Epidemics and Spreading Phenomena

In this chapter we mainly focus on the results of activation processes in networks and on various combinations of activation and deactivation processes. The bootstrap percolation problem is about the basic activation process on networks, in which vertices can be in active and inactive states. A vertex becomes active when the number of its active neighbours exceeds some threshold; and once active, a vertex never becomes inactive (Adler and Aharony, 1988; Adler, 1991). This is one of the spreading processes with discontinuous phase transitions (Bizhani, Paczuski, and Grassberger, 2012). Let us define bootstrap percolation on undirected graphs in more strict terms. In the initial state, a fraction f of vertices is active (seed vertices). These vertices are chosen uniformly at random. Each inactive vertex becomes active if it has at least kb active nearest neighbours. Here we introduce the subscript ‘b’ to distinguish this threshold from a threshold in the k-core percolation problem.