Efficient Topology Design Algorithms for Power Grid Stability

Monotonicity of f(A)=Tr(W̃(L̃e+ΔA)^{-1}) with respect to edge additions is handled correctly in the paper (via order arguments / first-order Neumann term) and also by the model; see the paper’s discussion around Lemma 1 and the objective’s monotonicity after (9) , and the supermodularity section’s first-order sign observation . However, the paper’s Theorem 2 claims supermodularity under conditions stated solely in terms of L̃e^{-1}, using the mixed second-derivative at the base point (27) with L̃e^{-1} rather than at general b; this checks only the b=0 coefficient in a Neumann-series expansion and does not control higher-order mixed terms or guarantee nonnegativity of cross-partials over the whole orthant, so it is incomplete for proving set-function supermodularity for all 0–1 additions . The model’s proof attempts to ensure nonnegative cross-partials everywhere via a “positivity preservation” inequality to propagate a_r^T K a_p > 0 under rank-one updates, but the key inequality (⟨x,s⟩⟨s,y⟩ ≤ ⟨s,s⟩⟨x,y⟩ for a K-induced inner product) is false in general; thus the claimed preservation and the resulting Topkis argument fail. Consequently, both the paper’s argument (insufficient control beyond second-order at b=0) and the model’s argument (incorrect inequality) are incomplete for establishing full supermodularity.