  Network embedding maps the nodes of a given network into a low-dimensional
space such that the semantic similarities among the nodes can be effectively
inferred. Most existing approaches use inner-product of node embedding to
measure the similarity between nodes leading to the fact that they lack the
capacity to capture complex relationships among nodes. Besides, they take the
path in the network just as structural auxiliary information when inferring
node embeddings, while paths in the network are formed with rich user
informations which are semantically relevant and cannot be ignored. In this
paper, We propose a novel method called Network Embedding on the Metric of
Relation, abbreviated as NEMR, which can learn the embeddings of nodes in a
relational metric space efficiently. First, our NEMR models the relationships
among nodes in a metric space with deep learning methods including variational
inference that maps the relationship of nodes to a gaussian distribution so as
to capture the uncertainties. Secondly, our NEMR considers not only the
equivalence of multiple-paths but also the natural order of a single-path when
inferring embeddings of nodes, which makes NEMR can capture the multiple
relationships among nodes since multiple paths contain rich user information,
e.g., age, hobby and profession. Experimental results on several public
datasets show that the NEMR outperforms the state-of-the-art methods on
relevant inference tasks including link prediction and node classification.
