Watts Threshold Cascade Model — Global Cascades

On a random network, each node adopts a behavior when a fraction φ of its neighbors already have. The (z, φ) phase diagram shows the cascade window: a region where global cascades (giant component adoption) are possible from a single seed.

Parameters

Node i adopts when: m_i/k_i ≥ φ m_i = adopted neighbors, k_i = degree ER network, Poisson degree P(k)=e^{-z}z^k/k! Cascade condition (Watts 2002): ∂z/∂z boundary: vulnerability χ diverges Window: z₁(φ) ≤ z ≤ z₂(φ)