Synergistic effects in threshold models on networks

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.