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.

中文翻译：

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加权网络的双曲嵌入方法

网络嵌入是学习顶点的低维表示的一项任务，最近引起了越来越多的关注。已经发现许多现实的复杂网络的隐藏度量空间是双曲线的。拓扑和权重自然是双曲线度量属性的反映。双曲嵌入的共同目标是最大化双曲网络模型的似然函数。困难在于似然函数是非凹面的，难以优化。在本文中，我们提出了一种加权网络的双曲嵌入方法。为了防止优化陷入众多局部最优，可以通过近似获得初始嵌入。然后，提出的梯度算法根据似然函数改进了嵌入。