论文标题
检测网络中的双曲几何形状：为什么三角形不够
Detecting hyperbolic geometry in networks: why triangles are not enough
论文作者
论文摘要
在过去的十年中，几何网络模型在文献中受到了广泛的关注。这些模型正式化了类似顶点可能连接的自然想法。因此，这些模型能够充分捕获现实世界网络的许多常见结构特性，例如自变异和高聚类。实际上，可以通过将网络图的顶点定位在双曲线空间中的顶点来准确地建模。然而，如果仅观察网络连接，几何的存在并不总是很明显。当前，三角计数和聚类系数是信号几何形状的标准统计数据。在本文中，我们表明三角计数或聚类系数不足，因为它们无法检测到由双曲线空间引起的几何形状。因此，我们引入了一种新型的基于三角形的统计量，该统计量根据其几何证据的强度来称重三角形。我们在分析上以及合成和现实世界中的数据表明，这是检测网络中双曲线几何形状的强大统计量。
In the past decade, geometric network models have received vast attention in the literature. These models formalize the natural idea that similar vertices are likely to connect. Because of that, these models are able to adequately capture many common structural properties of real-world networks, such as self-invariance and high clustering. Indeed, many real-world networks can be accurately modeled by positioning vertices of a network graph in hyperbolic spaces. Nevertheless, if one observes only the network connections, the presence of geometry is not always evident. Currently, triangle counts and clustering coefficients are the standard statistics to signal the presence of geometry. In this paper we show that triangle counts or clustering coefficients are insufficient because they fail to detect geometry induced by hyperbolic spaces. We therefore introduce a novel triangle-based statistic, which weighs triangles based on their strength of evidence for geometry. We show analytically, as well as on synthetic and real-world data, that this is a powerful statistic to detect hyperbolic geometry in networks.
