Seed size strongly affects cascades on random networks.

The average avalanche size in the Watts model of threshold dynamics on random networks of arbitrary degree distribution is determined analytically. Existence criteria for global cascades are shown to depend sensitively on the size of the initial seed disturbance. The dependence of cascade size upon the mean degree z of the network is known to exhibit several transitions-these are typically continuous at low z and discontinuous at high z; here it is demonstrated that the low- z transition may in fact be discontinuous in certain parameter regimes. Connections between these results and the zero-temperature random-field Ising model on random graphs are discussed.