Bootstrap percolation with inhibition

Bootstrap percolation is a prominent framework for studying the spreading of activity on a graph. We begin with an initial set of active vertices. The process then proceeds in rounds, and further vertices become active as soon as they have a certain number of active neighbors. A recurring feature in bootstrap percolation theory is an `all-or-nothing' phenomenon: either the size of the starting set is so small that the process stops very soon, or it percolates (almost) completely. Motivated by several important phenomena observed in various types of real-world networks we propose in this work a variant of bootstrap percolation that exhibits a vastly different behavior. Our graphs have two types of vertices: some of them obstruct the diffusion, while the others facilitate it. We study the effect of this setting by analyzing the process on Erd\H{o}s-R\'enyi random graphs. Our main findings are two-fold. First we show that the presence of vertices hindering the diffusion does not result in a stable behavior: tiny changes in the size of the starting set can dramatically influence the size of the final active set. In particular, the process is non-monotone: a larger starting set can result in a smaller final set. In the second part of the paper we show that this phenomenom arises from the round-based approach: if we move to a continuous time model in which every edge draws its transmission time randomly, then we gain stability, and the process stops with an active set that contains a non-trivial constant fraction of all vertices. Moreover, we show that in the continuous time model percolation occurs significantly faster compared to the classical round-based model. Our findings are in line with empirical observations and demonstrate the importance of introducing various types of vertex behaviors in the mathematical model.