In this paper, we first introduce the multilayer random dot product graph (MRDPG) model, which can be seen as an extension of the random dot product graph model to multilayer networks. The MRDPG model is convenient for incorporating nodes' latent positions when understanding connectivity. By modelling a multilayer network as an MRDPG, we further deploy a tensor-based method and demonstrate its superiority over the state-of-the-art methods. We then move from a static to a dynamic MRDPG and are concerned with online change point detection problems. At every time point, we observe a realisation from an $L$-layered MRDPG. Across layers, we assume shared common node sets and latent positions, but allow for different connectivity matrices. In this paper we unfold a comprehensive picture concerning a range of problems. For both fixed and random latent position cases, we propose efficient online change point detection algorithms, minimising the delay in detection while controlling the false alarms. Notably, in the random latent position case, we devise a novel nonparametric change point detection algorithm with a kernel estimator in its core, allowing for the case when the density does not exist, accommodating stochastic block models as special cases. Our theoretical findings are supported by extensive numerical experiments, with the code available online https://github.com/MountLee/MRDPG.

翻译：本文首先引入了多层随机点积图（MRDPG）模型，该模型可视为随机点积图模型向多层网络的扩展。MRDPG模型便于在理解连通性时纳入节点的潜位置。通过将多层网络建模为MRDPG，我们进一步采用基于张量的方法，并展示了其相较于现有先进方法的优越性。随后，我们从静态MRDPG转向动态MRDPG，并关注在线变化点检测问题。在每个时间点，我们观测到来自一个$L$层MRDPG的实例。在各层之间，我们假设共享共同的节点集和潜位置，但允许不同的连通性矩阵。本文全面阐述了涉及一系列问题的完整图景。针对固定与随机潜位置两种情况，我们提出了高效的在线变化点检测算法，在控制虚警的同时最小化检测延迟。值得注意的是，在随机潜位置情形下，我们设计了一种新颖的非参数变化点检测算法，其核心采用核估计器，允许密度不存在的情况，并将随机块模型作为特例纳入。我们的理论发现得到大量数值实验的支持，代码已在https://github.com/MountLee/MRDPG公开。