SIR model on networks: each infected node spreads with rate β, recovers at rate γ. Epidemic threshold: R₀ = β⟨k⟩/γ for Erdős–Rényi; R₀ = β⟨k²⟩/(γ⟨k⟩) for scale-free (diverges → no threshold!).
R₀ = — | S=100% I=0% R=0%
Epidemic threshold: for scale-free networks with degree distribution P(k)~k^{-γ_d} (γ_d≤3), ⟨k²⟩→∞ and R₀→∞ — the network is always supercritical. Herd immunity threshold: 1-1/R₀ of the population must be immune. The Fiedler eigenvalue (algebraic connectivity) controls spread speed.