  Many Graph Neural Networks (GNNs) perform poorly compared to simple
heuristics on Link Prediction (LP) tasks. This is due to limitations in
expressive power such as the inability to count triangles (the backbone of most
LP heuristics) and because they can not distinguish automorphic nodes (those
having identical structural roles). Both expressiveness issues can be
alleviated by learning link (rather than node) representations and
incorporating structural features such as triangle counts. Since explicit link
representations are often prohibitively expensive, recent works resorted to
subgraph-based methods, which have achieved state-of-the-art performance for
LP, but suffer from poor efficiency due to high levels of redundancy between
subgraphs. We analyze the components of subgraph GNN (SGNN) methods for link
prediction. Based on our analysis, we propose a novel full-graph GNN called
ELPH (Efficient Link Prediction with Hashing) that passes subgraph sketches as
messages to approximate the key components of SGNNs without explicit subgraph
construction. ELPH is provably more expressive than Message Passing GNNs
(MPNNs). It outperforms existing SGNN models on many standard LP benchmarks
while being orders of magnitude faster. However, it shares the common GNN
limitation that it is only efficient when the dataset fits in GPU memory.
Accordingly, we develop a highly scalable model, called BUDDY, which uses
feature precomputation to circumvent this limitation without sacrificing
predictive performance. Our experiments show that BUDDY also outperforms SGNNs
on standard LP benchmarks while being highly scalable and faster than ELPH.
