Fast Sequential Clustering in Riemannian Manifolds for Dynamic and Time-Series-Annotated Multilayer Networks

Online Classification of Dynamic Multilayer-Network Time Series in Riemannian Manifolds

“…, (CA m−1 ) ] ∈ R mTw|L|×ρ . An estimateÔt of O per time instance t is computed here by the same sequential way employed in the clustering frameworks of [20,31]. Due to space limitations, the description of such a procedure is omitted.…”

“…In community classification, where features per nodal time series need to be extracted, V := ν, for ν ∈ N , and for a time-window length τw ∈ Z>0, with q := τw, let y [20,31] adapts to this case as ϕ (l)…”

“…per node ν and layer l. As in state classification, the way by which estimates of an observability matrix are obtained sequentially in [20,31] is also used here to compute an estimateÔ…”

“…To obtain a concise description of the potentially massive in number Xtrain, landmark points L k are selected from each class C k . Although selecting points randomly from C k is attractive from a computational aspect, the non-random way of [20], via a Riemannian modification of the Maxmin algorithm [35], is adopted also here to collect geometric information that could be neglected by random sampling techniques. More details and comparisons between the non-random approach and random sampling techniques can be found in [20].…”

“…This work spins off from [20], introduced very recently to address sequential clustering (i.e., unsupervised classification) in Riemannian manifolds for dynamic multilayer networks. As a case study, features are extracted from nodal time series via a kernel autoregressive-moving-average model, and mapped into the Grassmann manifold (Grassmannian) [21,22]; a well-known Riemannian manifold.…”