Dynamical importance and network perturbations

The paper derives the near-onset Kuramoto order-parameter formula r^2 = (π^2 g(0)^2 η / 4α) (k/k_c − 1)(k/k_c)^{−3} with k_c = 2/(π λ g(0)) and then, for a single edge addition, writes r̂^2 in terms of Δλ and shows r̂^2 > r^2 when degrees remain approximately homogeneous; see Eqs. (24)–(30) and discussion of validity (k/k_c ≲ 1.3) and assumptions (Appendix C) . They then define β and γ, approximate Δλ by the unnormalized FoEDI ι† = 2 v_i v_j/(v^T v), and present ř^2 = β (kγ(λ+ι†) − 1)(γ(kλ+ι†))^{−3} (Eq. (32)) , consistent with their earlier derivation of FoEDI and its relation to Δλ and the Rayleigh quotient . The candidate solution reproduces these steps, including the β-constancy under approximate degree homogeneity and the substitution Δλ ≈ ι†, and gives a clean monotonicity argument near threshold using f(s) = (s − 1)s^{−3}, which aligns with the paper’s qualitative claim r̂^2 > r^2 for edge additions in the near-onset regime . A subtle point is the paper’s (and the candidate’s) mixed first-order modeling in Eq. (32): the cubic factor uses γ(kλ + ι†) rather than γk(λ + ι†). This is heuristically justifiable by viewing ι† as the first-order shift of ρ(kA) under an unscaled edge addition to A, but it would benefit from an explicit note; otherwise the logic and assumptions are consistent.