  Bootstrap percolation is a process that is used to model 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 novel
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 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. Among other
results, we quantify the notion of "systemic risk", that is, to what extent
local adverse shocks can propagate to large parts of the graph through a
cascade, and discover novel features that make graphs prone/resilient to
initially small infections.
