Node Similarities under Random Projections: Limits and Pathological Cases

Random Projections have been widely used to generate embeddings for various graph tasks due to their computational efficiency. The majority of applications have been justified through the Johnson-Lindenstrauss Lemma. In this paper, we take a step further and investigate how well dot product and cosine similarity are preserved by Random Projections. Our analysis provides new theoretical results, identifies pathological cases, and tests them with numerical experiments. We find that, for nodes of lower or higher degrees, the method produces especially unreliable embeddings for the dot product, regardless of whether the adjacency or the (normalized version) transition is used. With respect to the statistical noise introduced by Random Projections, we show that cosine similarity produces remarkably more precise approximations.