Every node is given a threshold: the fraction of its neighbours that must have adopted before it does. One node is nudged at the outset. Then the thing runs until it stops, and the question is how much of the network went.

Almost always, nothing. The nudge dies within a few nodes and the network never notices. Occasionally the whole thing goes.

Two regimes are identified in which the network is susceptible to very large cascades — herein called global cascades — that occur very rarely.

Duncan J. Watts, PNAS, 30 April 2002 — abstract

What decides it is not the size of the push. The push is one node either way. It is the connectivity: too sparse and there is no path for the thing to travel, too dense and every node has so many neighbours that no small group can ever meet its threshold. Between those two failures there is a band, and the band is narrow.

Inside it a single node, chosen arbitrarily and indistinguishable from any other, takes the entire network. Afterwards it will be possible to write an account in which that node was the reason. There is nothing about the node.

Outside the window nothing propagates. Inside it, something that is not special takes everything, and is then explained.