  Graph embedding based on random-walks supports effective solutions for many
graph-related downstream tasks. However, the abundance of embedding literature
has made it increasingly difficult to compare existing methods and to identify
opportunities to advance the state-of-the-art. Meanwhile, existing work has
left several fundamental questions -- such as how embeddings capture different
structural scales and how they should be applied for effective link prediction
-- unanswered. This paper addresses these challenges with an analytical
framework for random-walk based graph embedding that consists of three
components: a random-walk process, a similarity function, and an embedding
algorithm. Our framework not only categorizes many existing approaches but
naturally motivates new ones. With it, we illustrate novel ways to incorporate
embeddings at multiple scales to improve downstream task performance. We also
show that embeddings based on autocovariance similarity, when paired with dot
product ranking for link prediction, outperform state-of-the-art methods based
on Pointwise Mutual Information similarity by up to 100%.
