Spectral Embedding of Weighted Graphs

When analyzing weighted networks using spectral embedding, a judicious transformation of the edge weights may produce better results. To formalize this idea, we consider the asymptotic behavior of spectral embedding for different edge-weight representations, under a generic low rank model. We measure the quality of different embeddings—which can be on entirely different scales—by how easy it is to distinguish communities, in an information-theoretical sense. For common types of weighted graphs, such as count networks or p-value networks, we find that transformations such as tempering or thresholding can be highly beneficial, both in theory and in practice. Supplementary materials for this article are available online.