Complex contagion: node adopts when fraction of active neighbors ≥ threshold φ
Network State
Cascade Window (z vs φ)
Parameters
Final adopted: —%
Cascade steps: —
Network type: Erdős–Rényi
Vulnerable nodes: —%
In window: —
Watts 2002: global cascades possible only in "cascade window" — low z or low φ. Complex contagion ≠ simple (SIR): threshold dependence means hubs can block cascades (high degree → harder to tip).