Spectral Embedding of Directed Networks

Most relationships in a social network have an element of asymmetry: the strength of A's relationship to B need not be the same as B's to A; and relationships that are based on power or influence have a natural flow associated with them. It is therefore natural to model many kinds of social networks by directed graphs, with a node corresponding to each participant, and a weighted directed edge to each relationship. Spectral embeddings for directed graphs are known, but they have significant weaknesses. We design a new directed-graph embedding, show that its mathematical properties are appropriate, and demonstrate its application to some synthetic and real-world networks. As well as avoiding the weaknesses of known techniques, the new embedding also represents a property we call flow betweenness of each node, allowing for directed edge prediction.