Discontinuous Transition of a Multistage Independent Cascade Model on Networks

TL;DR

This paper introduces a multistage independent cascade model on networks, revealing complex phase transitions including both continuous and discontinuous fad percolation, with implications for understanding spread dynamics.

Contribution

It formulates a new multistage cascade model and analyzes its phase transitions on various networks, highlighting the conditions for discontinuous and continuous percolation.

Findings

Discontinuous fad percolation occurs when T1 exceeds a threshold.

Continuous percolation of susceptible nodes is observed under certain conditions.

Finite initial adopters significantly alter phase boundaries.

Abstract

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.