Title: Online change point localization in multi-layer random dot product graphs.

Abstract: We study the online change point localization problem in dynamic multilayer random dot product graphs (D-MRDPGs). To be specific, at every time point, we extend the nonparametric random dot product graph (RDPG) to the directed multilayer graph with the common node sets and the underlying distributions change when a change point occurs. The goal is to detect the change point as quickly as possible if it exists, subject to a constraint on the probability of false alarms. The change in our framework is in fact, a multivariate nonparametric distributional change. Instead of defining the jump size using a density, we use the expectation of the kernel density estimator (KDE). Our generic model setting allows the model parameters, including the number of nodes, the dimension of latent position, and the magnitude of the change, to vary as functions of the location of the change point. We propose a novel change point detection algorithm and proved an upper bound on the detection delay under a signal-to-noise ratio condition. Additionally, for a single multilayer random dot product graph (MRDPG), we study an estimation method based on the adjacency tensor and show it achieves the optimal statistical performance in estimation error. Finally, extensive numerical results including simulation studies and real data studies are provided to support our theoretical results.