Social Contagion — Complex Threshold Cascades

Watts (2002) threshold model: global cascades from local interactions

N = 80
⟨k⟩ = 5.0
φ = 0.18
ρ₀ = 2.5%

Watts Threshold Model (2002)

Each node i adopts if the fraction of its neighbors already adopted exceeds threshold φᵢ (here uniform φ). Unlike simple SIR contagion, adoption requires multiple social contacts — a "complex contagion". Global cascade condition:

Cascade window: 1/φ > ⟨k²⟩/⟨k⟩ − 1 (vulnerable cluster must span network)

On Erdős-Rényi graphs, a cascade window exists in (φ, ⟨k⟩) space. Outside it, cascades die locally. The vulnerable cluster — nodes whose degree k ≤ 1/φ — must percolate for global spread.