Spectral embedding of large graphs and dynamic networks

大规模网络几乎在每个领域都无处不在，它们所能提供的见解的价值也在不断增加，这促使了它们分析的原则统计方法的巨大增长。应用领域包括网络安全（例如入侵检测），生物学（例如超分辨率显微镜数据）和欺诈检测以及许多其他新方法可能解锁的领域。该项目将推进大规模网络的估计和推理方法，包括简单静态图之外的扩展。特别是，该项目将把单部网络的方法扩展到二部和多部设置，把静态网络的方法扩展到生成机制在连续时间内演变的动态网络。理论和实证分析将由实现新方法的开源软件的开发来补充。该研究项目将开发潜在位置网络模型的方法，其中图被表示为点云，以及一类被称为谱嵌入的估计程序，其中从图邻接或拉普拉斯矩阵的缩放特征向量中获得估计。广义随机点积图就是这样一种模型，它的理论结果对谱嵌入得到的估计误差有很好的统计控制。这些结果将被用于开发新的工具来模拟其他类型的矩阵值数据，并更好地利用所得点云的底层几何结构。

The ubiquity of large-scale networks in almost every domain area and the increasing value of insights they can provide, has motivated tremendous growth in principled statistical methodologies for their analysis. Application areas include cyber-security (e.g. intrusion detection), biology (e.g. super-resolution microscopy data) and fraud detection as well as many others which new methodologies might unlock. This project will advance methods for estimation and inference on large-scale networks, including extensions beyond simple, static graphs. In particular, the project will extend methods for unipartite networks to bipartite and multipartite settings, and for static networks to dynamic networks whose generative mechanism evolves in continuous time. Theoretical and empirical analyses will be complemented by the development of open-source software which implements the new methodologies. The research project will develop methods for latent position network models, in which graphs are represented as point clouds, and a class of estimation procedures known as spectral embedding, in which estimates are obtained from the scaled eigenvectors of the graph adjacency or Laplacian matrix. The generalised random dot product graph is one such model for which theoretical results give fine statistical control on the error of estimates obtained by spectral embedding. These results will be leveraged to develop new tools to model other types of matrix-valued data and to better exploit the underlying geometry of the resulting point clouds.