We study the multilayer random dot product graph (MRDPG) model, an extension of the random dot product graph to multilayer networks. By modelling a multilayer network as an MRDPG, we deploy a tensor-based method and demonstrate its superiority over existing approaches. Moving to dynamic MRDPGs, we focus on online change point detection problems. At every time point, we observe a realisation from an MRDPG. Across layers, we assume shared common node sets and latent positions but allow for different connectivity matrices. We propose efficient algorithms for both fixed and random latent position cases, minimising detection delay while controlling 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：我们研究了多层随机点积图（MRDPG）模型，该模型是随机点积图向多层网络的扩展。通过将多层网络建模为MRDPG，我们采用了一种基于张量的方法，并证明了其相较于现有方法的优越性。转向动态MRDPG后，我们聚焦于在线变点检测问题。在每个时间点，我们观测到MRDPG的一个实现。在不同层之间，我们假设共享共同节点集和潜在位置，但允许不同的连接矩阵。我们针对固定潜在位置和随机潜在位置两种情况提出了高效算法，在控制误报的同时最小化检测延迟。值得注意的是，在随机潜在位置情况下，我们设计了一种新颖的非参数变点检测算法，其核心为核估计器，适用于密度不存在的情形，并将随机块模型作为特例纳入其中。我们的理论发现得到了大量数值实验的支持，相关代码已在线公开于https://github.com/MountLee/MRDPG。