External field and critical exponents in controlling dynamics on complex networks

The paper derives equilibrium exponents β=1/(m−Γ0), δ=m−Λ0, γ=(m−Λ0−1)/(m−Γ0) under Hahn-series assumptions Γ0=Π0>Λ0 with a0<0<b0 and c0M2(Δ)>0, and verifies Widom’s identity β(δ−1)=γ; it also derives transient exponents ϕ=1/(m−1), φ=(m−1)/(m−Γ0), θ=(m−1)/(m−Λ0) using the mean‑field dynamics near criticality. The candidate solution reproduces the same exponents with the same hypotheses and essentially the same expansion-and-balance method, including the definition m=min{Γ1,Π1}, the steady-state expansion around S=Sc, the susceptibility calculation via χ=−(∂ρ f)/(∂x f)|ρ=0, and the transient analysis from ẋ≈K x^m at criticality. These match the paper’s statements and derivations for β, δ, γ and ϕ, φ, θ, respectively, and the same regularity/sign conditions are invoked; see the paper’s general result and assumptions, its δ and γ derivations, and the transient exponents, which align exactly with the model’s results and steps . The paper’s external-field smoothing condition (c0M2(Δ)>0, Λ0<Π0=Γ0) used in the expansions is also observed in the model’s solution .