Assume a unidirectional, unweighted network generated according to a degree distribution. Each node is given a value between 0 and 1 called threshold $\phi$. We topple some nodes, the neighbours will perish if $\phi$ of their neighbours die. We iterate this rule until equilibrium.
We are interested in the final size of this cascade, the total number of dead nodes in equilibrium.
There are certain nodes that will perish if a single of their neighbours dies. We call these nodes vulnerable.
Here is my question. Watts has shown that if there is a percolating cluster of vulnerable nodes there is a nonzero chance a a global cascade.
Can you direct me to some literature that shows the reverse? Meaning, there is a global cascade if and only if there is a percolating cluster of vulnerable nodes?