This work exploits Riemannian manifolds to introduce a geometric framework for online state and community classification in dynamic multilayer networks where nodes are annotated with time series. A bottom-up approach is followed, starting from the extraction of Riemannian features from nodal time series, and reaching up to online/sequential classification of features via geodesic distances and angular information in the tangent spaces of a Riemannian manifold. As a case study, features in the Grassmann manifold are generated by fitting a kernel autoregressive-moving-average model to the nodal time series of the multilayer network. The paper highlights also numerical tests on synthetic and real brain-network data, where it is shown that the proposed geometric framework outperforms state-of-the-art deep-learning models in classification accuracy, especially in cases where the number of training data is small with respect to the number of the testing ones. ——- © 20XX IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

This work exploits Riemannian manifolds to build a sequential-clustering framework able to address a wide variety of clustering tasks in dynamic multilayer (brain) networks via the information extracted from their nodal time-series. The discussion follows a bottom-up path, starting from feature extraction from time-series and reaching up to Riemannian manifolds (feature spaces) to address clustering tasks such as state clustering, community detection (a.k.a. network-topology identification), and subnetwork-sequence tracking. Kernel autoregressive-moving-average modeling and kernel (partial) correlations serve as case studies of generating features in the Riemannian manifolds of Grassmann and positive-(semi)definite matrices, respectively. Feature point-clouds form clusters which are viewed as submanifolds according to Riemannian multi-manifold modeling. A novel sequential-clustering scheme of Riemannian features is also established: feature points are first sampled in a non-random way to reveal the underlying geometric information, and, then, a fast sequential-clustering scheme is brought forth that takes advantage of Riemannian distances and the angular information on tangent spaces. By virtue of the landmark points and the sequential processing of the Riemannian features, the computational complexity of the framework is rendered free from the length of the available time-series data. The effectiveness and computational efficiency of the proposed framework is validated by extensive numerical tests against several state-of-the-art manifold-learning and brain-network-clustering schemes on synthetic as well as real functional-magnetic-resonance-imaging (fMRI) and electro-encephalogram (EEG) data.