SIR Epidemic on Heterogeneous Networks

On heterogeneous (scale-free) networks, the epidemic threshold λ_c = ⟨k⟩/⟨k²⟩ can approach zero, meaning any infection spreads. For homogeneous (Poisson) networks, λ_c = 1/⟨k⟩. Hubs accelerate spreading dramatically.

Epidemic Curve

λ_c = —

λ = β/γ = —

—

S: —

I: —

R: —

Node size ∝ degree. Scale-free networks (γ=2-3) have diverging ⟨k²⟩ → zero threshold — herd immunity is nearly impossible.

Network β (infection) 0.10 γ (recovery) 0.05 N nodes 150