  Learning representations of sets of nodes in a graph is crucial for
applications ranging from node-role discovery to link prediction and molecule
classification. Graph Neural Networks (GNNs) have achieved great success in
graph representation learning. However, expressive power of GNNs is limited by
the 1-Weisfeiler-Lehman (WL) test and thus GNNs generate identical
representations for graph substructures that may in fact be very different.
More powerful GNNs, proposed recently by mimicking higher-order-WL tests, only
focus on representing entire graphs and they are computationally inefficient as
they cannot utilize sparsity of the underlying graph. Here we propose and
mathematically analyze a general class of structure-related features, termed
Distance Encoding (DE). DE assists GNNs in representing any set of nodes, while
providing strictly more expressive power than the 1-WL test. DE captures the
distance between the node set whose representation is to be learned and each
node in the graph. To capture the distance DE can apply various graph-distance
measures such as shortest path distance or generalized PageRank scores. We
propose two ways for GNNs to use DEs (1) as extra node features, and (2) as
controllers of message aggregation in GNNs. Both approaches can utilize the
sparse structure of the underlying graph, which leads to computational
efficiency and scalability. We also prove that DE can distinguish node sets
embedded in almost all regular graphs where traditional GNNs always fail. We
evaluate DE on three tasks over six real networks: structural role prediction,
link prediction, and triangle prediction. Results show that our models
outperform GNNs without DE by up-to 15\% in accuracy and AUROC. Furthermore,
our models also significantly outperform other state-of-the-art methods
especially designed for the above tasks.

Machine Learning;Machine Learning