Abstract
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 through the Riemannian multi-manifold modeling. A novel sequential-clustering scheme of Riemannian features is also established: landmark points are first identified in a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">non-random</i> way to reveal the underlying geometric information of the feature point-cloud, 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.
Highlights
Nodes are usually annotated with time-series, and popular noninvasive modalities that acquire those time-series data are functional magnetic resonance imaging, which monitors blood oxygen-level dependent (BOLD) data [4], and electroencephalography (EEG), where data are collected by electrodes placed on the scalp of a subject
This paper introduced a computationally efficient framework to address all possible clustering tasks, i.e., state clustering, community detection, and subnetwork-sequence identification, in dynamic multilayer networks where nodes are annotated with real-valued time-series
A full learning path was offered, which starts from low-level feature extraction and reaches up to network clustering
Summary
Method [58] can perform subnetwork-sequence clustering by considering each state as a layer, under the assumption that state changes are reflected in a known and measurable property of nodal time series data. Another sequential clustering algorithm for multilayer networks, based on local linear embedding and spectral clustering, was proposed in [59]. C. CONTRIBUTIONS OF THIS MANUSCRIPT This work introduces a sequential clustering framework to address all possible clustering tasks in a time-series-annotated multilayer network: state clustering, community detection, and subnetwork-sequence identification/clustering. Figures and tables that do not fit in the main body of the manuscript, due to space constraints
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