Discontinuous transition of a multistage independent cascade model on networks

We propose a multistage version of the independent cascade model, which we call a multistage independent cascade (MIC) model, on networks. This model is parameterized by two probabilities: the probability T1 that a node adopting a fad increases the awareness of a neighboring susceptible node and the probability T2 that an adopter directly causes a susceptible node to adopt the fad. We formulate a tree approximation for the MIC model on an uncorrelated network with an arbitrary degree distribution pk. Applied on a random regular network with degree k = 6, this model exhibits a rich phase diagram, including continuous and discontinuous transition lines for fad percolation and a continuous transition line for the percolation of susceptible nodes. In particular, the percolation transition of fads is discontinuous (continuous) when T1 is larger (smaller) than a certain value. A similar discontinuous transition is observed in random graphs and scale-free networks. Furthermore, assigning a finite fraction of initial adopters dramatically changes the phase boundaries. (paper)