A multi-type branching process method for modelling complex contagion on clustered networks

Complex contagion adoption dynamics are characterised by a node being more likely to adopt after multiple network neighbours have adopted. We show how to construct multi-type branching processes to approximate complex contagion adoption dynamics on networks with clique-based clustering. This involves tracking the evolution of a cascade via different classes of clique motifs that account for the different numbers of active, inactive and removed nodes. This discrete-time model assumes that active nodes become immediately and certainly removed in the next time step. This description allows for extensive Monte Carlo simulations (which are faster than network-based simulations), accurate analytical calculation of cascade sizes, determination of critical behaviour and other quantities of interest.