  Large-scale power blackouts caused by cascading failure are inflicting
enormous socioeconomic costs. We study the problem of cascading link failures
in power networks modelled by random geometric graphs from a percolation-based
viewpoint. To reflect the fact that links fail according to the amount of power
flow going through them, we introduce a model where links fail according to a
probability which depends on the number of neighboring links. We devise a
mapping which maps links in a random geometric graph to nodes in a
corresponding dual covering graph. This mapping enables us to obtain the
first-known analytical conditions on the existence and non-existence of a large
component of operational links after degree-dependent link failures. Next, we
present a simple but descriptive model for cascading link failure, and use the
degree-dependent link failure results to obtain the first-known analytical
conditions on the existence and non-existence of cascading link failures.
