Dynamics in local influence cascading models

We study a general class of local influence cascading failure model, in which the failure of one node is directly caused by its neighbors, and propose a method to analytically evaluate the cascading size. The proposed method drastically reduces the dimension of the state space as compared to previous works. Here we use only one state variable, the probability that an arbitrary neighbor of a random node being vulnerable, to describe the system. It is the dimension reduction that makes the subsequent stability analysis feasible. The method is applied to two widely used models, i.e., the linear threshold model and the constant local load sharing model. In the former, comparing to previous analysis, we reveal that multiple stable states of the network may simultaneously exist. In the latter, we present the cascading size theoretically as opposed to previous works.