  We consider propagation models that describe the spreading of an attribute,
called "damage", through the nodes of a random network. In some systems, the
average fraction of nodes that remain undamaged vanishes in the large system
limit, a phenomenon we refer to as exhaustive percolation. We derive scaling
law exponents and exact results for the distribution of the number of undamaged
nodes, valid for a broad class of random networks at the exhaustive percolation
transition and in the exhaustive percolation regime. This class includes
processes that determine the set of frozen nodes in random Boolean networks
with indegree distributions that decay sufficiently rapidly with the number of
inputs. Connections between our calculational methods and previous studies of
percolation beginning from a single initial node are also pointed out. Central
to our approach is the observation that key aspects of damage spreading on a
random network are fully characterized by a single function specifying the
probability that a given node will be damaged as a function of the fraction of
damaged nodes. In addition to our analytical investigations of random networks,
we present a numerical example of exhaustive percolation on a directed lattice.
