We study the multilayer random dot product graph (MRDPG) model, an extension of the random dot product graph to multilayer networks. To estimate the edge probabilities, we deploy a tensor-based methodology and demonstrate its superiority over existing approaches. Moving to dynamic MRDPGs, we formulate and analyse an online change point detection framework. At every time point, we observe a realization from an MRDPG. Across layers, we assume fixed shared common node sets and latent positions but allow for different connectivity matrices. We propose efficient tensor algorithms under both fixed and random latent position cases to minimize the detection delay while controlling false alarms. Notably, in the random latent position case, we devise a novel nonparametric change point detection algorithm based on density kernel estimation that is applicable to a wide range of scenarios, including 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。为估计边概率，我们采用了一种基于张量的方法，并证明了其优于现有方法的性能。针对动态MRDPG，我们构建并分析了一个在线变点检测框架。在每个时间点，我们观察到MRDPG的一个实现。假设各层之间共享固定的公共节点集和潜在位置，但允许不同的连接矩阵。我们提出了在固定和随机潜在位置两种情况下高效的张量算法，以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，我们设计了一种基于密度核估计的新型非参数变点检测算法，该算法适用于包括随机块模型作为特例的广泛场景。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。