  We present the mathematical analysis of generalized complex contagions in
clustered multiplex networks for susceptible-infected-recovered (SIR)-like
dynamics. The model is intended to understand diffusion of influence, or any
other spreading process implying a threshold dynamics, in setups of
interconnected networks with significant clustering. The contagion is assumed
to be general enough to account for a content-dependent linear threshold model,
where each link type has a different weight (for spreading influence) that may
depend on the content (e.g., product, rumor, political view) that is being
spread. Using the generating functions formalism, we determine the conditions,
probability, and expected size of the emergent global cascades. This analysis
provides a generalization of previous approaches and is specially useful in
problems related to spreading and percolation. The results present non trivial
dependencies between the clustering coefficient of the networks and its average
degree. In particular, several phase transitions are shown to occur depending
on these descriptors. Generally speaking, our findings reveal that increasing
clustering decreases the probability of having global cascades and their size,
however this tendency changes with the average degree. There exists a certain
average degree from which on clustering favours the probability and size of the
contagion. By comparing the dynamics of complex contagions over multiplex
networks and their monoplex projections, we demonstrate that ignoring link
types and aggregating network layers may lead to inaccurate conclusions about
contagion dynamics, particularly when the correlation of degrees between layers
is high.
