Title: Multi-Stage Complex Contagions in Random Multiplex Networks,
Abstract: Complex contagion models have been developed to understand a wide range of
social phenomena such as adoption of cultural fads, the diffusion of belief,
norms, and innovations in social networks, and the rise of collective action to
join a riot. Most existing works focus on contagions where individuals' states
are represented by {\em binary} variables, and propagation takes place over a
single isolated network. However, characterization of an individual's standing
on a given matter as a binary state might be overly simplistic as most of our
opinions, feelings, and perceptions vary over more than two states. Also, most
real-world contagions take place over multiple networks (e.g., Twitter and
Facebook) or involve {\em multiplex} networks where individuals engage in
different {\em types} of relationships (e.g., acquaintance, co-worker, family,
etc.). To this end, this paper studies {\em multi-stage} complex contagions
that take place over multi-layer or multiplex networks. Under a linear
threshold based contagion model, we give analytic results for the probability
and expected size of \textit{global} cascades, i.e., cases where a randomly
chosen node can initiate a propagation that eventually reaches a {\em positive}
fraction of the whole population. Analytic results are also confirmed and
supported by an extensive numerical study. In particular, we demonstrate how
the dynamics of complex contagions is affected by the extra weight exerted by
\textit{hyper-active} nodes and by the structural properties of the networks
involved. Among other things, we reveal an interesting connection between the
assortativity of a network and the impact of \textit{hyper-active} nodes on the
cascade size.