Expansion properties of a random regular graph after random vertex deletions

Abstract

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 $\alpha =\alpha \left(n\right)$ is bounded away from 0. We show that a.a.s. the resulting graph has a connected component of size $n-o\left(n\right)$ 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.