Changing the ranking in eigenvector centrality of a weighted graph by small perturbations

The candidate solution reproduces the paper’s Theorem 5.1: it establishes that along the constrained flow Ė = −P+ Gε(E) + γ P+ E, the functional Fε is nonincreasing and that Ḟε=0, Ė=0, and colinearity P+Gε(E) || P+E are equivalent. The paper derives Ḟε via Lemma 5.1 and the KKT-based flow (Lemma 5.3), arriving at Ḟε/ε = −∥P+Gε∥^2 + ⟨P+Gε, P+E⟩^2/∥P+E∥^2 and the same equivalences (Theorem 5.1) . The candidate’s proof expands the same steps geometrically as a squared distance to the projection, and even treats the degenerate case ∥P+E∥=0. The only minor mismatch is a positive scaling factor (η=ε) in the paper’s chain rule, but this does not affect the sign or the equivalences .