Social behaviors and choices spread through interactions and may lead to a cascading behavior. Understanding how such social cascades spread in a network is crucial for many applications ranging from viral marketing to political campaigns. The behavior of cascade depends crucially on the model of cascade or social influence and the topological structure of the social network.
In this paper we study the general threshold model of cascades which are parameterized by a distribution over the natural numbers, in which the collective influence from infected neighbors, once beyond the threshold of an individual u, will trigger the infection of u. By varying the choice of the distribution, the general threshold model can model cascades with and without the submodular property. In fact, the general threshold model captures many previously studied cascade models as special cases, including the independent cascade model, the linear threshold model, and k-complex contagions.
We provide both analytical and experimental results for how cascades from a general threshold model spread in a general growing network model, which contains preferential attachment models as special cases. We show that if we choose the initial seeds as the early arriving nodes, the contagion can spread to a good fraction of the network and this fraction crucially depends on the fixed points of a function derived only from the specified distribution. We also show, using a coauthorship network derived from DBLP databases and the Stanford web network, that our theoretical results can be used to predict the infection rate up to a decent degree of accuracy, while the configuration model does the job poorly.