Random graphs are increasingly becoming objects of interest for modeling networks in a wide range of applications. Latent position random graph models posit that each node is associated with a latent position vector, and that these vectors follow some geometric structure in the latent space. In this paper, we consider random dot product graphs, in which an edge is formed between two nodes with probability given by the inner product of their respective latent positions. We assume that the latent position vectors lie on an unknown one-dimensional curve and are coupled with a response covariate via a regression model. Using the geometry of the underlying latent position vectors, we propose a manifold learning and graph embedding technique to predict the response variable on out-of-sample nodes, and we establish convergence guarantees for these responses. Our theoretical results are supported by simulations and an application to Drosophila brain data.

翻译：随机图正日益成为众多应用领域中建模网络的对象。潜位置随机图模型假设每个节点关联一个潜位置向量，且这些向量在潜空间中遵循某种几何结构。本文考虑随机点积图，其中两个节点之间形成边的概率由其各自潜位置的内积决定。我们假设潜位置向量位于未知的一维曲线上，并通过回归模型与响应协变量耦合。利用底层潜位置向量的几何性质，我们提出了一种流形学习与图嵌入技术来预测样本外节点的响应变量，并建立了这些响应变量的收敛性保证。我们的理论结果得到了仿真实验以及果蝇大脑数据应用的验证。