Pseudo-Euclidean Attract-Repel Embeddings for Undirected Graphs

Dot product embeddings take a graph and construct vectors for nodes such that dot products between two vectors give the strength of the edge. Dot products make a strong transitivity assumption, however, many important forces generating graphs in the real world lead to non-transitive relationships. We remove the transitivity assumption by embedding nodes into a pseudo-Euclidean space - giving each node an attract and a repel vector. The inner product between two nodes is defined by taking the dot product in attract vectors and subtracting the dot product in repel vectors. Pseudo-Euclidean embeddings can compress networks efficiently, allow for multiple notions of nearest neighbors each with their own interpretation, and can be `slotted' into existing models such as exponential family embeddings or graph neural networks for better link prediction.