The journal version of The Effects of Faults on Network Expansion, with Bagchi, Bhargava, Chaudhury, and Scheideler, in Theory of Computing Systems, has been put online by the publisher. It's about showing that certain network connectivity structures are robust against failures — if a small enough fraction of nodes fail, many of the remaining nodes can still be connected together to form a subgraph that is not just connected, but nearly as well connected as the original graph in terms of its expansion.

My contribution was a lemma about high dimensional grids: suppose \( S \) is a collection of points in the integer lattice \( \mathbb{Z}^d \), that \( S \) is connected orthogonally, and that the complement of \( S \) is also connected. Let \( B \) be the points in \( S \) that are adjacent to a point not in \( S \) (a discrete analog of the boundary of \( S \)). \( B \) may not be orthogonally connected (think of the boundary of an X pentomino) but what I could show is that, if one connects any two points in \( B \) that are at distance at most \( \sqrt{2} \) from each other, the resulting graph is connected, regardless of dimension. This turned out to be the key for showing the desired robustness results for grid-connected computer networks, and I was very pleased to be able to use some baby homology theory in the proof.