Several observations indicate the existence of a latent hyperbolic space behind real networks that makes their structure very intuitive in the sense that the probability for a connection is decreasing with the hyperbolic distance between the nodes. A remarkable network model generating random graphs along this line is the popularity-similarity optimisation (PSO) model, offering a scale-free degree distribution, high clustering and the small world property at the same time. These results provide a strong motivation for the development of hyperbolic embedding algorithms, that tackle the problem of finding the optimal hyperbolic coordinates of the nodes based on the network structure. A very promising recent approach for hyperbolic embedding is provided by the noncentered minimum curvilinear embedding (ncMCE) method, belonging to the family of coalescent embedding algorithms. This approach offers a high quality embedding at a low running time. In the present work we propose a further optimisation of the angular coordinates in this framework that seems to reduce the logarithmic loss and increase the greedy routing score of the embedding compared to the original version, thereby adding an extra improvement to the quality of the inferred hyperbolic coordinates.

中文翻译：

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复杂网络合并双曲线嵌入的优化

一些观察表明真实网络背后存在潜在的双曲线空间，这使得它们的结构非常直观，因为连接的概率随着节点之间的双曲线距离而降低。沿着这条线生成随机图的一个非凡的网络模型是流行-相似度优化 (PSO) 模型，它同时提供无标度度分布、高聚类和小世界属性。这些结果为双曲嵌入算法的发展提供了强大的动力，该算法解决了基于网络结构寻找节点最佳双曲坐标的问题。非中心最小曲线嵌入（ncMCE）方法提供了一种非常有前途的双曲线嵌入方法，属于聚结嵌入算法家族。这种方法在较短的运行时间内提供了高质量的嵌入。在目前的工作中，我们建议在这个框架中进一步优化角坐标，与原始版本相比，这似乎减少了对数损失并增加了嵌入的贪婪路由分数，从而对推断的双曲线的质量进行了额外的改进坐标。