  We study a simple embedding technique based on a matrix of personalized
PageRank vectors seeded on a random set of nodes. We show that the embedding
produced by the element-wise logarithm of this matrix (1) are related to the
spectral embedding for a class of graphs where spectral embeddings are
significant, and hence useful representation of the data, (2) can be done for
the entire network or a smaller part of it, which enables precise local
representation, and (3) uses a relatively small number of PageRank vectors
compared to the size of the networks. Most importantly, the general nature of
this embedding strategy opens up many emerging applications, where eigenvector
and spectral techniques may not be well established, to the PageRank-based
relatives. For instance, similar techniques can be used on PageRank vectors
from hypergraphs to get "spectral-like" embeddings.
