  We present an analytical approach to determining the expected cascade size in
a broad range of dynamical models on the class of random networks with
arbitrary degree distribution and nonzero clustering introduced in [M.E.J.
Newman, Phys. Rev. Lett. 103, 058701 (2009)]. A condition for the existence of
global cascades is derived as well as a general criterion which determines
whether increasing the level of clustering will increase, or decrease, the
expected cascade size. Applications, examples of which are provided, include
site percolation, bond percolation, and Watts' threshold model; in all cases
analytical results give excellent agreement with numerical simulations.
