  We study a cascading edge failure mechanism on a connected random graph with
a prescribed degree sequence, sampled using the configuration model. This
mechanism prescribes that every edge failure puts an additional strain on the
remaining network, possibly triggering more failures. We show that under a
critical loading mechanism that depends on the global structure of the network,
the number of edge failure exhibits scale-free behavior (up to a certain
threshold). Our result is a consequence of the failure mechanism and the graph
topology. More specifically, the critical loading mechanism leads to scale-free
failure sizes for any network where no disconnections take place. The
disintegration of the configuration model ensures that the dominant
contribution to the failure size comes from edge failures in the giant
component, for which we show that the scale-free property prevails. We prove
this rigorously for sublinear thresholds, and we explain intuitively why the
analysis follows through for linear thresholds. Moreover, our result holds for
other graph structures as well, which we validate with simulation experiments.
