Smith Anna L, Asta Dena M, Calder Catherine A
Department of Statistics, University of Kentucky, 725 Rose Street, Lexington, Kentucky 40536, USA.
Department of Statistics, The Ohio State University, 1958 Neil Avenue, Columbus, Ohio 43210, USA.
Stat Sci. 2019 Aug;34(3):428-453. doi: 10.1214/19-sts702. Epub 2019 Oct 11.
We review the class of continuous latent space (statistical) models for network data, paying particular attention to the role of the geometry of the latent space. In these models, the presence/absence of network dyadic ties are assumed to be conditionally independent given the dyads' unobserved positions in a latent space. In this way, these models provide a probabilistic framework for embedding network nodes in a continuous space equipped with a geometry that facilitates the description of dependence between random dyadic ties. Specifically, these models naturally capture homophilous tendencies and triadic clustering, among other common properties of observed networks. In addition to reviewing the literature on continuous latent space models from a geometric perspective, we highlight the important role the geometry of the latent space plays on properties of networks arising from these models via intuition and simulation. Finally, we discuss results from spectral graph theory that allow us to explore the role of the geometry of the latent space, independent of network size. We conclude with conjectures about how these results might be used to infer the appropriate latent space geometry from observed networks.
我们回顾了用于网络数据的连续潜在空间（统计）模型类别，特别关注潜在空间几何结构的作用。在这些模型中，假设给定二元组在潜在空间中未观察到的位置，网络二元关系的存在/不存在是条件独立的。通过这种方式，这些模型提供了一个概率框架，用于将网络节点嵌入到配备了有助于描述随机二元关系之间依赖性的几何结构的连续空间中。具体而言，这些模型自然地捕捉了同质性倾向和三元聚类以及观察到的网络的其他常见属性。除了从几何角度回顾关于连续潜在空间模型的文献外，我们还通过直觉和模拟强调了潜在空间几何结构对这些模型产生的网络属性所起的重要作用。最后，我们讨论了光谱图理论的结果，这些结果使我们能够探索潜在空间几何结构的作用，而不受网络大小的影响。我们以关于如何利用这些结果从观察到 的网络中推断出合适的潜在空间几何结构的猜想作为结论。
