  We investigate the following vertex percolation process. Starting with a
random regular graph of constant degree, delete each vertex independently with
probability p, where p=n^{-alpha} and alpha=alpha(n) is bounded away from 0. We
show that a.a.s. the resulting graph has a connected component of size n-o(n)
which is an expander, and all other components are trees of bounded size.
Sharper results are obtained with extra conditions on alpha. These results have
an application to the cost of repairing a certain peer-to-peer network after
random failures of nodes.
