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Bootstrap percolation is a process that is used to model the spread of aninfection on a given graph. In the model considered here each vertex isequipped with an individual threshold. As soon as the number of infectedneighbors exceeds that threshold, the vertex gets infected as well and remainsso forever. We perform a thorough analysis of bootstrap percolation on a novelmodel of directed and inhomogeneous random graphs, where the distribution ofthe 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 othervertices. Under the assumption that the limiting degree distribution of thegraph is integrable we determine the typical fraction of infected vertices. Ourmodel allows us to study a variety of settings, in particular the prominentcase in which the degree distribution has an unbounded variance. Among otherresults, we quantify the notion of "systemic risk", that is, to what extentlocal adverse shocks can propagate to large parts of the graph through acascade, and discover novel features that make graphs prone/resilient toinitially small infections.

Bootstrap percolation in directed and inhomogeneous random graphs