  Understanding cascading processes on complex network topologies is paramount
for modelling how diseases, information, fake news and other media spread. In
this paper, we extend the multi-type branching process method developed in
Keating et al., 2022, which relies on homogenous network properties, to a more
general class of clustered networks. Using a model of socially-inspired complex
contagion we obtain results, not just for the average behaviour of the cascades
but for full distributions of the cascade properties. We introduce a new method
for the inversion of probability generating functions to recover their
underlying probability distributions; this derivation naturally extends to
higher dimensions. This inversion technique is used along with the multi-type
branching process to obtain univariate and bivariate distributions of cascade
properties. Finally, using clique cover methods, we apply the methodology to
synthetic and real-world networks and compare the theoretical distribution of
cascade sizes with the results of extensive numerical simulations.
