  Threshold cascade models have been used to describe spread of behavior in
social networks and cascades of default in financial networks. In some cases,
these networks may have multiple kinds of interactions, such as distinct types
of social ties or distinct types of financial liabilities; furthermore, nodes
may respond in different ways to in influence from their neighbors of multiple
types. To start to capture such settings in a stylized way, we generalize a
threshold cascade model to a multiplex network in which nodes follow one of two
response rules: some nodes activate when, in at least one layer, a large enough
fraction of neighbors are active, while the other nodes activate when, in all
layers, a large enough fraction of neighbors are active. Varying the fractions
of nodes following either rule facilitates or inhibits cascades. Near the
inhibition regime, global cascades appear discontinuously as the network
density increases; however, the cascade grows more slowly over time. This
behavior suggests a way in which various collective phenomena in the real world
could appear abruptly yet slowly.
