Node2LV: Squared Lorentzian representations for node proximity

Recently, network embedding has attracted extensive research interest. Most existing network embedding models are based on Euclidean spaces. However, Euclidean embedding models cannot effectively capture complex patterns, especially latent hierarchical structures underlying in real-world graphs. Consequently, hyperbolic representation models have been developed to preserve the hierarchical information. Nevertheless, existing hyperbolic models only capture the first-order proximity between nodes. To this end, we propose a new embedding model, named Node2LV, that learns the hyperbolic representations of nodes using squared Lorentzian distances. This yields three advantages. First, our model can effectively capture hierarchical structures that come from the network topology. Second, compared with the conventional hyperbolic embedding methods that use computationally expensive Riemannian gradients, it can be optimized in a more efficient way. Lastly, different from existing hyperbolic embedding models, Node2LV captures higher-order proximities. Specifically, we represent each node with two hyperbolic embeddings, and make the embeddings of related nodes close to each other. To preserve higher-order node proximity, we use a random walk strategy to generate local neighborhood context. We conduct extensive experiments on four different types of real-world networks. Empirical results demonstrate that Node2LV significantly outperforms various graph embedding baselines.