The Forward-Backward Embedding of Directed Graphs

We introduce a novel embedding of directed graphs derived from the singular value decomposition (SVD) of the normalized adjacency matrix. Specifically, we show that, after proper normalization of the singular vectors, the distances between vectors in the embedding space are proportional to the mean commute times between the corresponding nodes by a forward-backward random walk in the graph, which follows the edges alternately in forward and backward directions. In particular, two nodes having many common successors in the graph tend to be represented by close vectors in the embedding space. More formally, we prove that our representation of the graph is equivalent to the spectral embedding of some co-citation graph, where nodes are linked with respect to their common set of successors in the original graph. The interest of our approach is that it does not require to build this co-citation graph, which is typically much denser than the original graph. Experiments on real datasets show the efficiency of the approach.