Watts Threshold Cascade Model

Each node in a random network adopts a new behavior when a fraction φ of its neighbors have adopted. A global cascade (nearly all nodes infected) is possible only when a "vulnerable cluster" spans the network — the cascade window in (z, φ) space. This model explains viral adoption, bank runs, and social fads: small shocks trigger large failures when the network is in the right regime.

Mean degree z 2.00 Threshold φ 0.18 Network N 200

Current state:
Adopted: 0 / 0
Cascade size: 0%
Global cascade: —

Cascade condition:
Need ρ₀(z,φ) > 0 where ρ₀ is
the vulnerable cluster density
ρ* = Σ_k P(k)·½[1−φ(k+1)]