  Synergistic interactions are ubiquitous in the real world. Recent studies
have revealed that, for a single-layer network, synergy can enhance spreading
and even induce an explosive contagion. There is at the present a growing
interest in behavior spreading dynamics on multiplex networks. What is the role
of synergistic interactions in behavior spreading in such networked systems? To
address this question, we articulate a synergistic behavior spreading model on
a double layer network, where the key manifestation of the synergistic
interactions is that the adoption of one behavior by a node in one layer
enhances its probability of adopting the behavior in the other layer. A general
result is that synergistic interactions can greatly enhance the spreading of
the behaviors in both layers. A remarkable phenomenon is that the interactions
can alter the nature of the phase transition associated with behavior adoption
or spreading dynamics. In particular, depending on the transmission rate of one
behavior in a network layer, synergistic interactions can lead to a
discontinuous (first-order) or a continuous (second-order) transition in the
adoption scope of the other behavior with respect to its transmission rate. A
surprising two-stage spreading process can arise: due to synergy, nodes having
adopted one behavior in one layer adopt the other behavior in the other layer
and then prompt the remaining nodes in this layer to quickly adopt the
behavior. Analytically, we develop an edge-based compartmental theory and
perform a bifurcation analysis to fully understand, in the weak synergistic
interaction regime where the dynamical correlation between the network layers
is negligible, the role of the interactions in promoting the social behavioral
spreading dynamics in the whole system.
