On scale-free networks (P(k)~k^-γ), the epidemic threshold β_c → 0 as N→∞. HMF theory groups nodes by degree class k and tracks i_k(t) = fraction infected at degree k.
Pastors-Satorras & Vespignani (2001): For scale-free networks with γ≤3, ⟨k²⟩→∞ and the epidemic threshold β_c = μ⟨k⟩/⟨k²⟩ → 0. Any nonzero infection spreads.