Single-Seed Cascades on Clustered Networks

We consider a dynamic network cascade process developed by Watts [20] applied to a random networks with a specified amount of clustering, belonging to a class of random networks developed by Newman [19]. We adapt tree-based methods from [20] to formulate an appropriate two-type branching process to describe the spread of a cascade started with a single active node, and obtain a fixed-point equation to implicitly express the extinction probability of such a cascade. In so doing, we also recover a special case of a formula of Hackett et al. [12] giving conditions for certain extinction of the cascade.