  Graph embedding learns low-dimensional representations for nodes in a graph
and effectively preserves the graph structure. Recently, a significant amount
of progress has been made toward this emerging research area. However, there
are several fundamental problems that remain open. First, existing methods fail
to preserve the out-degree distributions on directed graphs. Second, many
existing methods employ random walk based proximities and thus suffer from
conflicting optimization goals on undirected graphs. Finally, existing
factorization methods are unable to achieve scalability and non-linearity
simultaneously.
  This paper presents an in-depth study on graph embedding techniques on both
directed and undirected graphs. We analyze the fundamental reasons that lead to
the distortion of out-degree distributions and to the conflicting optimization
goals. We propose {\em transpose proximity}, a unified approach that solves
both problems. Based on the concept of transpose proximity, we design \strap, a
factorization based graph embedding algorithm that achieves scalability and
non-linearity simultaneously. \strap makes use of the {\em backward push}
algorithm to efficiently compute the sparse {\em Personalized PageRank (PPR)}
as its transpose proximities. By imposing the sparsity constraint, we are able
to apply non-linear operations to the proximity matrix and perform efficient
matrix factorization to derive the embedding vectors. Finally, we present an
extensive experimental study that evaluates the effectiveness of various graph
embedding algorithms, and we show that \strap outperforms the state-of-the-art
methods in terms of effectiveness and scalability.
