Dynamic theory of cascades on finite clustered random networks with a threshold rule

Cascade dynamics on networks are usually analyzed statically to determine existence criteria for cascades. Here, the Watts model of threshold dynamics on random Erdős-Rényi networks is analyzed to determine the dynamic time evolution of cascades. The network is assumed to have a specific finite number of nodes n and is not assumed to be treelike. All combinations of threshold ϕ, network average nodal degree z, and seed sizes |S| from a single node up are included. The analysis permits study of network size effects and increased clustering coefficient. Several size effects not found by infinite network theory are predicted by the analysis and confirmed by simulations. In the region of ϕ and z where a single node can start a cascade, cascades are expanding, in the sense that each step flips a larger group than the previous step did. We show that this region extends to larger values of z than predicted by infinite network analyses. In the region where larger seeds are needed (size proportional to n), cascades begin by contracting: at the outset, each step flips fewer nodes than the previous step, but eventually the process reverses and becomes expanding. A critical mass that grows during the cascade beyond an easily-calculated threshold is identified as the cause of this reversal