A Hyperbolic Embedding Method for Weighted Networks

Network embedding, which is the task of learning low-dimensional representations of vertices, has attracted increasing attention recently. Evidences have been found that the hidden metric space of many realistic complex networks is hyperbolic. The topology and weight emerge naturally as reflections of the hyperbolic metric property. A common objective of hyperbolic embedding is to maximize the likelihood function of the hyperbolic network model. The difficulty is that the likelihood function is non-concave which is difficult to optimize. In this paper, we propose a hyperbolic embedding method for weighted networks. To prevent the optimization from falling into numerous local optima, initial embedding is obtained by approximation. A proposed gradient algorithm then improves the embedding according to the likelihood function. Experiments on synthetic and real networks show that the proposed method achieves good embedding performance with respect to different quality metrics and applications.