Synergistic effects in threshold models on networks

Network structure can have a significant impact on the propagation of diseases, memes, and information on social networks. Different types of spreading processes (and other dynamical processes) are affected by network architecture in different ways, and it is important to develop tractable models of spreading processes on networks to explore such issues. In this paper, we incorporate the idea of 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 in a way that depends on the number of neighbors who have adopted that behavior. We illustrate that our model's synergy parameter 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 heterogeneous mean-field approximation, which we derive under the assumption that a network is locally tree-like, we are able to determine which synergy-parameter values allow degree-k nodes to be activated for many networks and for a broad family of synergistic models.