  Message Passing Graph Neural Networks are known to suffer from two problems
that are sometimes believed to be diametrically opposed: over-squashing and
over-smoothing. The former results from topological bottlenecks that hamper the
information flow from distant nodes and are mitigated by spectral gap
maximization, primarily, by means of edge additions. However, such additions
often promote over-smoothing that renders nodes of different classes less
distinguishable. Inspired by the Braess phenomenon, we argue that deleting
edges can address over-squashing and over-smoothing simultaneously. This
insight explains how edge deletions can improve generalization, thus connecting
spectral gap optimization to a seemingly disconnected objective of reducing
computational resources by pruning graphs for lottery tickets. To this end, we
propose a more effective spectral gap optimization framework to add or delete
edges and demonstrate its effectiveness on large heterophilic datasets.
