Bootstrap Percolation in Directed Inhomogeneous Random Graphs

Bootstrap percolation is a process that is used to describe the spread of an infection on a given graph. In the model considered here each vertex is equipped with an individual threshold. As soon as the number of infected neighbors exceeds that threshold, the vertex gets infected as well and remains so forever. We perform a thorough analysis of bootstrap percolation on a model of directed and inhomogeneous random graphs, where the distribution of the edges is specified by assigning two distinct weights to each vertex, describing the tendency of it to receive edges from or to send edges to other vertices. Under the mild assumption that the limiting degree distribution of the graph is integrable we determine the typical fraction of infected vertices. Our model allows us to study a variety of settings, in particular the prominent case in which the degree distribution has an unbounded variance. As a second main contribution, we quantify the notion of "systemic risk", that is, we characterize to what extent tiny initial infections can propagate to large parts of the graph through a cascade, and discover novel features that make graphs prone/resilient to initially small infections.