Spectral embedding of large graphs and dynamic networks

大规模网络在几乎每个领域的无处不在，以及它们可以提供的洞察力的日益增长的价值，推动了用于其分析的原则性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。这个项目将推进大规模网络上的估计和推理方法，包括简单、静态图形之外的扩展。特别是，该项目将把单方网络的方法扩展到两方和多方环境，并将静态网络的方法扩展到动态网络，动态网络的生成机制在连续时间内演变。理论和经验分析将得到实施新方法的开放源码软件开发的补充。该研究项目将开发潜在位置网络模型的方法，其中图形表示为点云，以及一类被称为谱嵌入的估计过程，其中估计值是从图的邻接或拉普拉斯矩阵的缩放特征向量获得的。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估计的误差给出了很好的统计控制。这些结果将被用来开发新的工具，以对其他类型的矩阵价值数据进行建模，并更好地利用所产生的点云的底层几何图形。

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.