  Network structure can have significant effects on the propagation of
diseases, memes, and information on social networks. Such effects depend on the
specific type of dynamical process that affects the nodes and edges of a
network, and it is important to develop tractable models of spreading processes
on networks to explore how network structure affects dynamics. In this paper,
we incorporate the idea of \emph{synergy} into a two-state ("active" or
"passive") threshold model of social influence on networks. Our model's update
rule is deterministic, and the influence of each meme-carrying (i.e., active)
neighbor can --- depending on a parameter --- either be enhanced or inhibited
by an amount that depends on the number of active neighbors of a node. Such a
synergistic system models social behavior in which the willingness to adopt
either accelerates or saturates depending on the number of neighbors who have
adopted that behavior. We illustrate that the synergy parameter in our model
has a crucial effect on system dynamics, as it determines whether degree-$k$
nodes are possible or impossible to activate. We simulate synergistic meme
spreading on both random-graph models and networks constructed from empirical
data. Using a local-tree approximation, we examine the spreading of synergistic
memes and find good agreement on all but one of the networks on which we
simulate spreading. We find for any network and for a broad family of
synergistic models that one can predict which synergy-parameter values allow
degree-$k$ nodes to be activated.
