  Representing nodes in a network as dense vectors node embeddings is important
for understanding a given network and solving many downstream tasks. In
particular, for weighted homophilous graphs where similar nodes are connected
with larger edge weights, we desire node embeddings where node pairs with
strong weights have closer embeddings. Although random walk based node
embedding methods like node2vec and node2vec+ do work for weighted networks via
including edge weights in the walk transition probabilities, our experiments
show that the embedding result does not adequately reflect edge weights. In
this paper, we propose ARGEW (Augmentation of Random walks by Graph Edge
Weights), a novel augmentation method for random walks that expands the corpus
in such a way that nodes with larger edge weights end up with closer
embeddings. ARGEW can work with any random walk based node embedding method,
because it is independent of the random sampling strategy itself and works on
top of the already-performed walks. With several real-world networks, we
demonstrate that with ARGEW, compared to not using it, the desired pattern that
node pairs with larger edge weights have closer embeddings is much clearer. We
also examine ARGEW's performance in node classification: node2vec with ARGEW
outperforms pure node2vec and is not sensitive to hyperparameters (i.e.
consistently good). In fact, it achieves similarly good results as supervised
GCN, even without any node feature or label information during training.
Finally, we explain why ARGEW works consistently well by exploring the
coappearance distributions using a synthetic graph with clear structural roles.
