent functional robustness observable.
• Figure 4: Higher-order functional collapse missed by graph benchmarks. (a) Watts-Strogatz clique complex at $f=0$ and at the realized collapse threshold $f_c\approx 0.037$, with the actually removed simplices shown as translucent red triangular regions. The node-edge skeleton is visually unchanged because only $2$-simplices are deleted. (b) The branch-consistent robustness $\widetilde{\mu}_1(f)$ collapses to zero at $f_c$, whereas the normalized graph observables normalized $\lambda_2(L_0)$ and GCC remain identically equal to $1$ throughout, since triangle deletion leaves the $1$-skeleton exactly invariant. The dashed vertical line marks the mode-collapse threshold.
• Figure 5: Localization and concentration of critical simplices in a modular bridge complex. (a) Triangle support map together with the top-simplex ranking table. The highest-ranked simplices concentrate on the intercommunity bridge rather than inside dense modules. (b) Sensitivity $MS_\tau$ versus triangle-overlap degree, showing weak correlation ($\rho=0.23$). (c) Cumulative captured support versus ranked simplex fraction; a small fraction of simplices carries a large portion of the total support.