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 $T_1$ that a node adopting a fad increases the awareness of a neighboring susceptible node, and the probability $T_2$ 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 $p_k$. 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 $T_1$ is larger (smaller) than a certain value. A similar discontinuous transition is also observed in random graphs and scale-free networks. Furthermore, assigning a finite fraction of initial adopters dramatically changes the phase boundaries. 
Takehisa Hasegawa, Koji Nemoto
physics.soc-ph cond-mat.dis-nn