Expansion properties of a random regular graph after random vertex deletions

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 − α and α = α ( 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 α . These results have an application to the cost of repairing a certain peer-to-peer network after random failures of nodes.