Herd immunity threshold on a heterogeneous network: p* = 1 − ⟨k⟩/(R₀·(⟨k²⟩−⟨k⟩)).
For scale-free networks (power-law degree, BA model), ⟨k²⟩ diverges → threshold approaches 0 as N→∞: even tiny outbreaks spread. Hub vaccination (targeting high-degree nodes) is far more efficient than random vaccination. Node color: blue=susceptible, red=infected, green=recovered/immune.